KEY CONCEPTS After completing this chapter you will be able to • distinguish between different types of forces and describe how they affect the velocity and acceleration of an object • explain how Galileo and Newton advanced our knowledge of forces and motion • state and apply Newton’s laws qualitatively • use free-body diagrams to calculate net force and acceleration • solve problems involving forces in one dimension using free- body diagrams and Newton’s laws • conduct an inquiry into the relationship among the acceleration, net force, and mass of an object, and analyze the resulting data • assess the environmental and social impact of technologies that involve forces What Effect Do Forces Have on the Motion of Objects? e skier on the opposite page has many different forces acting on him at the same time. Each force has an effect on his motion. e ground exerts forces on the skier, gravity is pulling on him, and even the air is pushing him back. When you combine all of these forces, you can determine how the skier will move. At times, the skier will move with a constant velocity, while at other times he will speed up or slow down. e sum of all the forces acting on the skier determines which of these will occur. To help increase the speed of the skier, the skis are designed to decrease the force of friction. e skier can push on the snow-covered ground with his poles to help him speed up. Even the skier’s clothing and safety equipment are designed to help increase speed and reduce drag (air resistance). An experi- enced skier will adjust his or her stance to reduce air resistance. Extreme velocities are dangerous even to experienced skiers. To help reduce speed, the skier can plow through the snow or skid with his skis across the snow, causing the skis to dig into the snow. Just reading about the physics of skiing will not make you a skilled skier, but understanding the physics behind skiing can make a good skier even better. An understanding of the physics of skiing also helps equipment designers create better skis, poles, and other skiing gear, which help skiers win races. In this chapter, you will explore different types of forces and discover how they affect motion. You will learn Newton’s three laws of motion and use them to explain how and why objects move. You will also solve problems related to forces and motion. Answer the following questions using your current knowledge. You will have an opportunity to revisit these questions later, applying concepts and skills from the chapter. 1. List as many forces as you can think of that might be acting on the skier in the photograph. For each force, give the direction of the force and suggest what might be exerting the force. 2. If the skier is moving slowly at a constant velocity across a horizontal surface, what forces do you think are acting on him? How would your answer change if the skier were moving quickly at a constant velocity? 3. What do you think is true about the direction of the total force acting on the skier if he is slowing down? What if he is speeding up? 4. What force or forces do you think would cause the skier to (a) speed up? (b) slow down? 5. The physics of skiing is similar to the physics of skateboarding. (a) List three forces that you think act on both a skier and a skateboarder. (b) How do you think a skateboarder can speed up and slow down? STARTING POINTS CHAPTER 3 Newton’s Laws of Motion 112 Chapter 3 • Newton’s Laws of Motion NEL
Transcript
Key concePTsAfter completing this chapter you will be able to
• distinguish between different types of forces and describe how they affect the velocity and acceleration of an object
• explain how Galileo and Newton advanced our knowledge of forces and motion
• state and apply Newton’s laws qualitatively
• use free-body diagrams to calculate net force and acceleration
• solve problems involving forces in one dimension using free-body diagrams and Newton’s laws
• conduct an inquiry into the relationship among the acceleration, net force, and mass of an object, and analyze the resulting data
• assess the environmental and social impact of technologies that involve forces
what Effect Do Forces Have on the Motion of Objects?Th e skier on the opposite page has many diff erent forces acting on him at the same time. Each force has an eff ect on his motion. Th e ground exerts forces on the skier, gravity is pulling on him, and even the air is pushing him back. When you combine all of these forces, you can determine how the skier will move. At times, the skier will move with a constant velocity, while at other times he will speed up or slow down. Th e sum of all the forces acting on the skier determines which of these will occur.
To help increase the speed of the skier, the skis are designed to decrease the force of friction. Th e skier can push on the snow-covered ground with his poles to help him speed up. Even the skier’s clothing and safety equipment are designed to help increase speed and reduce drag (air resistance). An experi-enced skier will adjust his or her stance to reduce air resistance.
Extreme velocities are dangerous even to experienced skiers. To help reduce speed, the skier can plow through the snow or skid with his skis across the snow, causing the skis to dig into the snow.
Just reading about the physics of skiing will not make you a skilled skier, but understanding the physics behind skiing can make a good skier even better. An understanding of the physics of skiing also helps equipment designers create better skis, poles, and other skiing gear, which help skiers win races.
In this chapter, you will explore diff erent types of forces and discover how they aff ect motion. You will learn Newton’s three laws of motion and use them to explain how and why objects move. You will also solve problems related to forces and motion.
Answer the following questions using your current knowledge. You will have an opportunity to revisit these questions later, applying concepts and skills from the chapter.
1. List as many forces as you can think of that might be acting on the skier in the photograph. For each force, give the direction of the force and suggest what might be exerting the force.
2. If the skier is moving slowly at a constant velocity across a horizontal surface, what forces do you think are acting on him? How would your answer change if the skier were moving quickly at a constant velocity?
3. What do you think is true about the direction of the total force acting on the skier if he is slowing down? What if he is speeding up?
4. What force or forces do you think would cause the skier to(a) speed up? (b) slow down?
5. The physics of skiing is similar to the physics of skateboarding.(a) List three forces that you think act on both a skier
and a skateboarder.(b) How do you think a skateboarder can speed up and
The SI unit of force is the newton. In this chapter, you will be required to measure forces using a spring scale or a force sensor. The following activity will help you improve your skills in estimating and measuring forces. Before performing this activity, make sure you know how to zero the spring scale or calibrate the force sensor.
Equipment and Materials: two spring scales or force sensors; one 100 g object; one 200 g object
1. Hang a 100 g object from a spring scale or a force sensor and record the reading.
2. Hold a 200 g object in your hand and estimate how much force is required to hold it up. Record your estimate. Hang the object from the spring scale or force sensor. Measure and record the force.
3. Predict the reading on the spring scale if you hang both the 100 g and the 200 g object from it. Record your prediction. Measure and record the force.
4. Predict the reading on each scale if you use two spring scales or two force sensors to hold up one 200 g object. Record your prediction. Test your prediction and record your results.
A. How accurate were your predictions? How could they be improved? T/I
B. What can you conclude about forces from your observations? Write one or two statements that summarize your observations. T/I C
Types of ForcesForces are all around you, acting on every object that you see. The motion of cars, trucks, planes, and boats is determined by the forces acting on them. Engineers must consider forces carefully when designing bridges and buildings. You are always using forces to move around, to lift objects, or to turn the pages of this book. Forces are involved in every type of sport and activity. For example, when a pitcher throws a ball, she exerts a force on the ball that causes the ball to move forward (Figure 1). If the batter hits the ball, then the bat exerts a force on the ball to change its motion. An understanding of forces is essential for a scientific description of our environment.
In simple terms, a force is a push or a pull. Forces can cause objects to change their motion. When you push on a chair to tuck it under a desk, you change the motion of the chair. The direction of a force is very important. If you push straight down on a book on a desk, the effect is usually very different than if you push sideways or pull up. This means that force has direction, making it a vector quantity.
In Unit 1, you studied a branch of mechanics called kinematics. Remember that kinematics is the study of how objects move without being concerned with why they move. In this unit, you will study dynamics. Dynamics explains why objects move the way they do. One way to understand why an object moves is to study the forces acting on it. These forces can cause the object to start moving, speed up, slow down, or remain stationary. In this chapter, you will be introduced to different types of forces and the laws that govern them.
Measuring ForcesIsaac Newton discovered many of the concepts in this chapter. For this reason the unit of force, the newton, is named after him. The newton (N) is a derived SI unit equal to 1 kg·m/s2.
To measure force in the laboratory, you can use either a spring scale or a force sensor. A spring scale has a spring that stretches more when greater forces pull on it. A needle is attached to the spring to indicate the amount of force. Usually a spring scale must be zeroed (the reading must be set to zero when not pulling) before use. Most spring scales can only measure a pulling force. A force sensor is an electronic device that can be attached to a computer or used independently. This device pro-vides an accurate digital reading of a force and can even graph how the force changes over time. A force sensor can measure both pushes and pulls.
Force DiagramsTo understand why an object will remain at rest, start moving, or change its motion, you need to be able to draw diagrams that show clearly which forces are acting on the object. These diagrams are essential, especially when several forces are acting on the object simultaneously. The first type of force diagram is called a system diagram. A system diagram is a simple sketch of all the objects involved in a situation. For example, if you are lifting a book up in the air, the system diagram will show your hand pulling up on the book (Figure 2).
Figure 1 Forces are all around you.
newton (N) the SI unit of force (1 N = 1 kg·m/s2)
system diagram a simple sketch of all objects involved in a situation
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Figure 2 A system diagram is a sketch showing all the objects involved in a situation. A system diagram helps you determine which objects push or pull on other objects.
Figure 3 The FBD for the book shown in Figure 2. Since an FBD shows all the forces acting on a single object, two forces are drawn: the force of the hand pulling up and the force of gravity pulling down.
The other type of force diagram is called a free-body diagram (FBD). An FBD is a simple drawing representing the object being analyzed and all the forces acting on it (Figure 3). Usually the object is drawn as a rectangle or large dot. The forces are drawn as arrows originating from the outline of the rectangle or dot and pointing away from the centre. FBDs are not drawn to scale, but larger forces can be drawn longer than smaller ones to help predict the motion of the object. Each force is labelled with the symbol F
S
and an appropriate subscript that indicates the type of force.
Everyday ForcesTo draw useful force diagrams, you need to be familiar with some common forces encountered every day. Imagine two children playing outside with a wagon. One child pulls forward on a rope tied to the front, while the other pushes on the wagon from behind (Figure 4(a)). What forces act on the wagon? To answer this question, we first need to study some everyday forces.
First consider the applied forces. An applied force results when one object is in contact with another object and either pushes or pulls on it. The symbol for an applied force is F
S
a. In our example, the child behind the wagon exerts an applied force forward on the wagon by pushing on the back. Another force is the tension force (often called tension). Tension is a pulling force exerted on an object by a rope or a string. The symbol for tension is F
S
T. Ropes and strings are not rigid, which means that they cannot push on an object. If you try to push with a rope, the rope will just sag down and have no effect on the motion of the object. An easy way to remember the direction of the tension is that it always pulls the object toward the rope or the string. The child at the front of the wagon pulls on the rope, causing tension in the rope. The rope exerts tension on the wagon, pulling it forward. Notice that in Figure 4(b) both the applied force vector and the tension vector start from the outline of the rectangle representing the wagon and are directed forward, indicating the direction of each force.
Whenever an object is in contact with a surface, the surface can exert two dif-ferent forces on the object. One is called the normal force. The normal force is a perpendicular force on an object exerted by the surface with which it is in contact. The normal force is given its name because this force always acts perpendicular (or normal) to the surface. The symbol for the normal force is F
S
N. The normal force is always a push from the surface onto the object. For this reason, the normal force always points away from a surface. In Figure 4(b), the normal force from the ground on the wagon starts from the outline of the rectangle and points up, perpendicular to the ground. In this case, the normal force supports the wagon against the force of gravity.
The other force exerted by a surface on an object is friction. Friction is a force that resists the motion or attempted motion of an object. Friction is always parallel to the surface and acts opposite to the object’s motion or attempted motion. The symbol for friction is F
S
f . If the wagon is moving to the right, then the friction on the wagon acts toward the left, opposite to the motion. If the wagon is at rest even if the children are pushing and pulling on it, then the friction is left, opposite to the tension and applied force keeping the wagon at rest.
applied force (F>a) a force that results
when one object makes contact with another and pushes or pulls on it
tension (F>T) a pulling force from a rope
or string on an object that always points toward the rope or string
normal force (F>N) a perpendicular force
exerted by a surface on an object in contact with the surface; the normal force always points away from the surface
Figure 4 (a) The system diagram of a wagon and the two children pushing and pulling on it (b) The FBD showing all the forces acting on the wagon
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F N
F T
Fa
Fg
F f
(a) (b)
friction (F>f) opposes the sliding of two
surfaces across one another; friction acts opposite to the motion or attempted motion
free-body diagram (FBD) a simple drawing of an object showing all the forces that are acting on it
Notice that all of the forces described on the previous page require one object to be in contact with another. For this reason, they are called contact forces. Some forces do not require contact. Th ese forces are known as action-at-a-distance forces (sometimes called non-contact forces).
Th e force of gravity, also called the gravitational force, is the force of attraction that exists between any two objects due to their mass. Th e force of gravity is an action-at-a-distance force. Th e symbol for gravity is F
S
g. In this course, you will only learn about the force of gravity between Earth and other objects close to Earth’s surface. At Earth’s surface, the force of gravity always points down toward Earth’s centre. Even if the surface is sloped, such as the side of a mountain, the force of gravity still points down toward Earth’s centre.
force of gravity (F>g) force of attraction
between any two objects
In this investigation, you will observe the relationship between mass and the force of gravity. You will also measure and calculate the magnitude of the force of gravity.
Equipment and Materials: spring scale or force sensor; set of objects of known mass
1. Create a table with the headings “Mass (kg)” and “Force of gravity (N)” to record your observations.
2. Select one of the objects. To measure the force of gravity on the object, hang the object from the spring scale or force sensor. Hold the object steady before taking a reading. Record your observations in your table.
3. Repeat Step 2 for all the objects in the set.
A. Graph your results with the force of gravity (N) on the y-axis and mass (kg) on the x-axis. T/I C
B. What is the slope of the line of best fi t on the graph? What does the slope represent? (Hint: Think back to the projectile problems from Unit 1.) T/I C
C. Using your observations, predict the force of gravity acting on each object below. Use the spring scale or force sensor to check your predictions. How accurate were your predictions? T/I C
(i) 0.30 kg (ii) 0.50 kg
D. Describe how you could calculate the force of gravity (N) acting on an object if you knew its mass (kg). T/I C
To calculate the magnitude of the force of gravity on an object, you multiply the mass of the object by the acceleration due to gravity. To calculate the force of gravity, you can use the equation
where gS
5 9.8 m/s2 [down]. Th e force of gravity is measured in newtons and the mass is in kilograms. You will learn more about gravity in Chapter 4.
In this course, you will usually be concerned with external forces. External forces are those that are caused by one object pushing or pulling on another. An internal force occurs when an object exerts a force on itself. For example, when skater A pushes on skater B, the force on skater B is external. If skater B pulls forward on her own arm, then it is an internal force.
In the following Tutorial, you will use what you have learned about forces to draw system diagrams and FBDs.
F>g 5 mg
>
To learn more about contact and action-at-a-distance forces,
Some force diagrams are easy to draw, while others require more care. The following Sample Problems will help you to sharpen your skills with force diagrams.
Sample Problem 1Draw both the system diagram and the FBD for each object in italics.
(a) A cup is sitting at rest on a table.
(b) A large trunk in the basement is pulled by a rope tied to the right side of the trunk by a person. The trunk does not move.
(c) A baseball player is sliding to the left across the ground.
(d) A desk is pushed to the left across the fl oor.
Solution(a) Step 1. Identify the objects in the system diagram.
The system diagram shows the cup and also the desk because the desk exerts a force on the cup.
Step 2. Identify the forces acting on the cup for the FBD.
The forces acting on the cup are the normal force and gravity.
Step 3. Determine the direction of each force.
The normal force exerted by the desk pushes up on the cup and the force of gravity pulls the cup down. No one is pushing the cup so there is no applied force. There is no friction acting on the cup because the cup is not moving and friction is not required to keep it stationary.
Step 4. Draw the FBD as a rectangle representing the cup and the arrows representing each force and its direction. Each arrow must be labelled with the appropriate force symbol.
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(b) Step 1. Identify the objects in the system diagram.
The system diagram shows the trunk, the rope, and the basement fl oor.
Step 2. Identify the forces acting on the trunk for the FBD.
The forces acting on the trunk are tension from the rope, the normal force, friction, and gravity.
Step 3. Determine the direction of each force.
The rope can only pull, exerting tension to the right. The basement fl oor exerts two forces on the trunk, the normal force and friction. The normal force is perpendicular to the surface and is directed up. The force of friction acts opposite to the applied force to keep the trunk stationary. The force of gravity on the trunk is directed down.
Step 4. Draw the FBD as a rectangle representing the trunk and the arrows representing each force and its direction. Each arrow must be labelled with the appropriate force symbol.
(c) Step 1. Identify the objects in the system diagram.
The system diagram shows the baseball player and the ground.The system diagram shows the baseball player and the ground.
Step 2. Identify the forces acting on the baseball player for the FBD.
The forces acting on the baseball player are the normal force, friction, and gravity.
Step 3. Determine the direction of each force.
The normal force exerted by the ground pushes up on the baseball player. The force of friction is directed opposite to the direction of motion. The force of gravity pulls down on the baseball player.
Step 4. Draw the FBD as a rectangle representing the baseball player and arrows representing each force and its direction.
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(d) Step 1. Identify the objects in the system diagram.
The system diagram shows the desk, the person pushing on it, and the fl oor.The system diagram shows the desk, the person pushing on it, and the fl oor.
Step 2. Identify the forces acting on the desk for the FBD.
The forces acting on the desk are an applied force, the normal force, friction, and gravity.
Step 3. Determine the direction of each force.
The desk is being pushed to the left, so the applied force is directed to the left. The normal force pushes up on the desk. The force of friction is directed to the right, opposite to the direction of motion. The force of gravity pulls down on the desk.
Step 4. Draw the FBD as a rectangle representing the desk and the arrows representing each force and its direction.
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Sample Problem 2
A player kicks a soccer ball that is sitting on the ground. The ball moves up into the air. Ignore air resistance.
(a) Draw the FBD of the ball after it has moved away from the player but is still in the air.
(b) A student draws the FBD shown in Figure 5 to represent the soccer ball while it is coming back down. What is wrong with the FBD? Explain your reasoning.
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Figure 5
Solution(a) Step 1. Identify the forces acting on the ball.
The soccer ball has moved away from the player so there is no applied force. The ball is in the air so there is no normal force exerted by the ground. The only force acting on the soccer ball is gravity.
Step 2. Determine the direction of the force.
The force of gravity pulls down on the soccer ball.
Step 3. Draw the FBD as a rectangle representing the ball and the arrows representing each force and its direction.
Practice 1. Draw a system diagram and an FBD for each object in italics. K/U C
(a) A book is at rest on top of a desk.(b) A basketball falls through the hoop. Ignore air resistance.(c) A large trunk is pushed horizontally from behind toward the east across a rough fl oor.
2. Figure 6(a) shows a system diagram of a block sliding up a ramp. A student draws the FBD for the block (Figure 6(b)). Discuss the validity of the FBD. K/U C
Calculating net ForcesIn many force problems, you must combine all the forces acting on a single object into one combined force. Th e total force is also called the net force or the resultant force. To determine the net force, you need to use FBDs. Keep in mind that force is a vector quantity and the direction of each individual force must be considered before determining the net force. Net force is represented by the symbol F
S
net. In the following Tutorial, you will draw FBDs to determine the net force acting on an object.
net force (F>net) the sum of all forces
acting on an object
The fl oor exerts a normal force of 36 N [up] on a stationary chair. The force of gravity on the chair is 36 N [down]. Draw the FBD of the chair and use the FBD to determine the net force on the chair.
SolutionStep 1. Draw the FBD of the object.
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Step 2. Identify which directions are positive.
Defi ne up as the positive y-axis and down as the negative y-axis. This means the normal force exerted by the fl oor on the chair is positive and the force of gravity on the chair is negative. The values of the forces on the FBD are now
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F N � �36 N
Fg � �36 N
To determine the net force from the FBD of an object, it is necessary to defi ne the x- and y-axes and to defi ne which directions are positive. You may omit one axis or the other, if there are no forces parallel to that axis. In the following Sample Problems, we will practise this technique.
Sample Problem 1: Calculating the Net Force on a Stationary Object
Tutorial 2 Using FBDs to Determine Net Force
(b) When the soccer ball is in the air, there is no applied force or normal force. Both of these forces are contact forces. Since the ball is no longer in contact with the player’s foot or the ground, these forces no longer act on the ball
and should not be included in the diagram. The FBD that the student drew in Figure 5 would be appropriate at the start when the ball is still on the ground and the player is kicking it.
Step 3. Add the forces on each axis to determine the net force.
FS
net 5 FS
N 1 FS
g 5 136 N 1 1236 N2 Fnet 5 0 N
There is no net force on the chair.
Sample Problem 2: Calculating the Net Force When the FBD Is GivenFigure 7 shows all the forces acting on an object. Use the FBD to calculate the net force.
Figure 7
SolutionSince the FBD is given, you can skip the drawing step.
Step 1. Identify which directions are positive.
Choose up and east as positive. So down and west are negative. The values of the forces on the FBD are now
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�52 N
�52 N
�62 N�45 N
Step 2. Define east and west forces as being along the x-axis. So up and down forces are along the y-axis.
Add the forces along each axis to determine the net force.
First, find the sum of all the forces parallel to the y-axis.
1Fnet2 y 5 152 N 1 1252 N2 5 0 N
The net force on the y-axis is 0 N.
Now find the sum of all the forces parallel to the x-axis.
1Fnet2 x 5 162 N 1 1245 N2 5 117 N
The net force along the x-axis is 17 N [E].
Therefore, the net force on the object is 17 N [E].
Practice 1. Calculate the net force when each set of forces acts on the same object.
Draw an FBD for each object. K/U C
(a) 5.5 N [W], 3.4 N [W], 4.2 N [E] [ans: 4.7 N [W]]
(b) 92 N [up], 35 N [down], 24 N [down] [ans: 33 N [up]]
(c) 15 N [up], 15 N [down], 35 N [E], 12 N [W] [ans: 23 N [E]]
2. A chain exerts a force of 1200 N [up] on a beam which experiences a force of gravity of 1100 N [down]. Draw the FBD of the beam and determine the net force on the beam. K/U C [ans: 100 N [up]]
3. You push a book across a table to your friend with a force of 6.5 N [E]. The force of friction on the book is 4.5 N [W]. The normal force and the force of gravity have a magnitude of 7.5 N. Draw the FBD of the book and calculate the net force on the book. K/U C [ans: 2.0 N [E]]
The Four Fundamental ForcesIt might seem like there are many different kinds of forces in everyday life and even more involved with research and technology. Physicists have grouped all known nat-ural forces into four categories called fundamental forces. These four fundamental forces are the gravitational force, the electromagnetic force, the strong nuclear force, and the weak nuclear force.
You have already learned a little about the force of gravity, and you will expand that knowledge in Chapter 4. Not only is the force of gravity responsible for pulling objects down toward Earth’s centre, but it also exists between any two masses in the universe. The force of gravity keeps the Moon in orbit around Earth and Earth in orbit around the Sun. You do not notice the force of gravity between smaller objects such as a pair of basketballs because the masses are small and the force of gravity is too weak compared to Earth’s gravity.
The electromagnetic force is caused by electric charges. Electric force exists between charges, and magnetic force exists between magnets. The electromagnetic force is an action-at-a-distance force. Unlike gravity, this force can both attract and repel. For this reason, these forces often cancel each other out. The electromagnetic force holds atoms and molecules together: it makes concrete hard and a feather soft and governs the properties of chemical reactions.
In the nucleus of atoms, the positively charged protons are very close and repel each other. They do not fly apart because the strong nuclear force of attraction between the neutrons and the protons keeps them in place. In some cases, protons and neutrons can transform into other particles. The weak nuclear force is respon-sible for the interactions involved during these particle transformations.
Table 1 gives a brief comparison of the four fundamental forces. The electromagnetic, strong nuclear, and weak nuclear forces are studied in later physics courses.
Table 1 Comparing the Fundamental Forces
Type of forceApproximate
relative strength Range Effect
gravitational 1 infinite attract only
electromagnetic 1020 infinite attract and repel
strong nuclear 1038 less than 10215 m attract and repel
weak nuclear 1025 less than 10218 m attract and repel
of the situation showing the objects involved. A free-body diagram (FBD) shows all the forces acting on a single object.
• YoucanuseFBDstocalculatethenetforceonanobject.Indicatethedirection of the positive y-axis and the positive x-axis to determine the signs of each force acting on the object. Add up the forces parallel to each axis to determine the net force.
1. Assume that you are sitting at your desk facing north with a book on the desk directly in front of you. K/U
(a) If you push on the side of the book that is closest to you, in which direction is the applied force?
(b) If you pull on the side of the book that is closest to you, in which direction is the applied force?
2. Name the force or forces that cause these objects to experience changes in motion. Neglect air resistance. K/U
(a) A ball falls toward the ground.(b) A person accelerates up in an elevator.(c) A car gradually slows down while approaching a red
light.
3. A student forgets to zero the force sensor before performing an investigation. What effect, if any, will this have on the data collected during the investigation? T/I
4. Some everyday forces are gravity, tension, friction, an applied force, and the normal force. It is important that you be able to determine the direction of each of these forces when drawing an FBD. Create a table or a graphic organizer to help you remember the direction of these forces. K/U
5. Draw both the system diagram and the FBD for each object in italics. K/U C
(a) A car is parked on the road.(b) A small fish is hanging from a fishing line.(c) A football is falling toward a player.(d) A puck is being pushed forward across the ice by a
hockey stick.
6. Create a poster to teach students about common everyday forces. Include at least two system diagrams in your poster and the corresponding FBDs. You may include several everyday forces in your poster or concentrate on just one. K/U C
7. Determine the force of gravity acting on each object. T/I C
(a) A 2.0 kg object falls straight down.(b) A 62 kg person stands on the floor.
8. Your teacher says, “Any applied force can also be called a normal force.” Discuss the validity of this statement. K/U C
9. You tie a long string to a cart on the ground. Explain why you can only pull the cart forward but cannot push the cart away with the string. Draw diagrams to help explain your answer. K/U C
10. Describe the main difference between contact forces and action-at-a-distance forces. What implication does this difference have when drawing an FBD? K/U C
11. Explain the difference between system diagrams and FBDs. Describe why both are used to solve force problems. K/U C
12. For a physics project, you are given a small block of wood, a paper clip, some masking tape, a balloon, a magnet, and an elastic band. The objective is to move the block from one end of a long hallway to the other in the least amount of time using only the materials listed, without touching the wood directly. Draw a system diagram showing how you would complete the task. Draw an FBD of the block of wood. Include a brief description of what you would do. T/I C
13. Examine Figure 8, showing a spider in a web. The web is currently under construction and consists of only a few fine strands. Draw the FBD of the spider. T/I C
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14. A doctor states, “The bones in the human arm can exert forces that can either push or pull other objects. The muscles are made of small fibres and can only cause tension forces.” C A
(a) What does the doctor mean when he says muscles can only cause tension forces?
(b) Why can bones both push and pull whereas muscles cannot?
15. Calculate the net force when each set of forces acts on a single object. T/I
(a) 56 N [up], 35 N [down](b) 12.3 N [right], 14.4 N [right], 32.7 N [left](c) 45 N [up], 45 N [down], 21 N [W], 21 N [E]
16. (a) List the four fundamental forces from weakest to strongest.
(b) Describe two ways that gravity is different from the other three fundamental forces.
(c) Explain why friction and tension are not fundamental forces. K/U C
newton’s First law of MotionEvery year in Canada, there are about 160 000 car accidents. These accidents cause many injuries, and about 3000 are fatal. Many of these injuries and deaths involve motorists who are not wearing seat belts or are driving too fast. An understanding of Newton’s first law will help you appreciate the importance of safety features in cars such as seat belts and airbags. Figure 1 shows a crash test designed to help engineers test and improve the safety features of new cars. What will happen to a crash test dummy during a collision if it is not wearing a seat belt and no airbag is present? Newton’s first law will help us answer this question.
InertiaIf you have ever played air hockey, you may know that a small plastic puck moves with close to uniform velocity after you hit it. In other words, there is very little fric-tion acting on the puck to slow it down. In fact, the net force on the puck is zero because the upward force on the puck exerted by the air and the downward force of gravity cancel each other and there is almost no friction. If you do not hit the puck at all, it will just sit there at rest if the air table is level.
Unfortunately, early scientists and philosophers did not have access to air hockey tables. They noticed that moving objects would spontaneously slow down for no apparent reason. They did not know about friction and incorrectly attributed the decrease in velocity to “lazy” objects. They concluded that a constant net force was needed to keep an object moving. They thought that larger net forces made things move at a higher constant velocity and smaller net forces made them move at a lower constant velocity. If no net force was acting on the object, they thought that the object would stop.
It was not until the 1600s that Galileo was able to perform experiments to help clear up these misconceptions. To help explain the results of his experiments, Galileo used the following thought experiment (Figure 2). Figure 2(a) shows a ball rolling down an incline, onto a horizontal surface, and up another incline. Galileo reasoned that if there was no friction acting on the ball, it would continue to roll up the second incline until it reached the same height as its starting position. If friction were present, it would not go quite as high on the second slope.
Figure 2(b) shows a situation similar to the first, but the steepness of the second incline is decreased. Galileo reasoned that the ball would have to roll farther to reach the same initial height on the second incline. In Figure 2(c), the second incline has been elimin-ated. In this experiment, the ball will never reach the same height. Galileo concluded that if there is no friction, the ball will continue to roll forever. It will never reach the same initial height since it cannot go up another incline. We can verify Galileo’s thought experiment today using equipment such as air tables, where little friction is present.
3.2
Figure 1 Crash tests are designed to test and improve automotive safety.
Figure 2 (a) The ball rolls up the steep second ramp until it rises to the same initial height.(b) The ball rolls up the less steep second ramp, but it still rises to the same height.(c) There is no second ramp and the ball never stops because it cannot get to the same initial height.
Galileo concluded that once an object starts moving, it will continue moving at a constant velocity if there is no friction present. Galileo used the concept of inertia to help explain his conclusion. Inertia is the property of matter that causes it to resist changes in motion. Th e inertia of an object depends on the mass of the object. An object with more mass has more inertia, whereas an object with less mass has less inertia. In other words, inertia is directly proportional to the mass of the object.
Imagine placing a stuff ed animal on the dashboard of a stationary car. If the car speeds up rapidly, the stuff ed animal will resist this change in motion due to its inertia and end up in your lap. Th is does not mean the stuff ed animal is moving backwards. Th e car is moving forward and the stuff ed animal is stationary. In other words, the dashboard is being pulled out from under the stuff ed animal. If the car is moving with uniform velocity and the stuff ed animal is placed on the dashboard, the stuff ed animal will stay on the dashboard because there is no change in the velocity of the car. In this case, inertia does not upset the stuff ed animal because inertia only resists changes in motion. If the stuff ed animal is placed on the dashboard and the car suddenly slows down, the stuff ed animal will continue to move forward with uniform velocity due to its inertia. Again, we are not implying that the stuff ed animal jumps forward. It is the car that is slowing down, not the stuff ed animal speeding up. Th e dashboard slides under the stuff ed animal, causing the windshield to move toward the stuff ed animal.
Newton was born in 1642, the year that Galileo died. Newton published Principia Mathematica, a set of three books which included much of his own work about physics, as well as a description of Galileo’s law of inertia. Th e law of inertia is now called Newton’s fi rst law of motion because it was included with Newton’s other laws of motion.
inertia the property of matter that causes it to resist changes in motion; inertia is directly proportional to the mass of the object
First Law of MotionIf the net external force on an object is zero, the object will remain at rest or continue to move at a constant velocity.
Below are some of the important implications of Newton’s fi rst law:• Anon-zeronetforcewillchangethevelocityofanobject.Th evelocitycan
change in magnitude, direction, or both.• Anetforceisnotrequiredtomaintainthevelocityofanobject.• Externalforcesarerequiredtochangethemotionofanobject.Internalforces
have no eff ect on the motion of an object.
fi rst law of motion an object will remain at rest or continue to move at constant velocity when the net force on the object is zero
Tutorial 1 FBDs and the First Law of Motion
The following Sample Problems will help to clarify the meaning of Newton’s fi rst law by using FBDs.
Sample Problem 1Use Newton’s fi rst law to explain each situation below. (a) Why does a computer sitting on a desk remain at rest?
(b) Why does a hockey puck moving across smooth ice move at a constant velocity?
(c) Why does a wagon pulled across a rough surface by a child move at a constant velocity?
Solution (a) Examine the system diagram and FBD of the computer
shown in Figure 3. Notice that the desk exerts a normal force up on the computer and the force of gravity pulls down on the computer. These two forces cancel to give a net force of zero. According to Newton’s fi rst law, the computer will remain at rest.
(a)
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(b)
Figure 3 (a) System diagram of a computer on a desk (b) FBD of the computer
Sample Problem 2: Newton’s First Law Applied to HeadrestsOlder cars did not have headrests, but all new cars do. How do headrests help prevent injuries during a rear-end collision? Use Newton’s fi rst law to explain your answer.
SolutionConsider the forces acting on a person’s body during a collision. During a rear-end collision, the car will suddenly accelerate forward and so will your body because the seat exerts a force directed forward on your torso. In a vintage car with no headrest
(Figure 6(a)), there is no force applied to the head. According to Newton’s fi rst law, your head will continue to remain at rest. Your head will initially appear to snap backwards relative to your body as your body accelerates forward, possibly resulting in a neck injury known as whiplash. The headrest in a modern car helps push the head forward with the rest of the body (Figure 6(b)). This helps to prevent whiplash since your neck does not bend backwards as far during a rear-end collision.
Figure 5 (a) System diagram of a child and a wagon (b) FBD of the child and the wagon
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Fg�
F T�F f
�
Figure 6 (a) No headrest (b) Headrest present
F
F
F
(a) (b)
(a)
(b)
Figure 4 (a) System diagram of a hockey puck moving on ice (b) FBD of the hockey puck
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(c) Examine the system diagram and FBD of the wagon shown in Figure 5. The child applies tension forward on the wagon. The ground exerts a normal force up, and the force of friction acts backwards on the wagon. The force of gravity acts down on the wagon. The two vertical forces cancel each other and the two horizontal forces cancel each other to produce a net force of zero. According to Newton’s fi rst law, the wagon will move at a constant velocity.
(b) Examine the system diagram and FBD of the hockey puck shown in Figure 4. Notice that the ice exerts a normal force up on the hockey puck and the force of gravity pulls down on the hockey puck. These two forces cancel to give a net force of zero in the vertical direction. Since no external force acts in the horizontal direction, according to Newton’s fi rst law, the hockey puck will continue to move at a constant velocity.
What is the missing force on each FBD shown in Figure 7?
Solution (a) According to Newton’s first law, since the object is at rest,
the net force must be zero. Choose up as positive. So down is negative.
Given: two upward forces of 112 N and 18 N
Required: FS
g
Analysis: FS
net 5 0
Solution: Fnet 5 112 N1 118 N21Fg
0 5 20 N 1 Fg
Fg 5 220 N
FS
g 5 20 N 3down 4Statement: The force of gravity on the object is 20 N [down].
(b) According to Newton’s first law, the object must have a net force of zero since it is moving at a constant velocity. Choose up and right as positive. So down and left is negative. We can look at (Fnet)y and (Fnet)x separately. Start with the forces along the y-axis.
Given: 222 N
Required: FS
N
Analysis: 1Fnet 2 y 5 0
Solution: 1F net 2 y 5 F N 1 1222 N2 0 5 F N 2 22 N
FN 5 122 N
FS
N 5 22 N [up]
Now examine the forces along the x-axis.
Given: 128 N; 215 N
Required: FS
1
Analysis: 1Fnet 2 x 5 0
Solution: 1Fnet2 x 5 128 N 1 1215 N2 1 F1
0 5 13 N 1 F1
F1 5 213 N
FS
1 5 13 N [left]
Statement: The normal force on the object is 22 N [up].
FS
1 is 13 N [left].
Sample Problem 3: Determining the Missing Force
Practice 1. Explain why it is unsafe to stand when riding a bus or subway without holding onto
something. K/U C
2. In a demonstration, a teacher places some plates on top of a smooth tablecloth. The teacher then pulls quickly on the tablecloth. If the tablecloth is a frictionless surface, predict what will happen to the plates. Explain your reasoning. K/U
3. A driver in a car passes over some black ice, which exerts almost no friction on the wheels. Explain why the car cannot slow down when the driver pushes on the brakes. K/U
4. You have some snow stuck on your shovel. Explain how you could apply Newton’s first law to get the snow off. K/U
5. The FBD in Figure 8 is for a car at rest on the ground. T/I
(a) Determine FS
1 and FS
g. [ans: FS
1 = 2650 N [R]; FS
g = 13 000 N [down]]
(b) How would your answers change if the car was moving at a constant velocity? Assume none of the given forces change.
Figure 7 (a) FBD for an object at rest (b) FBD for an object moving left at a constant velocity
Your teacher will present several situations and ask you to predict what will happen and explain your reasoning. You will then observe the demonstration and record what happens. Finally, you will explain what happens and why in terms of Newton’s first law.
Equipment and Materials: coin; playing card; ballistics cart and ball; skateboard; thread; two standard masses
1. Create a table in your notebook similar to Table 1. The second column in Table 1 shows a sketch of each demonstration. Summarize your results in the appropriate column of your table. T/I C
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Table 1
Situation Sketch of situationPrediction/explanation Observations Explanation
A. A coin is on top of a playing card on the left fist. Hit the card.
B. A moving ballistics cart fires a ball by exerting a force straight up.
C. A moving skateboard with an object on top hits a wall.
D. A thread supports an object. Another thread is underneath. Pull slowly on the bottom thread.
E. A thread supports an object. Another thread is underneath. Pull quickly on the bottom thread.
You can apply what you learned about Newton’s fi rst law to the Unit Task described on page 204.
Applications of newton’s First LawA wide range of technologies take advantage of Newton’s fi rst law including auto-mobiles, planes, rockets, and construction. Th e key idea is that the net force must be zero for any object to remain at rest or to keep moving with uniform motion.
Consider the physics of a typical seat belt. What will happen if you are not wearing a seat belt and the car suddenly stops? According to Newton’s fi rst law, you will con-tinue to move forward at a constant velocity until an object exerts a net force on you. Th is could be the dashboard or the windshield. If you are wearing a seat belt, the seat belt will exert a net force on you to slow you down.
One design feature of a seat belt uses the sudden decrease in velocity of the car to activate a gear mechanism. In this design, the seat belt strap is attached to a spool, which in turn is attached to a gear (Figure 9). Beneath this gear is a pendulum that is free to swing back and forth. When the car comes to a sudden stop, the pendulum swings forward due to inertia. Th is causes the pendulum to move a metal stop into the teeth of the gear, locking the seat belt in place.
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3.2 Summary
• Inertiaisthepropertyofmatterthatcausesittoresistchangesinmotion.Inertia is directly proportional to the mass of the object.
• Newton’sfirstlawcanalsobecalledthelawofinertia.Newton’sfirstlawstates that if the net force acting on an object is zero, the object will either remain at rest or continue to move at a constant velocity.
• Th erearemanyimplicationsofNewton’sfirstlaw:objectsatresttendto remain at rest; objects in motion tend to remain in motion; if the velocity of an object is constant, then the net force is zero; if a net force acts on an object, the velocity will change in magnitude, direction, or both.
• Newton’sfirstlawofmotioncanbeappliedinmanysituationstohelpincrease our understanding of motion.
Space and fl ight technology also takes advantage of Newton’s fi rst law. During high accelerations in a jet or a rocket, a pilot or astronaut might experience pooling of blood in certain parts of the body such as the legs. Th e inertia of the blood oft en causes it to move out of the head and into the legs. A G-suit helps to prevent this condition by exerting pressure on the legs and lower torso. In eff ect, it is squeezing the blood out of the legs back up toward the brain.
pendulum
After coming toa sudden stop
Before coming toa sudden stop
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(b)(a)
Figure 9 (a) When the vehicle is moving at constant velocity the seat belt pendulum (shown in green) hangs straight down. (b) After a sudden stop, the pendulum swings, causing the metal stop to lock into the seat belt gear.
1. Explain how modern technology can help demonstrate Galileo’s thought experiments. K/U
2. Which has the most inertia, a truck, a desk, or a feather? Which has the least inertia? How do you know? K/U
3. Skater 1 has a mass of 45 kg and is at rest. Skater 2 has a mass of 50 kg and is moving slowly at a constant velocity of 3.2 m/s [E]. Skater 3 has a mass of 75 kg and is moving quickly at a constant velocity of 9.6 m/s [E]. Which skater experiences the greater net force? Explain your reasoning. K/U
4. Explain each statement using inertia or Newton’s fi rst law. K/U
(a) You should not sit in the back (bed) of a pickup truck when it is moving.
(b) It is hard to get a car moving on very slippery ice.(c) You should not put objects on the ledge of a car
between the rear windshield and the rear seat.(d) During liftoff, astronauts are placed horizontally in the
capsule rather than vertically.
5. Headrests and seat belts are two important pieces of safety equipment used in automobiles. K/U
(a) Will both technologies signifi cantly improve safety if the car suddenly slows down but keeps moving in a straight line? Explain your reasoning.
(b) Will both technologies signifi cantly improve safety if the car suddenly speeds up but keeps moving in a straight line? Explain your reasoning.
6. Many people buy a coffee or other hot beverage on their way to work. For safety, the cup usually has a lid and is placed in a cup holder. Using Newton’s fi rst law, explain why both of these precautions are necessary. C A
7. Figure 10 shows a string tied to a spike at one end and a puck at the other. The puck is moving around in a circle on the ice. Describe what the puck will do if the string is suddenly cut at the red line. Explain your reasoning using a diagram. K/U C
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Figure 10
8. Use Newton’s fi rst law to explain why the normal force must be equal in magnitude to the force of gravity for an object to remain at rest on a horizontal surface when no other forces are acting on the object. K/U C
9. You are inside a car moving fast along a sharp turn in the road. Use Newton’s fi rst law to describe what happens to you as you safely make it around the curve. T/I C
10. Use Newton’s fi rst law to explain why you should slow down when going around a curve on an icy highway. K/U C
11. Figure 11 shows some simple equipment used to test Newton’s fi rst law. In the experiment, the ring is suddenly pulled horizontally. T/I C
(a) Predict what will happen to the piece of chalk. Explain your reasoning.
(b) Why does it help to put some water in the container?
container
chalk
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12. While on the bus, you throw an apple straight up into the air. What will happen if the bus(a) moves at a constant velocity?(b) slows down?Explain your reasoning using diagrams. T/I C
13. Determine the indicated forces on each FBD shown in Figure 12. T/I (a) statue at rest on a shelf (b) sled pulled right at a constant velocity
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14. Your physics teacher challenges the class to the following puzzle. Place eight quarters into a single stack on a desk. Your task is to take the stack apart one quarter at a time using only a thin ruler. You cannot make any contact with the quarters but the ruler can. Only quarters at the bottom of the pile may be removed. T/I C
(a) Describe how you would complete the task. Carry out your plan to test whether it works.
(b) Explain why it works.
15. Studies reveal that many people do not use their headrest properly. Research what most people do wrong and how it can be fi xed. Prepare a small pamphlet that can be used to make people aware of the problem and how to fi x it. Include any useful statistics that might encourage people to make the necessary changes. T/I C
newton’s second law of MotionNewton’s fi rst law of motion applies to situations when the net force on an object is zero. When the net force is zero, an object that is at rest will remain at rest and an object that is in motion will continue to move in a straight line at a constant velocity. What do these cases have in common? In both cases, the acceleration is zero.
The variables Involved in newton’s Second LawOne way to experience Newton’s second law of motion is in a moving car. A light car with a powerful engine that can exert a large force will experience a large accelera-tion (Figure 1(a)). A massive truck experiences small accelerations even if the force is large (Figure 1(b)). A small economy car experiences smaller accelerations because the force its engine exerts is smaller (Figure 1(c)). Th e acceleration is aff ected by changing either the mass or the net force. To understand this relationship among acceleration, net force, and mass, we need to consider these variables more carefully.
Figure 1 (a) A race car speeds up quickly because it has a small mass and experiences a large net force. (b) A truck accelerates more slowly. Even though a large net force acts on it, the mass is large. (c) A small economy car also accelerates more slowly. It has a small mass and a small applied force.
We will fi rst examine the relationship between the net force and the acceleration. Imagine pushing a large sled with a few small boxes on top across a nearly friction-less icy surface. If you push as hard as you can, the sled starts from rest and attains a high velocity in little time (Figure 2(a)). If you push with less force on the sled, the velocity will increase more gradually (Figure 2(b)). Th ese two scenarios imply that acceleration increases as the net force increases. Now we will examine the eff ect of mass on acceleration. If you remove the boxes, the lighter sled is easier to speed up from rest (Figure 2(c)). If you put more boxes on the sled, the heavier sled is harder to speed up at the same rate (Figure 2(d)). In other words, acceleration decreases as mass increases.
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3.3
Figure 2 The relationship among acceleration, net force, and mass (a) A large net force results in a large acceleration. (b) A small net force results in a small acceleration. (c) A small mass results in a large acceleration. (d) A large mass results in a small acceleration.
Figure 3 The graph of net force versus acceleration is linear, and the slope represents the mass. The line must pass through the origin, since when the acceleration is zero the net force must also be zero.
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F net
F net2
aa2
Representing newton’s Second Law MathematicallyFrom Newton’s second law, we can derive an equation that connects acceleration, net force, and mass.
aS
~ FS
net when m is constant and aS
~1m
when FS
net is constant
Combining the two proportionalities, we get
aS
~FS
net
mWe can create an equation relating all three variables if we insert a proportionality
constant k:
aS
5 k FS
net
mIf we define the newton as the net force required to accelerate 1 kg at 1 m/s2 or
1 N = 1 kg·m/s2, then k = 1. We can write the equation as
aS
5 FS
net
m
Solving for the net force, we have
FS
net 5 maS
Consider a situation where the mass is kept constant but the net force gradually increases. From the equation F
S
net 5 m aS, if mass is kept constant and the net force on the object increases, the acceleration also increases. If you graph net force versus acceleration, the relationship is linear (Figure 3).
Consider the slope of the line:
slope 5riserun
5D F
S
net
DaS
The line must pass through the origin. From Newton’s first law, when the accelera-tion is zero, the net force must also be zero. If we use the origin as the initial point on the graph, the above slope equation simplifies to
slope 5FS
net22 F
S
net1
aS
2 2 aS
1
5FS
net22 0
aS
2 2 0
5FS
net2
aS
2
Second Law of MotionIf the net external force on an object is not zero, the object will accelerate in the direction of this net force. The magnitude of the acceleration is directly proportional to the magnitude of the net force and inversely proportional to the mass of the object.
Newton’s second law of motion is summarized as follows: second law of motion an object will accelerate in the direction of the net force; the magnitude of the acceleration is directly proportional to the magnitude of the net force and inversely proportional to the object’s mass
Newton’s second law can be used to solve a variety of problems involving many different situations. In the following Sample Problems, we will demonstrate how to apply Newton’s second law.
Sample Problem 1: Determining AccelerationA net force of 36 N [forward] is applied to a volleyball of mass 0.24 kg. Determine the acceleration of the volleyball.
Given: FS
net 5 36 N [forward]; m 5 0.24 kg
Required: aS
Analysis: FS
net 5 maS
. Choose forward as positive.
Solution: FS
net 5 maS
aS
5 FS
net
m
a 5136 N0.24 kg
a 5 1150 m/s2
a>5 150 m/s2 3forward 4
Statement: The acceleration of the volleyball is 150 m/s2 [forward].
Sample Problem 2: Calculating Net ForceA 64 kg runner starts walking at 3.0 m/s [E] and begins to speed up for 6.0 s, reaching a final velocity of 12.0 m/s [E]. Calculate the net force acting on the runner.
Given: m 5 64 kg; vS
1 5 3.0 m/s [E]; vS
2 5 12.0 m/s [E]; ∆t 5 6.0 s
Required: FS
net
Analysis: FS
net 5 maS
, but first we have to calculate the
acceleration using the kinematics equation a>5
v>2 2 v
>1
Dt.
Choose east as positive.
Solution: a>5
v>2 2 v
>1
Dt
a 5112.0 m/s 2 113.0 m/s2
6.0 s
a 5 11.5 m/s2
a>5 1.5 m/s2 3E 4
Now we can calculate the net force.
F>net 5 ma
>
Fnet 5 164 kg2 111.5 m/s2 2 Fnet 5 196 N
F>net 5 96 N 3E 4
Statement: The net force on the runner is 96 N [E].
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m
a
Figure 4 The graph of acceleration versus mass is a reciprocal function.
Physically, what does this slope represent? Using Newton’s second law, we haveFS
net 5 maS
m 5FS
net
aS
The expressions for slope and mass can be equated, so we can conclude that the slope of the graph of net force versus acceleration is the mass of the object.
Now consider a situation where the net force is kept constant but the mass
gradually increases. From the equation aS 5 FS
net
m , if the net force is kept constant
and the mass of an object increases, the acceleration decreases. If you graph accelera-tion versus mass, the graph shows a reciprocal relationship (Figure 4).
Practice 1. A net force of 126 N [S] is applied to a 70 kg sprinter. Determine the acceleration of
the sprinter. T/I [ans: 1.8 m/s2 [S]]
2. A car accelerates at 1.20 m/s2 [forward]. The net force on the car is 1560 N [forward]. What is the mass of the car? T/I [ans: 1300 kg]
3. A cyclist starts to pedal vigorously, increasing her velocity from 6.0 m/s [E] to 14.0 m/s [E] in 6.0 s. The total mass of the cyclist and the bicycle is 58 kg. Find the net force acting on the cyclist and bicycle. T/I [ans: 77 N [E]]
4. During a road test, a driver brakes a 1420 kg car moving at 64.8 km/h [W]. The car slows down and comes to a stop after moving 729 m [W]. T/I
(a) Calculate the net force acting on the car. [ans: 316 N [E]]
(b) What is the force of friction acting on the car while it is slowing down? Explain your reasoning. [ans: 316 N [E]]
5. For each FBD shown below, determine the net force applied to the object and its acceleration. T/I
(a) m 5 8.0 kg [ans: 36 N [left], 4.5 m/s2 [left]] (b) m 5 125 kg [ans: 1000 N [up], 8 m/s2 [up]]
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24 N
31 N
19 N
6. In a two-person bobsled competition, athlete 1 pushes forward on the sled with 310 N and athlete 2 pushes forward with 354 N. A force of friction of 40 N [backwards] is acting on the bobsled. The mass of the bobsled is 390 kg. Calculate the acceleration of the bobsled. T/I [ans: 1.6 m/s2 [forward]]
Sample Problem 3: Calculating Net Force and Acceleration Using an FBDA 9100 kg jet moving slowly on the ground fires its engines, resulting in a force of 22 000 N [E] on the jet. The force of friction on the jet is 3800 N [W].
(a) Draw the FBD for the jet.
(b) Calculate the net force acting on the jet.
(c) Calculate the acceleration of the jet.
Solution (a) Choose up and east as positive. So down and west
are negative.
(b) The normal force and gravity will cancel when the jet is on horizontal ground. The net force is east since the applied force is greater than the force of friction. To find the net force, we add all the horizontal forces.
F>net 5 F
>a 1 F
>f
Fnet 5 122 000 N 1 1238002 N Fnet 5 118 200 N
F>net 5 18 200 N 3E 4
The net force on the jet is 18 200 N [E].
(c) Given: FS
net 5 18 200 N [E]; m 5 9100 kg
Required: aS
Analysis: F>net 5 ma
>
Solution: F>net 5 ma
>
aS
5FS
net
m
a 5118 200 N
9100 kg a 5 12.0 m/s2
aS
5 2.0 m/s2 3E 4Statement: The acceleration of the jet is 2.0 m/s2 [E].
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Fa � �22 000 NF f � �3800 N
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The line weight of the arrows is the same as those in part (a).PDF glitch only.
Recall that strings can only pull on objects when they exert tension. A pulley is a small wheel that changes the direction of the tension in a rope or string without changing its magnitude. We will assume throughout the text that all pulleys are frictionless and light.
Sample Problem 1In an investigation, students place a 0.80 kg cart on a table. They tie one end of a light string to the front of the cart, run the string over a pulley, and then tie the other end to a 0.20 kg hanging object (Figure 6). Assume that no friction acts on either object.
(a) Determine the magnitude of the acceleration of the cart and the hanging object.
(b) Calculate the magnitude of the tension.
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0.20 kg
0.80 kg
Figure 6
(a) Step 1. Draw the FBD for each object.
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FBD of cart
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FBD of hanging object
Step 2. Identify which directions are positive. Determine the net force acting on each object.
The cart is on the table, so the normal force and gravity cancel each other. The cart will accelerate to the right, so choose right as positive. The equation for the net force acting on the cart is
Fnet 5 FT
Because Fnet 5 m1a, m1a 5 FT (Equation 1)
The hanging object will accelerate down, so choose down as positive. The equation for the net force acting on the hanging object is
Fnet 5 Fg2 2 FT
Because Fnet 5 m2a and Fg2 5 m2g,
m2a 5 m2g 2 FT (Equation 2)
Step 3. Add the equations to solve for the acceleration.
The tension acting on the cart and on the hanging object is the same. Adding equations 1 and 2 will cancel the tension.
The magnitude of the acceleration of the cart is 2.0 m/s2.
Figure 5 This FBD shows an object in free fall, with no air resistance acting on it. The net force on the object is the force of gravity. The acceleration of the object is equal to the acceleration due to gravity.
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newton’s Second Law and GravityIn Section 3.1, you learned that you can calculate the force of gravity by multiplying the mass by the acceleration due to gravity. We can now justify the equationFS
g 5 m gS using Newton’s second law.Consider an object in free fall with no air resistance acting on it (Figure 5).
Free fall is motion of an object toward Earth with no other forces acting on it. Th e only force acting on the object is the force of gravity. In this situation, the force of gravity is equal to the net force. We also know that the object accelerates down at aS 5 9.8 m/s2 [down]. Combining these two facts, we have
FS
g 5 FS
net
5 maS
Th en, because aS
5 gS
,FS
g 5 mgS
You will use the equation for the force of gravity to help you complete Investigation 3.3.1. To prepare for the investigation, read the following Tutorial and complete the Practice questions.
(b) To calculate the tension, substitute the acceleration into equation 1.
m1a 5 FT FT 5 10.80 kg2 11.96 m/s22FT 5 0.16 N
The magnitude of the tension in the string is 0.16 N.
Practice 1. Calculate the acceleration of the cart in Figure 7, given the following assumptions. T/I
(a) No friction is acting on the cart. [ans: 3.3 m/s2 [right]]
(b) A force of friction of 0.50 N acts on the cart opposite to the motion. [ans: 3.0 m/s2 [right]]
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0.60 kg
1.20 kg
Figure 7
2. In an experiment, objects are placed on top of a cart as shown in Figure 8. T/I
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motion sensor 2.0 kg
Figure 8
(a) Calculate the acceleration of the cart. Assume that no friction acts on the cart. [ans: 1.6 m/s2 [right]]
(b) What will happen to the acceleration of the cart if the mass of the objects on top of the cart increases? Explain your reasoning.
(c) What will happen to the acceleration of the cart if an object is taken from the top of the cart and tied to the hanging object? Explain your reasoning.
3.3 Summary
• AccordingtoNewton’ssecondlaw,whenanon-zeronetforceactsonan object, the object will accelerate in the direction of the net force. The acceleration is directly proportional to the net force and inversely proportional to the mass. The equation representing Newton’s second law is F
S
net 5 maS.• Ifthenetforceonanobjectincreasesandthemassisconstant,thenthe
acceleration increases. If the net force on an object is constant and the mass increases, then the acceleration decreases.
• Whensolvingproblemswheremorethanoneforceactsonasingleobject,make sure you draw an FBD. You can add all the forces to calculate the net force. You can also use the equation F
Investigating Newton’s Second Law (p. 150)Now that you have learned about Newton’s second law, you can perform Investigation 3.3.1. You will explore how changing the net force acting on a system affects the acceleration of the system. You will also explore how changing the mass of the system affects the acceleration of the system.
(a) A 72 kg rugby player accelerates at 1.6 m/s2 [forward].
(b) A 2.3 kg model rocket accelerates at 12 m/s2 [up].
2. Calculate the acceleration in each situation. T/I
(a) A cannon exerts a force of 2.4 3 104 N [E] on a 5.0 kg shell.
(b) A hockey stick hits a 160 g puck forward with a force of 24 N.
3. Determine the mass of the object in each situation. T/I
(a) A driver brakes and the car accelerates at 1.2 m/s2
[backwards]. The net force on the car is 1400 N [backwards].
(b) A woman throws a shot put with a net force of 33 N [forward] with an acceleration of 6.0 m/s2 [forward].
4. A 54 kg skier starts from rest at the top of a snow-covered hill, reaching a velocity of 12 m/s in 5.0 s. Calculate the net force acting on the skier. T/I
5. A dynamics cart is pulled from rest by a net force of 1.2 N [forward]. The cart moves 6.6 m, reaching a velocity of 3.2 m/s [forward]. Determine the mass of the cart. T/I
6. During a parachute jump, a 58 kg person opens the parachute and the total drag force acting on the person is 720 N [up]. T/I
(a) Calculate the net force acting on the person.(b) Determine the acceleration of the person.
7. A net force of magnitude 36 N gives an object of mass m1 an acceleration of 6.0 m/s2. The same net force gives m1 and another object of mass m2 fastened together an acceleration of 2.0 m/s2. What acceleration will m2 experience if the same net force acts on it alone? T/I
8. A 1300 kg car accelerates at 1.6 m/s2 [E]. A frictional force of 3800 N [W] is acting on the car. T/I C
(a) Draw the FBD of the car.(b) Determine the applied force acting on the car.
9. A long, heavy, metal chain is held at rest on a table with part of the chain hanging over the edge (Figure 9). The chain is released and it starts to accelerate. K/U
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Figure 9
(a) In which direction will the chain accelerate? What causes the acceleration? Explain your reasoning.
(b) What will happen to the acceleration of the chain as more chain moves over the edge of the table? Explain your reasoning.
10. Three students push horizontally on a large 80 kg crate sitting on the floor. Two of them push to the left on the crate, each with a force of 170 N. The third pushes to the right on the crate with a force of 150 N. Assume that no friction acts on the crate. T/I
(a) What is the acceleration of the crate?(b) What will happen to the net force and acceleration if a
fourth student jumps on top of the crate? Explain your reasoning.
11. A string can hold up 12 kg without breaking. You tie the string to a 30 kg object sitting on ice and use it to pull the object horizontally for 22 m. Calculate the minimum possible time to complete the task. T/I
12. Examine the data in Table 1. T/I C
(a) Copy and complete the table.
(b) Graph FS
a versus aS
and draw the line of best fit. What does the y-intercept represent? Explain.
(c) Graph FS
net versus aS
and draw the line of best fit. Calculate the slope of the line. What does the slope represent? Explain.
Table 1
Mass (kg) Friction (N) [W] Applied force (N) [E] Net force (N) [E] Acceleration (m/s2) [E]
newton’s Third law of MotionExamine the photograph in Figure 1 of an Atlas V rocket during liftoff. What forces are acting on the rocket? An obvious one is gravity pulling the rocket down. The rocket is accelerating upward in the picture. This means that the net force must be up according to Newton’s second law. What is this upward force? You might suggest that the ground is pushing up on the rocket with the normal force. This cannot be the answer because the normal force is a contact force and the rocket is no longer in contact with the ground. In this section, you will learn about this upward force as you learn about Newton’s third law of motion.
Action and Reaction ForcesIf you reach out and push this book away from you, you can actually feel the book pushing back on you. This is an example of a set of action and reaction forces. The action force is you pushing on the book, and the reaction force is the book pushing back on you. An easy way to experience action and reaction forces is by pushing on the boards when you are skating in an ice rink. When you push on the boards (the action force), the boards push back on you (reaction force). It is the reaction force that causes you to accelerate away from the boards.
These action and reaction forces are all around us and go unnoticed most of the time. When you push a door open, the door pushes back on you. When you walk forward across the floor, you push backwards on the ground and the ground pushes forward on you. In fact, whenever one object exerts a force on another, the second object exerts another force back.
Another simple way to demonstrate action and reaction forces is to have a person standing at rest on a skateboard suddenly step forward (Figure 2). When this hap-pens, the person and the skateboard will accelerate in opposite directions. According to Newton’s second law, both the person and the skateboard have net forces acting on them because they are accelerating. What causes these forces?
The net force on the skateboard is caused by the person’s feet pushing backwards on the skateboard. The person’s feet exert the action force on the skateboard. The net force on the person is caused by the skateboard pushing forward on the person. The skateboard exerts the reaction force on the person.
Keep in mind that these action and reaction forces are simultaneous. It is incorrect to imply that the action force happens first and then the reaction force happens. These are contact forces, and both the action and reaction forces start whenever contact is made.
Newton’s third law of motion deals with these action and reaction forces. For this reason, it is sometimes called the action–reaction law.
We can now explain how a rocket can accelerate up off the launch pad. During liftoff, the rocket engines burn rocket fuel, creating tremendous pressure, which causes the expanding hot gases to accelerate from the bottom of the rocket (action force). At the same time, the expanding hot gases push up on the rocket (reaction force), causing the rocket to accelerate up. According to Newton’s third law, the action force caused by the expanding hot gases is equal in magnitude to the reaction force of the expanding hot gases on the rocket. If the reaction force up on the rocket from the expanding hot gases is greater than the force of gravity pulling the rocket down, then the rocket will accelerate up. In the following Tutorial, you will apply Newton’s third law to problems involving motion.
Third Law of MotionFor every action force, there is a simultaneous reaction force that is equal in magnitude, but opposite in direction.
Figure 1 An Atlas V rocket was launched in June 2009. As the rocket lifted up off the launch pad, what force caused it to accelerate upward?
Figure 2 As a person jumps off a skateboard in one direction, the skateboard will accelerate in the other direction.
V of person
V of skateboard
third law of motion each action force has a reaction force that is equal in magnitude and opposite in direction
Explain each event in terms of Newton’s third law. (a) A swimmer moves through the water. (b) A small balloon releases air and fl ies around the classroom. (c) You start walking across the fl oor.
Solution (a) The swimmer’s arms and legs exert an action force
backwards on the water, causing the water to accelerate backwards. According to Newton’s third law, the water exerts an equal but opposite force forward on the swimmer, propelling the swimmer forward through the water.
(b) As the rubber in the balloon contracts, it forces air out the back of the balloon (action force). According to Newton’s third law, the reaction force is caused by the air pushing back on the balloon, which accelerates the balloon forward.
(c) The bottom of your shoe exerts a horizontal backward action force on the fl oor. According to Newton’s third law, the reaction force is caused by friction when the fl oor pushes forward on your feet. Friction causes you to accelerate forward.
In the following Sample Problem, we will clarify Newton’s third law and demonstrate how it can be used to explain motion when objects exert forces on each other.
Sample Problem 1
Tutorial 1 Explaining Motion Using Newton’s Third Law
Practice
1. Explain each event in terms of Newton’s third law. K/U
(a) A rocket can accelerate in outer space.(b) A motorboat accelerates east in the water.(c) Football player 1 tackles football player 2.
In this investigation, you will perform and observe several actions that involve Newton’s third law. Your task is to predict what will happen at each station, observe carefully what actually happens, and explain why it happens.
Equipment and Materials: offi ce chair with wheels; bathroom scale; desk; ballistics cart; fan cart with removable sail
1. Create a table similar to Table 1 to record your observations. Add six rows to the table.
Table 1
Activity Prediction ObservationAction and reaction forces
2. Read through the descriptions of activities (a) through (g). For each activity, predict what you think will happen and record your predictions in your table.
3. Perform activities (a) to (g) (or observe a classmate performing these activities). Record your observations.(a) Sit on an offi ce chair and push gently against a wall with
both hands.(b) Sit on the chair, not touching the wall. Use your hands to
pull horizontally on your own shoulders, away from the wall.
(c) Stand on a bathroom scale placed beside a desk. Put your hands on the desk and push down.
(d) Stand on a bathroom scale. Push down on your own head with your hands.
(e) Use a spring-loaded ballistics cart to fi re a ball horizontally.
(f) Obtain a fan cart (Figure 3(a)) with no sail. Direct the fan away from the cart. Turn the fan cart on.
(g) Obtain a fan cart (Figure 3(b)) with a sail. Direct the fan toward the sail. Turn the fan cart on.
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sail
Figure 3
A. Use Newton’s third law to describe the action and reaction forces you observed in each activity. T/I
You may have noticed that objects will not always accelerate or the reading on a scale will not always change when a force is applied to an object. What is diff erent about these situations? Th e diff erence is that in each of these situations there is no external net force acting on the object.
Does this mean that action and reaction forces cancel and we can never accelerate? Defi nitely not. Action and reaction forces always act on diff erent objects, or diff erent parts of a single object. When two objects are involved, the two forces are not added together and each object can accelerate.
For example, if a ballistics cart pushes backwards on a ball (action force), the ball accelerates backwards because the applied force causes a net force on the ball. According to Newton’s third law, the ball will also cause a reaction force on the cart in the opposite direction. Th is reaction force causes a net force on the cart, making it accelerate the other way. Th e action and reaction forces do not cancel because they do not act on the same object. When you draw the FBD for each object, the action and reaction forces will appear in separate FBDs (Figure 4).
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FBD of cartFBD of ball
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FBD of cartFBD of ball
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Tutorial 2 Using Newton’s Third Law in Calculations
In the following Sample Problem, we will clarify Newton’s third law and demonstrate how it can be used to explain motion when objects exert forces on each other.
Sample Problem 1: One Skater Pushing on AnotherTwo skaters are standing on ice facing each other (Figure 5). Skater 1 pushes on skater 2 with a force of 70 N [E]. Assume that no friction acts on either skater. The mass of skater 1 is 50 kg and the mass of skater 2 is 70 kg. (a) State the action and reaction forces. (b) Draw the FBD of each skater. (c) Describe what will happen to each skater. (d) Calculate the acceleration of each skater. skater 1 skater 2
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Figure 4 (a) Diagram and FBDs of a ball inside a ballistics cart before fi ring (b) Diagram and FBDs after fi ring the ball. The action and reaction forces do not cancel because they act on different objects.
Solution (a) The action force is skater 1 pushing with 70 N [E] on
skater 2. The reaction force is skater 2 pushing with 70 N [W] on skater 1.
(b) Choose east as positive. So west is negative.
(c) Skater 1 will accelerate west and skater 2 will accelerate east.
(d) For each skater, the normal force and the force of gravity cancel. This means that the applied force is equal to the net force. For skater 1,
FS
net 5 FS
a
m1a1 5 270 N
150 kg2a1 5 270 N
a1 5 21.4 m/s2
The acceleration of skater 1 is 1.4 m/s2 [W].
Similarly, for skater 2,
FS
net 5 FS
a
m2a2 5 170 N
170 kg2a2 5 70 N
a2 5 11.0 m/s2
The acceleration of skater 2 is 1.0 m/s2 [E].
Notice that the two skaters accelerate in opposite directions and with different accelerations. The accelerations do not have the same magnitude because the skaters’ masses are different.
1. Given the action force, describe the reaction force for each situation. K/U C
(a) You push forward on a book with 5.2 N.(b) A boat exerts a force of 450 N [W] on the water.(c) A hockey player hits the boards with a force of 180 N
[toward the boards].
2. Nobel and Maaham are wearing inline skates. Nobel has a mass of 62 kg and pushes on Maaham, whose mass is 54 kg. Maaham accelerates at 1.2 m/s2 [left]. Assume that no friction acts on either person. T/I
(a) Determine the force that Nobel exerts on Maaham. [ans: 65 N [left]]
3. A horse is tethered to a cart to pull it forward. A student claims, “According to Newton’s third law, when the horse pulls forward on the cart, the cart will also pull backwards on the horse. The two objects are attached together, so they cannot accelerate.” Discuss the validity of this statement. K/U C
4. A student stands on a skateboard and pushes on a wall with a force of 87 N [S]. The total mass of the student and the skateboard is 58 kg. K/U T/I
(a) Calculate the acceleration of the student. [ans: 1.5 m/s2 [N]]
(b) Explain why the wall does not move.
Practice
3.4 Summary
• Newton’sthirdlawalwaysinvolvessituationswheretwoobjectsexertforceson each other.
• Newton’sthirdlawstatesthatforeveryactionforcethereisasimultaneousreaction force equal in magnitude but opposite in direction.
• WhenapplyingNewton’sthirdlaw,theactionandreactionforceswillappearonseparate FBDs. Since they are on separate FBDs, they are not added together.
UNiT TASK BOOKMARK
You can apply what you have learned about Newton’s third law to the Unit Task on page 204.
1. Given each action force, state the reaction force. K/U
(a) A tire pushes on the road with a force of 240 N [backwards].
(b) You pull on a desk with a force of 25 N [N].
2. Explain each event below in terms of Newton’s third law. K/U
(a) A squid moves through the water by taking in water and expelling it.
(b) Walking on a wagon is dangerous.(c) A helicopter can hover above the ground.
3. During a space walk, an astronaut notices that her tether is not attached and she is drifting away from the space station. Explain each statement below using Newton’s third law. K/U C
(a) She can pull herself back to the space station if she can reach the space station and pull on it.
(b) She cannot push herself back to the station by pulling forward on her space suit.
(c) She can get back to the space station even if she cannot reach it if she is holding a tool.
4. Figure 6 shows a photograph of an early naval cannon tied to a ship. Explain why the ropes are necessary. A
5. Figure 7 shows a fan cart. K/U
(a) Explain why the fan cart cannot accelerate.(b) Explain why the fan cart can accelerate if the rigid sail
is removed.
6. A toy car (200 g) moves by shooting a plastic ball (50 g) horizontally out the back. The average acceleration of the car is 1.2 m/s2 [E] and there is negligible friction acting on each part of the toy. T/I C
(a) Draw an FBD for each object.(b) Identify the action and reaction forces on each part
of the toy.
7. Two fi gure skaters are moving east together during a performance. Skater 1 (78 kg) is behind skater 2 (56 kg) when skater 2 pushes on skater 1 with a force of 64 N [W]. Assume that no friction acts on either skater. T/I
(a) Determine the acceleration of each skater.(b) What will happen to the motion of each skater?
Explain your reasoning.
8. A milk carton fi lled with water is hanging from a string (Figure 8). What will happen if you punch two holes in opposite sides of the carton at the opposite corners? Explain your reasoning. T/I
Figure 8
9. A male astronaut (82 kg) and a female astronaut (64 kg) are fl oating side by side in space. T/I
(a) Determine the acceleration of each astronaut if the woman pushes on the man with a force of 16 N [left].
(b) How will your answers change if the man pushes with 16 N [right] on the woman instead?
(c) How will your answers change if they both reach out and push on each other’s shoulders with a force of 16 N?
using newton’s lawsNewton’s three laws of motion apply to many diff erent situations everywhere around you. Th ey can be used to determine if a train locomotive can make it through a mountain pass while pulling many cars behind it. Th e laws can also be used to design devices such as prosthetic limbs and bridges (Figure 1). To understand how to design these types of devices, you need to know how to apply all three of Newton’s laws, and you need to know how they are related to each other. For example, forces act on a prosthetic leg to accelerate it so a person can walk. Th e tension in suspension bridge cables helps to hold up the roadway against the force of gravity.
Figure 1 Newton’s three laws of motion are used to design devices such as (a) prosthetic limbs and (b) bridges. (a) (b)
In this section, you will practise using all three laws together to solve more com-plex and interesting problems involving forces and motion. In addition, you will learn more about forces, such as tension, and how to use kinematics to understand motion problems more clearly.
Tension and newton’s LawsRecall that tension is a pulling force exerted by a device such as a rope or a string. In this course, you may assume that the ropes or strings are light and do not stretch. Th is means that you do not have to include the mass of the string when calculating the acceleration of two objects tied together. It also means that the tension is uniform throughout the rope or string and pulls with the same force at both ends. Keep in mind that ropes and strings can only pull. Th is means that tension always acts on the object directed toward the rope or the string.
For example, two masses are tied together with a string, and a horizontal applied force pulls m1 to the right (Figure 2). If the string exerts a tension of 20 N [left ] on m1, then it also exerts a tension of 20 N [right] on m2. Th is is a direct consequence of Newton’s third law and the fact that the string does not stretch and has negligible mass.
How can we measure tension directly? One way is with a force sensor or a spring scale. If you tie opposite ends of a string to two separate force sensors and then pull on the string in opposite directions, you can easily see that the readings on both sensors are the same (Figure 3). Again, this is a direct consequence of Newton’s third law and our assumptions about strings. A single force sensor or spring scale can be used to measure the tension in strings if it is tied between two strings (Figure 4).
3.5
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Figure 2 Two objects tied together with a string. The tension force at both ends of the string is the same.
m2 m1FT FT Fa
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Figure 3 Two force sensors tied together with a string. The readings on the sensors will be equal according to Newton’s third law.
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Figure 4 Force sensors can be used to measure tension in strings.
force sensor
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Keep in mind that when two objects are tied together with a single string, if you drag one object forward, the other will also move forward once the string becomes taut. Th is means that the two objects will move with the same acceleration when they are pulled in a straight line. In this case, tension is an internal force and can be ignored if you are calculating the acceleration of both objects.
Since the tension is the same throughout the string and the tension pulls in opposite directions at the ends, we oft en just calculate the magnitude of the tension and refer to the FBD for the direction. Th e following Tutorial will help you to practise solving problems that involve tension and Newton’s laws.
Tutorial 1 Solving Tension ProblemsTo solve the following Sample Problems, we will use Newton’s laws and our knowledge of tension. Each problem will help clarify some of the concepts about tension.
Sample Problem 1: Objects Hanging from StringsEach object in Figure 5 has a force of gravity of 120 N [down] acting on it. Determine the tension in each string.
Solution
(a) First draw the FBD of the object. Choose up as positive.
The object is at rest, so it is not accelerating. Newton’s fi rst law implies that the net force must be zero.
F>net 5 F
>T 1 F
>g
Fnet 5 FT 1 12120 N2 0 5 FT 2 120 N
FT 5 1120 N
The tension in the string is 120 N [up].
(b) In this diagram, the force of gravity has not changed and the object is at rest. This means that the FBD is the same and we will fi nd the same tension. This example reinforces the concept that pulleys only change the direction of force without changing the magnitude of the force.
(c) In this balanced system, both objects are at rest. By drawing an FBD for either object, you will get exactly the same result for the tension. This result is contrary to what most people would expect. Most people would incorrectly say the string is holding up twice as much mass and should have twice the tension. Others incorrectly think the tension is zero since both forces of gravity pull the string at each end and they should cancel. Neither statement is true. The second object is just providing the force necessary to hold up the fi rst object. In other words, the second object is just doing the job of the wall or beam, but otherwise the situation is unchanged.
Three sleds are tied together and pulled east across an icy surface with an applied force of 120 N [E] (Figure 6). The mass of sled 1 is 12.0 kg, the mass of sled 2 is 11.0 kg, and the mass of sled 3 is 7.0 kg. You may assume that no friction acts on the sleds.
(a) Determine the acceleration of the sleds.
(b) Calculate the magnitude of the tension in rope A.
(c) Calculate the magnitude of the tension in rope B.
Solution
(a) All three sleds will move together with the same acceleration, so we can treat them as one single object. The total mass of all three sleds is mT = 12.0 kg + 11.0 kg + 7.0 kg = 30.0 kg. There is no need to consider the tension at this point because it is an internal force and will not contribute to the acceleration of the total mass. Choose east as positive. So west is negative.
FS
net 5 mT aS
1 120 N 5 130.0 kg2a a 5
1120 N30.0 kg
a 5 14.0 m/s2
The acceleration of all three sleds is 4.0 m/s2 [E].
(b) We could use the FBD for either sled 2 or sled 3 to calculate the tension in rope A. We will use the FBD for sled 3 because it is slightly simpler—it only has rope A pulling on it, whereas sled 2 has both rope A and rope B pulling on it.
FS
net 5 FS
TA
m3 aS
5 FS
TA
17.0 kg2 114.0 m/s22 5 F TA
FTA 5 128 N
The magnitude of the tension in rope A is 28 N.
(c) To calculate the tension in rope B, we can use the FBD of sled 2, but our answer will depend on the accuracy of the calculated tension in rope A. To avoid this problem, you can either use the FBD of sled 1 or the FBD of sleds 2 and 3 (Figure 7). We will choose the latter since it is slightly simpler. In this calculation, we will use a mass of m2 + m3 = 18.0 kg.
Figure 7 (a) FBD of sled 1 (b) FBD of sleds 2 and 3
FS
net 5 FS
TB
1m2 1 m32 aS
5 FS
TB
118.0 kg2 114.0 m/s22 5 F TB
FTB 5 172 N
The magnitude of the tension in rope B is 72 N.
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120 N [E]sled 1sled 2sled 3rope A rope B
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Practice 1. Examine each diagram in Figure 8. In each situation,
which rope will have the greater tension? Explain your reasoning. K/U
2. A locomotive with a mass of 6.4 3 105 kg is accelerating at 0.12 m/s2 [W] while pulling a train car with a mass of 5.0 3 105 kg. Assume that negligible friction is acting on the train. T/I
(a) Calculate the net force on the entire train. [ans: 1.4 3105 N [W]]
(b) Determine the magnitude of the tension between the locomotive and the train car. [ans: 6.0 3104 N] Figure 8
rope A rope BFa
(b)
Sample Problem 2: Objects Connected Horizontally by Strings
Kinematics and newton’s LawsWhen using any of the kinematics equations from Unit 1, it is essential that the acceleration remain constant. Now we can extend this restriction by stating that the net force on an object must also remain constant if you use one of the kinematics equations. This is a direct consequence of Newton’s second law, F
S
net 5 m aS, which shows that net force is constant when the acceleration is constant.
Imagine that you start moving east due to a constant net force acting on you. You might start walking, but you gradually speed up and start running. Then your net force drops to zero and you move at a constant velocity. Finally, another constant net force acts in the opposite direction and slows you down until you eventually come to rest.
During the three separate parts of the trip, your acceleration was constant because the net force was constant. This means that you can use a kinematics equation during one part of the trip but not a single equation for the entire trip. The following Tutorial will clarify how to use kinematics concepts with Newton’s laws.
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Tutorial 2 Newton’s Laws and Kinematics
When solving the following Sample Problems, keep in mind that the acceleration is constant when the net force is constant, according to Newton’s second law.
Sample Problem 1: Skater Pushing on the Boards
Starting from rest, an ice skater (54.0 kg) pushes the boards with a force of 130.0 N [W] and moves 0.704 m. He then moves at a constant velocity for 4.00 s before he digs in his skates and starts to slow down. When he digs in his skates, he causes a net force of 38.0 N [W] to slow him down until he stops.
(a) Determine the acceleration of the skater (i) when he is pushing on the boards (ii) just after he stops pushing on the boards (iii) when he starts to slow down (b) How far does he move?
SolutionUse FBDs to solve this problem.
(a) (i) When the skater pushes the boards with a force of 130.0 N [W], the boards push back on the skater with an equal and opposite force of 130.0 N [E]. Choose east as positive. So west is negative.
F>net 5 ma
>
1130.0 N 5 154.0 kg2a
a 51130.0 N
54.0 kg a 5 2.407 m/s2 1one extra digit carried2
The acceleration of the skater is 2.41 m/s2 [E] when he is pushing on the boards.
(ii) When he stops pushing on the boards, the net force acting on him is zero. According to Newton’s first law, his acceleration will also be zero.
(iii) Now the skater is slowing down and the net force is opposite to the direction of motion.
F>net 5 ma
>
238.0 N 5 154.0 kg2a
a 5238.0 N54.0 kg
a 5 20.704 m/s2
The acceleration of the skater is 0.704 m/s2 [W] when he is slowing down.
(b) During part (i), the skater moves 0.704 m [E], but we need to calculate his final velocity in order to calculate his displacement during the other sections of the motion.
Given: vS
1 5 0; DdS
5 0.704 m [E]; aS
5 2.407 m/s2 [E]
Required: vS
2
Analysis: v 22 5 v 2
1 1 2aDdSolution: v 2
2 5 v 21 1 2aDd
v 22 5 102 2 1 2 112.407 m/s22 110.704 m2
v2 5 1.841 m/s (one extra digit carried)
Now we can calculate the displacement for the other sections of the motion. For part (ii),
D dS
5 vS
Dt 5 (11.841 m/s)(4.00 s)
Dd 5 7.364 m (one extra digit carried)
For part (iii),
v 22 5 v
21 1 2aDd
02 5 (11.841 m/s)2 1 2(20.704 m/s2)Dd Dd 5 1.981 m (one extra digit carried)Statement: The total distance travelled by the skater is 0.704 m 1 7.364 m 1 1.981 m 5 10.1 m
A worker pushes two large boxes across the fl oor from rest with an applied force of 160.0 N [right] on the larger box (Figure 9). The boxes are touching. The mass of the larger box is m1 5 32.0 kg and the mass of the smaller box is m2 5 8.0 kg. The force of friction on the large box is 80.0 N [left] and the force of friction on the smaller box is 20.0 N [left].
(a) Calculate the acceleration of the two boxes. Assume that the boxes start to move.
(b) Determine the force exerted by the larger box on the smaller box. (c) Determine the velocity of the boxes after 4.0 s.
(a) Both boxes must move together with the same acceleration, so for now we will treat them like one single object with a total mass of mT 5 32.0 kg 1 8.0 kg 5 40.0 kg. From the FBD for both boxes, the normal force and gravity cancel. Choose right as positive. So left is negative.
FS
net 5 m T aS
1160.0 N 1 1280.0 N2 1 1220.0 N2 5 m T a
160.0 N 5 140.0 kg2a
a 5160.0 N40.0 kg
5 11.5 m/s2
The acceleration of both boxes is 1.5 m/s2 [right].
(b) The force from the large box on the smaller box is an internal force. To calculate this force, we need to draw the FBD for just one box. Either box will do but we will use the smaller one because it has fewer forces acting on it. Again, the normal force and the force of gravity cancel.
Fnet 5 F 1 on 2 1 1220.0 N2 m
2a 5 F 1 on 2 2 20.0 N
18.0 kg2 111.5 m/s22 5 F1 on 2 220.0 N
F 1 on 2 5 132.0 N
The force exerted by the larger box on the smaller box is 32.0 N [right].
(c) Given: vS
1 5 0; ∆t 5 4.0 s; aS
5 1.5 m/s2 [right]
Required: vS
2
Analysis: vS
2 5 vS
1 1 aS
Dt
Solution: vS
2 5 vS
1 1 aS
Dt
5 0 1 111.5 m/s22 14.0 s2 v
2 5 16.0 m/s
Statement: The fi nal velocity of both boxes is 6.0 m/s [right].
m1m2
160.0 N
Science Physics 11
ISBN: ISBN # 0176504338FNCO Dave McKay Illustration
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Practice 1. Two dynamics carts are placed end to end. Cart 1 (1.2 kg) is stuck to cart 2 (1.8 kg).
Cart 1 is pushed with a force of 18.9 N [W], causing cart 1 to push cart 2 forward. Ignore the force of friction. T/I
(a) Calculate the acceleration of each cart. [ans: 6.3 m/s2 [W]]
(b) Calculate the force that cart 1 exerts on cart 2. [ans: 11 N [W]]
(c) Would your answers change if cart 2 were pushed with an equal but opposite force instead of cart 1? If your answers change, calculate the new results. [ans: 6.3 m/s2 [E]; 7.6 N [W]]
2. A 1200 kg car is moving at 95 km/h when the driver notices a deer down the road. He immediately moves his foot toward the brake pedal, taking only 0.50 s before the car starts slowing down. The brakes cause a net force of 2400 N [backwards] on the car for 2.0 s. The deer then jumps out of the way and the driver lifts his foot off the brake pedal. How far does the car move in the 2.5 s starting from when the driver sees the deer? T/I [ans: 62 m]
1. You hold one end of a rope and pull horizontally with a force of 65 N. Calculate the tension in the rope if the other end is(a) tied to a wall(b) held by a friend who pulls with 65 N in the opposite
direction(c) tied to a 12 kg object on smooth ice T/I
2. A 72 kg sled is pulled forward from rest by a snowmobile and accelerates at 2.0 m/s2 [forward] for 5.0 s. The force of friction acting on the sled is 120 N [backwards]. The total mass of the snowmobile and driver is 450 kg. The drag force acting on the snowmobile is 540 N [backwards]. T/I
(a) Determine the tension in the rope.(b) Calculate the force exerted by the snowmobile that
pushes the sled forward.
3. Two people, each with a mass of 70 kg, are wearing inline skates and are holding opposite ends of a 15 m rope. One person pulls forward on the rope by moving hand over hand and gradually reeling in more of the rope. In doing so, he exerts a force of 35 N [backwards] on the rope. This causes him to accelerate toward the other person. Assuming that the friction acting on the skaters is negligible, how long will it take for them to meet? Explain your reasoning. T/I
4. A 1200 kg car pulls an 820 kg trailer over a rough road. The force of friction acting on the trailer is 650 N [backwards]. Calculate the force that the car exerts on the trailer if(a) the trailer is moving at a constant velocity of
30 km/h [forward](b) the trailer is moving at a constant velocity of
60 km/h [forward](c) the trailer is moving forward at 60 km/h and starts
accelerating at 1.5 m/s2 [forward](d) the trailer is moving forward at 60 km/h and starts
accelerating at 1.2 m/s2 [backwards] T/I
5. An old rope can now only safely suspend 120 kg. When the rope is tied to a beam, it hangs down with a vertical length of 12.0 m. Calculate the minimum time required for an 85 kg person starting from rest to climb the entire length of the rope without breaking it. T/I
6. Three dynamics carts have force sensors placed on top of them. Each force sensor is tied to a string that connects all three carts together (Figure 10). You use a sixth force sensor to pull the three dynamics carts forward. The reading on force sensor 2 is 3.3 N. Assume that the force sensors are light and that there is negligible friction acting on the carts. T/I
Figure 10
(a) What is the acceleration of all the carts?(b) What is the reading on each force sensor?(c) What force are you applying to force sensor 6?
7. A locomotive (6.4 × 105 kg) is used to pull two railway cars (Figure 11). Railway car 1 (5.0 × 105 kg) is attached to railway car 2 (3.6 × 105 kg) by a locking mechanism. A railway engineer tests the mechanism and estimates that it can only withstand 2.0 × 105 N of force. Determine the maximum acceleration of the train that does not break the locking mechanism. Explain your reasoning. Assume that friction is negligible. T/I C
Figure 11
8. A skier (68 kg) starts from rest but then begins to move downhill with a net force of 92 N for 8.2 s. The hill levels out for 3.5 s. On this part of the hill, the net force on the skier is 22 N [backwards]. T/I
(a) Calculate the speed of the skier after 8.2 s.(b) Calculate the speed of the skier at the end of the
section where the hill levels out.(c) Calculate the total distance travelled by the skier before
On July 5, 1687, Newton (Figure 1) first published his landmark book Principia Mathematica. This work consisted of three volumes and included many mathematical discov-eries, as well as his famous laws of motion. He also used some of Galileo’s earlier work to help formulate some of his own ideas, such as the concept of inertia.
The primary focus of Principia Mathematica is on mas-sive objects in motion, such as planets in our solar system. Newton realized that the rules governing the motion of large objects in the universe could be used to study how other fundamental forces act on objects.
In Newton’s time, the other fundamental forces were either not well understood (such as electromagnetic force) or completely unknown (such as the strong and weak nuclear forces). Yet in Principia Mathematica, Newton pre-dicted that the laws of physics would somehow be able to account for all physical phenomena.
newton and the Foundations of PhysicsToday we know that not only was Newton on the right track with his prediction, but his discoveries have had
impacts on fields of study and technology that no one could have predicted in Newton’s time. His laws of motion and his ideas about gravity are used to construct buildings and bridges, design cars and planes, and launch satellites into orbit.
Newton’s laws of motion are also the foundation of physics wherever forces are a factor. Albert Einstein wrote, “To the Master’s honour all must turn, each in its track, without a sound, forever tracing Newton’s ground.” This is an eloquent way of stating that Newton’s laws of motion are so all-inclusive that any other endeavour in physics can be traced back in some way to Newton’s work.
Einstein: Extending newton’s IdeasEinstein’s work on relativity followed a path very similar to Newton’s laws. Einstein’s special theory of relativity extended Newton’s first law to deal with the special case of objects moving at a very fast, constant velocity. Einstein’s theory of general relativity extended Newton’s second and third laws to deal with accelerating objects and gravity.
Newton’s third law deals with action and reaction forces. The most obvious example of these is contact forces. If you push on an object, it will push back on you with an equal but opposite force. Newton also stated that this was true of gravity, which is an action-at-a-distance force. For example, when you jump up into the air, the force of gravity exerted by Earth pulls you back down. There is also a reaction force on Earth exerted by you, pulling up on Earth.
What about the other action-at-a-distance forces? Does Newton’s third law still apply to those forces? For example, when a positive charge exerts a force on another charge, is there an equal and opposite reaction force? What about magnetism? When a north pole of one magnet exerts a force on a north pole of another magnet, is there also a reac-tion force? This is what Newton was thinking about in the preface of Principia Mathematica. The answer to all of these questions is yes. Newton’s third law works for any action-at-a-distance force, as well as any contact force. This makes the third law even more general and all-encompassing than he had imagined.
Galileo, newton, and HawkingABSTRACTIn his book Principia Mathematica, Newton realized that the laws governing the motion of planets could be applied to all physical phenomena. Newton’s predictions have had an impact on many fields of science and technology. Einstein’s theory of relativity and Stephen Hawking’s work on the nature of gravity and black holes are grounded in Newton’s laws. Newton’s work in Principia Mathematica may be the greatest achievement in physics.
Stephen Hawking: Standing on the Shoulders of GiantsPhysicist Stephen Hawking has continued with the work started by Galileo, Newton, and Einstein (Figure 2) in his study of the universe. His work deals mainly with extending the concepts explored by Newton and Einstein to topics such as black holes and the nature of gravity. One of his most widely known accomplishments is the idea that even black holes can lose mass through Hawking radiation (Figure 3).
To put the contributions and accomplishments of Newton into perspective, Hawking wrote,
Einstein is the only figure in the physical sciences with a stature that can be compared with Newton. Newton is reported to have said: “If I have seen further than other men, it is because I stood on the shoulders of giants.” This remark is even more true of Einstein who stood on the shoulders of Newton. Both Newton and Einstein put forward a theory of mechanics and a theory of gravity but Einstein was able to base General Relativity on the mathematical theory of curved spaces that had been constructed by Riemann while Newton had to develop his own mathematical machinery. It is therefore appropriate to acclaim Newton as the greatest figure in mathematical physics and the Principia is his greatest achievement.
Further ReadingGreene, B. (1999). The Elegant Universe. New York: W. W.
Norton & Company. Hawking, S. (1988). A Brief History of Time. New York:
Bantam Dell.Hawking, S. (2001). The Universe in a Nutshell. New York:
Bantam Books. Hawking, S., & Israel, W. (Eds.). (1987). 300 Years of
Gravitation. New York: Cambridge.
Figure 2 (a) Albert Einstein (b) Stephen Hawking
(a) (b)
Figure 3 Hawking radiation from a black hole. Until Hawking showed otherwise, people thought that nothing could escape a black hole.
3.6 Questions
1. Explain the meaning of the statement, “Great physicists are often influenced by the great physicists of the past.” K/U C
2. Describe, using an example, how Newton’s laws of motion have affected your own life. C A
3. Explain why Hawking places Newton ahead of Einstein as the greatest figure in mathematical physics. K/U C
4. Research some of the other topics and concepts covered in Principia Mathematica not outlined in this section. For one of these topics, write a short report on its significance and applications. T/I C A
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NEL 3.6 Physics Journal: Galileo, Newton, and Hawking 149
In this investigation, you will perform two controlled experiments. In Part A, you will measure the acceleration of a cart using diff erent net forces while keeping the total mass constant. In Part B, you will measure the acceleration of a cart using diff erent total masses while keeping the net force constant.
force acting on the cart if the total mass is constant?• Howdoestheaccelerationofacartdependonthe
total mass if the net force is constant?
Hypothesis/Prediction SKILLSHANDBOOK A2.2
Aft er reading through the experiment, write a hypothesis to answer each Testable Question.
variablesIdentify the independent (sometimes called manipulated) and dependent (sometimes called responding) variables in this experiment. Describe how you will measure these variables. What variables must be controlled?
Experimental DesignTh ere are many diff erent ways to perform this activity. Figure 1 shows one simple way to apply a constant force to a cart.
Figure 1
Keep in mind that the force of gravity acting on the hanging object is the net force on the total mass. Th e total mass includes the hanging object, the cart, the string, and any objects on top of the cart. If you want to change the force and keep the total mass constant, take an object off the cart and hang it from the string. If you want to change the mass and keep the net force constant, change the objects on top of the cart but do not change the hanging objects.
Discuss with your group members how you will safely stop the cart before performing the investigation.
As you plan to carry out the Unit Task (page XXX), consider what you have learned about synthesis and decomposition reactions in this section.
2. Measure the mass of the cart. 3. Set up the equipment as shown in Figure 1 or as
directed by your teacher. Make sure everything is working properly by allowing the cart to accelerate once or twice.
4. Put one 1.0 kg object and two 100 g objects on top of the cart. You might need to tape them down. Hang a 100 g object from the string. Allow the motion to occur and obtain the data required to find the acceleration ( aS1).
5. Take one of the 100 g objects from on top of the cart and hang it from the string. Allow the motion to occur and obtain the data required to find the new acceleration ( aS2).
6. Take the last 100 g object from the top of the cart and hang it from the string. Allow the motion to occur and obtain the data required to find the new acceleration ( aS3).
Part B: Acceleration and Mass 7. Use the data for aS3 from Step 6 as the first set of data
for this experiment. 8. Using the same cart setup as in Step 6, add one
1.0 kg object to the cart. Allow the motion to occur and obtain the data required to find the new acceleration ( aS4).
9. Add an additional 1.0 kg object to the cart. Allow the motion to occur and obtain the data required to find the new acceleration ( a
S
5).
Analyze and Evaluate(a) In terms of the variables in this investigation, what
type of relationship was being tested? T/I
(b) Calculate the acceleration for each trial. T/I
(c) Calculate the ratio of the net force to the total mass. What does this ratio represent? Explain your reasoning. K/U T/I
(d) Use your results from Part A to plot a graph of net force (y-axis) versus acceleration (x-axis). Draw a line of best fit and calculate its slope. What does this graph indicate about the relationship between acceleration and net force? What does the slope represent? T/I C
(e) Use your results from Part B to plot a graph of acceleration (y-axis) versus total mass (x-axis). Draw a smooth curve through the points. What does this graph indicate about the relationship between acceleration and total mass? T/I C
(f) Use your results from Part B to plot a graph of acceleration (y-axis) versus the reciprocal of the total mass (1/m; x-axis). Draw a line of best fit and calculate its slope. What does this graph indicate about the relationship between acceleration and total mass? What does the slope represent? T/I C
(g) Answer the Testable Questions. T/I
(h) Comment on the accuracy of your hypothesis. C (i) List some possible sources of error. How could you
modify the investigation to avoid or reduce these sources of error? T/I
Apply and Extend(j) Describe how you could determine if friction had any
effect on the results of this investigation. T/I
(k) Explain why a graph of net force versus acceleration must pass through the origin when the total mass is constant. T/I
(l) Explain why a graph of acceleration versus the reciprocal of the total mass must pass through the origin when net force is constant. T/I
(m) Commercial airlines are limiting the number of pieces and the mass of luggage that passengers can bring onto an aircraft. Use what you have learned in this investigation to explain why. C A
1. Create a study guide for this chapter based on the Key Concepts listed in the margin on page 112. For each point, create three or four subpoints that provide further information, relevant examples, explanatory diagrams, or general equations.
2. Look back at the Starting Points questions on page 112. Answer these questions using what you have learned in this chapter. Compare your latest answers with those that you wrote at the beginning
of the chapter. Note how your answers have changed.
3. You may use several diff erent forms of active transportation, such as bicycles, skateboards, and inline skates. Using the concepts of this chapter, analyze the safety issues of one of these forms of transportation. Make sure you use all of Newton’s laws in your explanations. Also provide some safety tips that will help reduce the chance of serious injury.
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SKILLSHANDBOOK A7
chaPTer 3 SUMMARY
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Grade 11 Physics can lead to a wide range of careers. Some require a college diploma or a bachelor of science degree. Others require specialized or post-graduate degrees. This graphic organizer shows a few pathways to careers related to topics covered in this chapter. 1. Select an interesting career that relates to forces and Newton’s laws of motion.
Research the educational pathway you would need to follow to pursue this career. 2. What is involved in the career you have chosen? Research at least two programs,
and design a poster or pamphlet to display your fi ndings.
3. Which object, when stationary, has the least inertia? (3.2) K/U
(a) a desk(b) a pen(c) a computer(d) a feather
4. A 60.0 kg object accelerates with a magnitude of 2.0 m/s2. What is the magnitude of the net force acting on this object? (3.3) T/I
(a) 30 N(b) 60 N(c) 120 N(d) 180 N
5. What is the mass of a rock that experiences a gravitational force of magnitude 11.8 N? (3.1) T/I
(a) 1.1 kg(b) 1.3 kg(c) 1.2 kg(d) 1.0 kg
6. Which statement best describes Newton’s third law? (3.4) K/U
(a) For every action force, there is a reaction force of lesser magnitude and in the same direction.
(b) For every action force, there is a reaction force of equal magnitude but opposite in direction.
(c) For every action force, there is a reaction force of equal magnitude and in the same direction.
(d) For every action force, there is a reaction force of lesser magnitude but in the opposite direction.
7. An ice skater pushes off a railing. Which of the following is the reaction force? (3.4) K/U
(a) the force of the skater pushing off the railing(b) the force of the railing pushing back on the skater(c) the force of friction from the ice(d) the force of gravity
8. What important realization did Newton describe in Principia Mathematica? (3.6) K/U
(a) Forces are governed by the theory of relativity.(b) The rules governing the motion of large objects
in the universe can be used to study how fundamental forces act on objects.
(c) Black holes lose mass through a special type of radiation.
(d) Apples fall from trees due to gravitational force.
Indicate whether each statement is true or false. If you think the statement is false, rewrite it to make it true.
9. Forces are measured in newtons. (3.1) K/U
10. The normal force is the force resisting the motion or attempted motion of an object. (3.1) K/U
11. A free-body diagram is a simple sketch of all the objects involved in a situation. (3.1) K/U
12. All forces require objects to be in contact and cannot act at a distance. (3.1) K/U
13. The net force on an object is the sum of all the forces acting on it. (3.1) K/U
14. Massive objects have more inertia than less massive objects. (3.2) K/U
15. An object can change its motion even if the net force acting on it is zero. (3.2) K/U
16. If the net force on an object is constant and the mass decreases, then the acceleration increases. (3.3) K/U
17. Newton’s third law states that for every action force, there is a simultaneous reaction force of equal magnitude acting in the same direction. (3.4) K/U
5. You have your seat belt on in the passenger seat of a car. The car suddenly accelerates. Which statement best describes your motion? (3.2) K/U
(a) Your body will suddenly move forward with respect to the seat.
(b) Your head will suddenly move forward with respect to your body.
(c) Both your body and your head will move forward at the same rate.
(d) Your head and your body will suddenly move backwards with respect to the seat.
6. Which of the following correctly states Newton’s second law? (3.3) K/U
(a) F>net 5
a>
m
(b) mF>net 5 a
>
(c) F>net 5 ma
>
(d) F>net , ma
>
7. A boy jumps off a skateboard. What is the reaction force? (3.4) K/U
(a) the force exerted by the boy pushing off the skateboard
(b) the force exerted by the skateboard pushing back on the boy
(c) the force of friction exerted by the road on the skateboard
(d) the force of gravity causing the boy to land on the ground
8. Which device could be used to measure tension? (3.5) K/U
(a) a bathroom scale(b) a balance(c) a spring scale(d) a sliding scale
Indicate whether each statement is true or false. If you think the statement is false, rewrite it to make it true. 9. One newton is equal to 1 kg·m/s. (3.1) K/U
10. A tension force is a perpendicular force acting on an object that is exerted by the surface with which it is in contact. (3.1) K/U
11. Friction is a force that resists the motion or attempted motion of an object. (3.1) K/U
12. Forces that do not require contact are called action-at-a-distance forces. (3.1) K/U
13. To determine the net force, you do not need to consider the direction of each force acting on an object. (3.1) K/U
14. An object with less mass has more inertia. (3.2) K/U
15. If there is a net force acting on an object, then the motion of the object will change. (3.2) K/U
16. If the net force on an object is constant and the mass increases, then the acceleration decreases. (3.3) K/U
17. Newton’s third law states that for every action force there is a simultaneous reaction force of lesser magnitude in the opposite direction. (3.4) K/U
Match each term on the left with the most appropriate description on the right.
18. (a) weak nuclear (i) a pulling force from a rope or a string
(b) normal (ii) a force that resists the motion or attempted motion of an object
(c) friction (iii) the force responsible for interactions during particle transformations
(d) tension (iv) the force of attraction between two objects because of their mass
(e) gravity (v) a perpendicular force exerted by a surface on an object (3.1) K/U
Write a short answer to each question. 19. What is the rate of acceleration due to gravity? (3.1) K/U
20. In your own words, describe what it means for an object to have inertia. (3.2) K/U C
21. In your own words, describe Newton’s third law and give an example that demonstrates this law. (3.4) K/U C
Understanding 22. A book sits on a desk. Draw a free-body diagram of
all the forces acting on the book. (3.1) K/U
23. Th e free-body diagram in Figure 1 represents a block being dragged at constant velocity across a rough surface. Copy Figure 1 into your notebook and draw in the missing force vector. Be sure to indicate its direction and magnitude relative to the other forces shown. (3.1) K/U
Figure 1
24. Th e free-body diagram in Figure 2 represents a block sitting motionless on a ramp. Is the FBD complete? Explain. Include an FBD in your answer. (3.1) T/I C
Figure 2
25. Draw a free-body diagram for the fi rst two situations. Th en answer (c). (3.1) T/I C
(a) A student pulls a large box across a smooth fl oor to the right.
(b) A student pushes a large box across a smooth fl oor to the right.
(c) Are there any diff erences between your diagrams for (a) and (b)? Explain.
26. Th e engines of a plane exert a horizontal force of magnitude 37 850 N while the plane is fl ying northward. A wind blowing to the south also exerts a horizontal force of magnitude 850 N on the plane. What is the net horizontal force on the plane? (3.1) T/I
27. Th e engines of a plane fl ying westward exert a horizontal force of magnitude 35 000 N. A wind blowing westward exerts a horizontal force of magnitude 1200 N. What is the net horizontal force on the plane? (3.1) T/I
28. A trap door that opens in the middle is stuck half open with a box sitting on it (Figure 3). Draw a free-body diagram for the box. (3.1) T/I
29. Use Newton’s fi rst law to explain why the normal force must be equal in magnitude to the force of gravity for an object to remain at rest on a horizontal surface when no other forces are acting on the object. (3.2) K/U C
30. Four children are playing tug-of-war. Th e two children on the right pull with forces of magnitude 84 N and 86 N, and one of the children on the left is able to pull with a force of magnitude 83 N. If the rope remains stationary, how hard is the second child on the left pulling? (3.2) T/I
31. A student is trying to push a box across the fl oor with 20 N of force, but the box does not move. What is the magnitude of the frictional force exerted by the ground on the box? (3.2) T/I
32. Answer each question below using Newton’s second law. Explain your answers. (3.3) K/U C
(a) Two cars have the same net force acting on them, but one has more mass than the other. Which car will have the greater acceleration?
(b) A heavy box of sand is on top of a cart. A person pulls the cart forward with a constant net force while the sand leaks out of the box and onto the ground. What will happen to the acceleration of the cart?
33. Calculate the net force in each situation. (3.3) T/I
(a) A 69 kg sprinter accelerates at 2.1 m/s2 [forward].(b) A 620 g basketball falls through the net.
34. Calculate the net acceleration in each situation. (3.3) T/I
(a) A 260 kg boat crew paddles north with a net force of magnitude 468 N.
(b) A 70.0 kg skydiver opens his parachute and experiences a net force of 236 N [up].
35. A student pushes a 10 kg box and lets it slide across the fl oor. Th e magnitude of the frictional force acting on the box is 40 N. What is the acceleration of the box? (3.3) T/I
36. A 175 g hockey puck slows down at a rate of 1.5 m/s2 as it slides across the ice. Determine the frictional force acting on the puck. (3.3) T/I
37. Th e engines of an airplane exert a net force of magnitude 800 000 N during takeoff , causing the plane to accelerate at a rate of 8.0 m/s2 [forward]. What is the mass of the plane? (3.3) T/I
38. A batter hits a baseball with a mass of 145 g. Aft er impact, the ball experiences a net force of 1.80 3 103 N [S]. What is the acceleration of the ball? (3.3) T/I
39. Th e cart in Figure 4 has a mass of 2.3 kg and is attached to a 1.7 kg object. Calculate the acceleration of the cart given the following assumptions: (3.3) T/I
(a) Th e force of friction is negligible.(b) Th e frictional force acting on the wheels of the
cart has a magnitude of 0.6 N.
Figure 4
40. Assume the cart in Figure 4 has an acceleration of magnitude 2.5 m/s2 and a mass of 1.8 kg. Calculate the mass of the attached object given the following assumptions: (3.3) T/I
(a) Th e force of friction is negligible.(b) A frictional force of magnitude 0.4 N acts on the
wheels of the cart. 41. Explain each situation using Newton’s third law. (3.4) K/U
(a) A boat is able to fl oat on water.(b) A dolphin can jump out of the water by pushing
down on the water.(c) A raft glides to the left when a student jumps off
the raft to the right. 42. When a cannon fi res, it rolls backwards. Use your
knowledge of Newton’s laws to explain why this happens. (3.4) K/U C
43. A student stands on a skateboard and pushes on a wall with a force of magnitude 89 N. Th e total mass of the student and the skateboard is 58 kg. Assume that the force of friction is negligible. (3.4) T/I C
(a) Calculate the acceleration of the student.(b) Explain why the wall does not appear to move.
44. A girl is ice skating and pushes off a rail with a force of magnitude 75 N. Th e total mass of the girl and her skates is 62 kg. (3.4) T/I C
(a) Calculate the acceleration of the girl while she is pushing if the ice exerts a frictional force of magnitude 4.0 N on the skates.
(b) Explain why the rail does not appear to move. 45. A 3.0 kg object hangs from a string inside an elevator.
Calculate the tension in the string for the following situations. (3.5) T/I
(a) Th e elevator is stationary.(b) Th e elevator is accelerating at 1.2 m/s2 [up].(c) Th e elevator is accelerating at 1.4 m/s2 [down].
Analysis and Application 46. Wind is blowing a fl ag to the east. What other forces
are acting on the fl ag if it remains attached to a pole? Include an FBD of the fl ag in your answer. (3.1, 3.2) T/I
47. A skydiver has a mass of 71.5 kg. What is the force of gravity acting on him(a) when he jumps?(b) aft er he has landed? (3.1) T/I
48. A 45.0 kg girl is sitting on a bench. What is the magnitude of the force with which the bench pushes against the girl? (3.1, 3.4) K/U
49. A boulder rolls off the edge of a cliff and initially falls with a force of magnitude 1100 N. What is the mass of the boulder? (3.1) K/U
50. A student weighs out 7.6 N of water into a beaker. What is the mass of the water? (3.1) K/U
51. A T-shirt launcher is capable of shooting shirts at 35 m/s horizontally. If the time it takes to accelerate the shirts to this speed is 0.50 s and each shirt has a mass of 250 g, how much force does the launcher exert on the shirts? Assume that the friction in the tube is negligible. (3.3) T/I
52. A 72 kg runner is jogging at a rate of 6.0 m/s and then begins to sprint, reaching a full speed of 15 m/s aft er 3.0 s. What is the net force acting on the runner? (3.3) T/I
53. Two boxes of mass 30.0 kg and 10.0 kg are at rest side by side (Figure 5). You apply a force of 3.0 3 102 N on the fi rst box for 5.0 s, and they both slide across the fl oor. Th e larger box has a force of friction of 180 N, and the smaller box has a force of friction of 60.0 N. (3.3, 3.4) T/I C
Figure 5
(a) Calculate the acceleration of the boxes during the fi rst 5.0 s.
(b) If the applied force is removed, the boxes will slow down and stop. Explain why.
(c) Calculate the total distance travelled by the boxes. 54. Four students are playing tug-of-war. Th e two
students on the right pull with forces of magnitude 55 N and 65 N, and weigh 60 kg and 62 kg. Th e two students on the left pull with forces of magnitude 58 N and 70 N, and weigh 59 kg and 64 kg. What are the magnitude and direction of the acceleration of the students? (3.3) T/I
55. Th e box in Figure 6 has a mass of 85 kg. F>a = 10 N [up],
F>b 5 30 N [right], F
>c 5 25 N [right], F
>d = 10 N [down],
and F>e 5 22 N [left ]. (3.3) T/I
Figure 6
(a) What is the net force acting on the box in both the vertical and horizontal directions?
(b) What is the acceleration of the box? 56. Suppose the box in Figure 6 has a net acceleration
of magnitude 5.5 m/s2. F>a 5 13 N [up], F
>b 5 12 N
[right], F>c 5 19 N [right], F
>d 5 26 N [down], and
F>e 5 31 N [left ]. (3.3) T/I
(a) What is the net force acting on the box in both the vertical and horizontal directions? In which direction does the box move?
(b) What is the mass of the box? 57. Suppose the box in Figure 6 has a mass of 12 kg.
F>a 5 220 N [up], F
>b 5 82 N [right], F
>d 5 220 N
[down], and F>e 5 112 N [left ]. Th e magnitude of the
acceleration of the box is 1.5 m/s2. (3.3) T/I C
(a) If the box is moving to the right, what is the magnitude of F
>c?
(b) If the box is moving to the left , what is the magnitude of F
>c?
58. A dogsled team has four dogs that pull a person and a sled with a combined mass of 100 kg. (3.3) T/I (a) Th ey start from rest and reach a speed of 45 km/h
in 2.5 s. What is the average force applied by each dog?
(b) Suppose each dog can pull with a force of magnitude 150 N. What is the frictional force acting on the sled?
59. Th e Moon is much smaller than Earth, so the gravitational strength of the Moon is only about one-sixth that of Earth. (3.3) T/I A
(a) What is the acceleration due to gravity on the Moon?(b) How much would a 72 kg person weigh on the
Moon?(c) If an object falls with a force of 700 N [down] on
Earth, what would the force on the same object be if it fell on the Moon?
60. A girl is standing on a raft fl oating on the water and jumps off to the right with a force of magnitude 100 N. Th e mass of the girl is 55 kg, and the mass of the raft is 120 kg. (3.4) T/I C
(a) Describe the action and reaction forces in this situation.
(b) Determine the acceleration of both the girl and the raft .
61. A boy and a girl are standing on skateboards. Th e boy pushes off the girl to the left with a force of magnitude 74 N. Th e mass of the boy is 62 kg and the mass of the girl is 59 kg. Ignore friction. (3.4) T/I C
(a) State the action and reaction forces in this situation.(b) Determine the magnitude and direction of the
acceleration of each skateboarder. 62. Two ice skaters are playing a game. Skater A pushes off
skater B. Skater A has a mass of 75 kg and accelerates with a magnitude of 1.2 m/s2. Skater B accelerates with a magnitude of 0.80 m/s2. Ignore friction. (3.4) T/I C
(a) Based on their accelerations, which skater has more mass?
(b) State the action and reaction forces and calculate their values.
(c) Calculate the mass of skater B. 63. Students are performing an experiment about Newton’s
third law using skateboards. Student A has a mass of 58 kg and pushes off student B with a force of magnitude 80.0 N. Student B has a mass of 55 kg and has placed a block of unknown mass with him on his skateboard. (3.4) T/I C
(a) Calculate the acceleration of student A.(b) If the magnitude of the acceleration of student B
is greater than that of student A, what range of values could the mass of the block have? What if the magnitude of the acceleration of student B is less than that of student A?
(c) If student B accelerates with a magnitude of 1.25 m/s2, what is the mass of the block?
64. A male astronaut (82 kg) and a female astronaut (64 kg) are fl oating side by side in space. (3.4) T/I C
(a) Determine the acceleration of each if the woman pushes on the man with a force of 16 N [left ].
(b) How will your answers change if the man pushes with 16 N [right] on the woman instead?
65. A locomotive with a mass of 6.4 3 105 kg is accelerating with a magnitude of 0.12 m/s2 while pulling a train car of mass 5.3 3 105 kg. Assume that negligible friction is acting on the train. (3.5) T/I
(a) Calculate the net force on the entire train.(b) Determine the tension between the locomotive
and the train car.
66. Two objects are hung from strings (Figure 7). Th e top object (m1) has a mass of 18 kg and the bottom object (m2) has a mass of 12 kg. (3.5) T/I
(a) Calculate the tension in each string.(b) Calculate the tension in each string if you pull
down on m1 with a force of 45 N [down].(c) Calculate the tension in each string if you pull
down on m2 with a force of 45 N [down].(d) Compare your answers to parts (b) and (c) and
explain any diff erences.(e) If you keep increasing the downward force on
m2, which string will most likely break fi rst? Explain your reasoning by assuming that both strings are identical.
Figure 7
67. Th e blocks in Figure 8 are pulled to the right and accelerate with a magnitude of 1.1 m/s2. If m1 5 4.0 kg, m2 5 2.3 kg, and m3 5 3.4 kg, calculate the tension in each string. Assume that friction is negligible. (3.5) T/I
Figure 8
68. Suppose the blocks in Figure 8 are pulled to the right with a force of magnitude 15 N. If m1 5 4.3 kg, m2 5 5.5 kg, and m3 5 3.1 kg, calculate the tension in each string and the acceleration of the blocks. Ignore friction. (3.5) T/I
69. Suppose the blocks in Figure 8 are pulled to the right with a force of magnitude 24 N. Th e tension in string B is twice that of the tension in string A. If m1 5 10 kg and m3 5 8 kg, calculate the tension in each string and the mass of m2. Ignore friction. (3.5) T/I
70. A student has tied together two sleds with a rope. Two of his friends, with masses of 55.0 kg and 60.0 kg, are riding in the sleds, one in each (55.0 kg in the back, 60.0 kg in front). Th e student uses another rope to pull the sleds with a horizontal force of magnitude 230 N. Th e sleds accelerate with a magnitude of 1.02 m/s2. Th e front sled experiences a force of friction of magnitude 58.8 N. Assume that the masses of the sleds are negligible. (3.5) T/I
(a) What is the frictional force on the back sled?(b) What is the tension in the rope connecting the sleds?(c) Th e student pulling the sleds starts from rest,
runs for 3.00 s, and then lets the sleds go. How far will the sleds travel aft er he lets them go?
Evaluation 71. Th e planets are constantly changing position around
the Sun and have been closely following the same paths for millennia. However, the Sun has a small wobble in its position. Use Newton’s fi rst law and the concept of inertia to explain why this is so. (3.2) T/I C
72. Th e gravity of Earth attracts the Moon, just as it does objects near Earth. By Newton’s third law, the force that attracts the Moon to Earth is the same force that attracts Earth to the Moon. (3.4) T/I A
(a) Are objects on Earth attracted to the Moon?(b) Why does nothing fl y off toward the Moon?
73. Figure 9 shows a fan cart with the fan running but without the cart accelerating. (3.4) K/U
Figure 9
(a) Explain why the fan cart cannot accelerate.(b) Th e sail is removed and the cart begins to
accelerate. Explain why.
74. Two blocks are hung on opposite ends of a pulley system, as shown in Figure 10. Th e left block has a mass of 35.2 kg and the right block has a mass of 36.4 kg. (3.5) T/I
Figure 10
(a) Is it possible for the blocks to remain stationary? What forces could be causing this to happen?
(b) What would happen if one of the blocks were tapped downward?
Refl ect on Your Learning 75. A student claims, “If action and reaction forces are
equal in magnitude and opposite in direction, then when you add them together, you will always get zero and nothing will ever accelerate.” Discuss the validity of this statement. C
Research 76. When two positively charged particles are close
together, they repel each other, and the closer the particles are, the stronger this repulsion is. However, in atoms, protons are so close together that the strong nuclear force is able to overcome the electromagnetic repulsion. Write a one-page report describing the strong nuclear force and how strong it has to be in order to hold the nucleus of atoms together. Compare this with the strength of the electromagnetic repulsion acting at the same distance. What happens in heavier elements where there are a lot of protons and thus an even stronger electromagnetic repulsion? T/I C
77. Newton is reported to have said, “If I have seen further than other men, it is because I stood on the shoulders of giants,” meaning that he could not have made the discoveries that he did without the previous work of other scientists. Prepare a presentation in a format of your choice describing the knowledge of science at the time and who the primary infl uences were for Isaac Newton. What discoveries did they make, what observations did they make, and what experiments did they perform that helped lead Newton to his discoveries? T/I C
