Unit 4: Energy

For this unit you must:1. Make predictions about the changes in kinetic energy of an object based on considerations of the

direction of the net force on the object as the object moves2. Use net force and velocity vectors to determine qualitatively whether kinetic energy of an object would

increase, decrease, or remain unchanged3. Use force and velocity vectors to determine qualitatively or quantitatively the net force exerted on an object and

qualitatively whether kinetic energy of that object would increase, decrease, or remain unchanged.4. Apply mathematical routines to determine the change in kinetic energy of an object given the forces on the

object and the displacement of the object5. Calculate the total energy of a system and justify the mathematical routines used in the calculation of

component types of energy within the system whose sum is the total energy6. Predict changes in the total energy of a system due to changes in position and speed of objects or frictional

interactions within the system7. Make predictions about the changes in the mechanical energy of a system when a component of an external

force acts parallel or antiparallel to the direction of the displacement of the center of mass8. Apply the concepts of conservation of energy and the work-energy theorem to determine qualitatively and/or

quantitatively that work done on a two-object system in linear motion will change the kinetic energy of the center of mass of the system, the potential energy of the system, and/or the internal energy of the system

9. Define open and closed systems for everyday situations and apply conservation concepts for energy, charge, and linear momentum to those situations.

10. Set up a representation or model showing that a single object can only have kinetic energy and use information about that object to calculate its kinetic energy

11. Translate between a representation of a single object, which can only have kinetic energy, and a system that includes the object, which may have both kinetic and potential energies

12. Calculate the expected behavior of a system using the object model (i.e. by ignoring changes in internal structure) to analyze a situation. Then, when the model fails, the student can justify the use of conservation of energy principles to calculate the change in internal energy due to changes in internal structure because the object is actually a system

13. Describe and make qualitative and/or quantitative predictions about everyday examples of systems with internal potential energy

14. Make quantitative calculations of the internal potential energy of a system from a description or diagram of that system

15. Apply mathematical reasoning to create a description of the internal potential energy of a system from a description or diagram of the objects and interactions in that system

16. Describe and make predictions about the internal energy of systems17. Calculate changes in kinetic and potential energy of a system, using information from representations of that

system18. Design an experiment and analyze date to examine how a force exerted on an object or system does work on

the object or system as it moves through a distance19. Design an experiment and analyze graphical data in which interpretations of the area under a force-distance

curve are needed to determine the work done on or by the object or system20. Predict and calculate from graphical data the energy transfer to or work done on an object or system from

information about a force exerted on the object or system through a distance21. Make claims about the interaction between a system and its environment in which the environment exerts a

force on the system, thus doing work on the system and changing the energy of the system (kinetic energy plus potential energy)

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22. Predict and calculate the energy transfer to (i.e., the work done) an object or system from information about a force exerted on the object or system through a distance

23. Make qualitative predictions about natural phenomena based on conservation of linear momentum and restoration of kinetic energy in elastic collisions

24. Apply the principles of conservation of momentum and restoration of kinetic energy to reconcile a situation that appears to be isolated and elastic, but in which data indicate that linear momentum and kinetic energy are not the same after the interaction, by refining a scientific question to identify interactions that have not been considered. You are expected to solve qualitatively and/or quantitatively for one-dimensional situations and only qualitatively for two-dimensional situations

25. Apply mathematical routines appropriately to problems involving elastic collisions in one dimension and justify the selection of those mathematical routines based on conservation of momentum and restoration of kinetic energy

26. Design an experimental test of an application of the principle of conservation of linear momentum, predict an outcome of the experiment using the principle, analyze data generated by that experiment whose uncertainties are expressed numerically, and evaluate the match between the prediction and the outcome

27. Classify a given collision situation as elastic or inelastic, justify the selection of conservation of linear momentum and restoration of kinetic energy as the appropriate principles for analyzing an elastic collision, solve for missing variables, and calculate their values

28. Qualitatively predict, in terms of linear momentum and kinetic energy, how the outcome of a collision between two objects changes depending on whether the collision is elastic or inelastic

29. Apply the conservation of linear momentum to a closed system of objects involved in an inelastic collision to predict the change in kinetic energy

Chapter 7: Work and Kinetic Energy

Section 7-1 Work Done by a Constant Force

Energy can be transferred by an external force exerted on an object or system that moves the object or system through a distance; this energy transfer is call work.

If the force is constant during a given displacement, then the work done is the product of the displacement and the component of the force parallel or antiparallel to the displacement

Example 1: Helen had 12.4 kg of groceries in her cart which is 1.0 m high. How much work must Helen do to lift the groceries into her trunk which is 0.8 m above the cart?

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Example 2: In a gravity escape system (GES), an enclosed lifeboat on a large ship is deployed by letting it slide down a ramp and then continue in free fall to the water below. Suppose a 4970 kg lifeboat slides a distance of 5.00 m on a ramp, dropping through a vertical height of 2.50 m. How much work does gravity do on the boat?

Example 3: You want to load a box into the back of a truck. One way is to lift it straight up with constant speed through a height h doing a work W1. Alternatively, you can slide the box up a loading ramp with constant speed a distance L, doing work W2. Assuming the box slides on the ramp without friction, which of the following statements is correct: (a) W1 < W2, (b) W1 = W2, (c) W1 > W2? Justify your response.

Example 4: A car of mass m coasts down a hill inclined at an angle φ below the horizontal. The car is acted upon by three forces: (i) the normal force exerted by the road, (ii) air resistance, and (iii) the force of gravity. Find the total work done on the car as it travels a distance d along the road.

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Section 7-2 Kinetic Energy and the Work-Energy Theorem

The work energy theorem comes from the equations of motion:

Kinetic energy is given by:

The change in kinetic energy depends on the force exerted on the object and on the displacement of the object One the component of the net force exerted on an object parallel or antiparallel to the displacement of the

object will increase or decrease the kinetic energy of the object The magnitude of the change in the kinetic energy is given by:

The component of the net force exerted on an object perpendicular to the direction of the displacement of the object can change the direction of the motion of the object without changing the kinetic energy (exs. Uniform circular motion and projectile motion)

The kinetic energy of a rigid system may be translational, rotational or a combination of both. The change in the rotational kinetic energy of a rigid system is the product of the angular displacement and the net torque (studied later)

Classically, an object can only have kinetic energy since potential energy requires an interaction between two or more objects

Example 5: A 0.80 kg ball is thrown vertically from the top of a building 12m high with a velocity of 18 m/s. What is the ball’s kinetic energy when the ball is at its maximum height and just before the ball hits the ground?

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Example 6: A two-man bobsled has a mass of 390 kg. Starting from rest, the two racers push the sled for the first 50 m with a net force of 270 N. Neglecting friction, what is the sled’s speed at the end of the 50 m?

Example 7: A 4.1 kg box of books is lifted vertically from rest a distance of 1.6 m by an upward applied force of 60.0 N. Find the final speed of the box.

Example 8: To accelerate a certain car from rest to the speed v requires the work W1. The work needed to accelerate the car from v to 2v is W2. Which of the following statements is correct: W2 = W1, W2 = 2W1, W2 = 3W1, W2 = 4W1? Justify your response.

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Section 7-3 Work Done by a Variable Force

Work (change in energy) can be found from the area under a graph of the magnitude of the force component parallel to the displacement versus displacement

Graphical representation of work done by

A Constant Force A Variable Force

The work needed to stretch a spring a distance x is found by calculating the area under the F vs x graph for a spring

Example 9: An archer pulls back the string on her bow to a distance of 70 cm from its equilibrium position. To hold the string at this position takes a force of 140 N. How much elastic potential energy is stored in the bow?

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Example 10: If 30.0 J of work is required to stretch a spring from 4.0 cm to 5.0 cm, how much work is needed to stretch it from 5.0 cm to 6.0 cm?

Example 11: A block with mass of 1.5 kg and an initial speed of 2.2 m/s slides on a frictionless, horizontal surface. The block comes into contact with a spring that is in its equilibrium position, and compresses it until the block comes to rest momentarily. Find the maximum compression of the spring, assuming its force constant is 475 N/m.

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Section 7-4 Power

Energy transfer in mechanical or electrical systems may occur at different rates. Power is defined as the rate of energy transfer into, out of, or within a system.

Example 12: To pass a slow moving truck your 1.30 x 103 kg car must accelerate from 13.4 m/s to 17.9 m/s. If it takes 3.0 s to accelerate the car, what is the minimum power required by the car to pass the truck?

Example 13: If takes a force of 1280 N to keep a 1000.0kg car moving with a constant speed up a slope of 5.0°.a. If the engine delivers 50.0 hp to the drive wheels, what is the maximum speed of the car? (746 W = 1 hp)b. What is the frictional force experienced by the car?

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Example 14: A kayaker paddles with a power output of 50.0 W to maintain a steady speed of 1.50 m/s.a. Calculate the resistive force exerted by the water on the kayak.b. If the kayaker doubles her power output, and the resistive force due to the water remains the same, by

what factor does the kayaker’s speed change?

Chapter 8: Potential Energy and Conservation of Energy

Recall that a system is an object or a collection of objects. The objects are treated as having no internal structure.

For all systems under all circumstances, energy, charge, linear momentum, and angular momentum are conserved. For an isolated or closed system, conserved quantities are constant. An open system is one that exchanges any conserved quantity with its surroundings.

An interaction can be either a force exerted by objects outside the system or the transfer of some quantity with objects outside the system

The placement of a boundary between a system and its environment is a decision made by the person considering the situation in order to simplify or otherwise assist in analysis

Through this chapter you will identify the types of energy within a system. Energy of one kind can be transformed into another kind within a system. In addition, energy can be transferred between a system and its boundary. When forces result in work being done on the system, energy transfer occurs.

1. A child is on a playground swing, motionless at the highest point of his arc. What energy transformation takes place as he swings back down to the lowest point of his motion?

a. K Ug

b. Ug Kc. Eth Kd. Ug Eth

e. K Eth

2. A skier is gliding down a gentle slope at a constant speed. What energy transformation is taking place?a. K Ug

b. Ug Kc. Eth Kd. Ug Eth

e. K Eth

3. A tow rope pulls a skier up the slope at constant speed. What energy transfer (or transfers) is taking place?a. W Ug

b. W Kc. W Eth

d. Both A and B.e. Both A and C.

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4. A crane lowers a girder into place at constant speed. Consider the work Wg done by gravity and the work WT

done by the tension in the cable. Which is true?a. Wg > 0 and WT > 0b. Wg > 0 and WT < 0c. Wg < 0 and WT > 0d. Wg < 0 and WT < 0e. Wg = 0 and WT = 0

5. Robert pushes the box to the left at constant speed. In doing so, Robert does ______ work on the box.a. positiveb. negativec. zero

Section 8-1 Conservative and Nonconservative Forces

Conservative Forces The work done by a conservative force is stored in a form that can be released as kinetic energy Examples include gravity and spring force A conservative force does zero total work on a closed path The work done by a conservative force between two points is independent of the path taken

Nonconservative Forces The work done by nonconservative forces is converted to other forms of energy and therefore cannot be

recovered Examples include friction and tension A nonconservative force does work on a closed path The work done by a conservative force between two points depends on the path taken

Example 15: The pendulum of a grandfather clock is 0.994 m long and has a 0.76 kg bob at the bottom. Calculate the work required to move the pendulum. Calculate the work required to do move the pendulum from its lowest point to its highest point if the pendulum swings through an angle of 6.0° from vertical.

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Example 16: A 0.30 kg block of wood is rubbed back and forth against a wood table 30 times in each direction. The coefficient of friction between the block and the table is 0.20. The block is moved 8.0 cm during each stroke and pressed against the table with a force of 22 N. How much thermal energy is created in this process?

Example 17: A 4.57 kg box is moved with constant speed from A to B along two paths shown in part (a). Calculate the work done by gravity on each of these paths. The same box is pushed across a floor from A to B along path 1 and path 2 in part (b). If the coefficient of friction between the box and the surface is 0.63, how muh work is done by friction along each path?

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Section 8-2 Potential Energy and the Work Done by Conservative Forces

A system with internal structure can have potential energy. Potential energy exists within a system if the objects within that system interact with conservative forces

The work done by a conservative force is independent of the path taken. The work description is used for forces external to the system. Potential energy is used when the forces are internal interactions between parts of the system

Changes in the internal structure can result in changes in potential energy. Examples include mass-spring oscillators, objects falling in a gravitational field/simple pendulums

Giancoli defines potential energy as the energy associated with forces that depend on the position or configuration of a body (or bodies) and the surroundings. Various types of potential energy can be defined and each type is associated with a particular force.

The work done by a conservative force is equal to the negative of the change in the corresponding potential energy:

What does this mean? As potential energy is released, work is done by a conservative force.The change in potential energy associated with a particular force, when an object is moved from one position to another, is equal to the work that would be done by that force in moving the object from the second position back to the first.

Gravitational Potential Energy:

Alternative equation for Gravitational Potential Energy:

Elastic Potential Energy (Spring Energy):

Section 8-3 Conservation of Mechanical Energy

The energy of a system includes it kinetic energy, potential energy, and microscopic internal energy. Examples include gravitational potential energy, elastic potential energy and kinetic energy.

A rotating, rigid body may be considered to be a system, and may have both translational and rotational energy During an inelastic collision, some of the mechanical energy dissipates as thermal energy

Mechanical energy (the sum of kinetic and potential energy) is transferred into or out of a system when an external force is exerted on a system such that a component of the force is parallel to its displacement (ie. when work is done).

The internal energy of a system includes the kinetic energy of the objects that make up the system and the potential energy of the configuration of the objects that make up the system

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Since energy is constant in a closed system, changes in a system’s potential energy can result in changes to the system’s kinetic energy

The changes in potential and kinetic energies in a system may be further constrained by the construction of the system

In systems involve only conservative forces, mechanical energy is conserved. Mechanical energy is written as:

and our conservation of energy equation comes from the work energy theorem whereby the work done is work done by conservative forces:

Example 18: The image below shows 3 excited Gonzaga grads tossing their cap in the gym. Label the diagram from left to right as A, B and C. Rank the final speed of the graduation cap at position y f in each case and justify your ranking.

Example 19: A car at rest at the top of a hill begins rolling down the hill. After its height has dropped by 5.0 m, it is moving at a good clip. Note the choice of the system, what energy transformation(s) has taken place, and write the equation for conservation of energy.

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Example 20: In the bottom of the ninth inning, a player hits a 0.15 kg baseball over the outfield fence. The ball leaves the bat with a speed of 36 m/s, and a fan in the bleachers catches it 7.2m above the point where it was hit. Assuming frictional forces can be ignored, find the speed of the ball just before it’s caught.

Example 21: Swimmers at a water park can enter a pool using one of the two frictionless slides of equal height. Slide 1 approaches the water with a uniform slope, slide 2 dips rapidly at first, then levels out. Is the speed at the bottom of slide 1 greater than, less than, or the same as the speed at the bottom of slide 2? Justify your response.

Example 22: A snow boarder coasts on a smooth track that rises from one level to another. If the snowboarder’s initial speed is 4 m/s, the snowboarder just makes it to the upper level and comes to rest. With a slightly greater initial speed of 5 m/s, the snowboarder is still moving to the right on the upper level. Is the snowboarder’s final speed in this case (a) 1 m/s, (b) 2 m/s, or (c) 3 m/s?

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Example 23: A spring loaded toy gun is used to launch a 10.0 g plastic ball. The spring, which has a spring constant of 10.0 N/m, is compressed by 10.0 cm as the ball is pushed into the barrel. When the trigger is pulled, the spring is released and shoots the ball back out. What is the ball’s speed as it leaves the barrel? Assume friction is negligible.

Section 8-4 Work Done by Nonconservative Forces

Nonconservative forces change the amount of mechanical energy in a system. The total mechanical energy is not conserved and can be written as:

Example 24: A child slides down a slide at a constant speed of 1.5 m/s. The height of the slide is 3.0 m. Write down the equation for conservation of energy, noting the choice of system, the initial and final states, and what energy transformation has taken place.

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Example 25: A skier drops her glove from rest at a height of h into fresh powder snow. The glove falls to a depth of 0.147 below the surface of the snow before coming to rest. If the work done by the snow is 1.44 J, what is the drop height of the glove?

Example 26: A golfer badly misjudges a putt, sending the ball only one-quarter of the distance to the hole. The original putt gave the ball an initial speed of v. If the force of resistance due to the grass is constant, would an initial speed of (a) 2v, (b) 3v, or (c) 4v be needed to get the ball to the hole from its original position?

Example 27: Monica pulls her daughter Jessie in a bike trailer. The trailer and Jessie together have a mass of 25 kg. Monica starts up a 100-m-long slope that’s 4.0 m high. On the slope, Monica’s bike pulls on the trailer with a constant force of 8.0 N. They start out at the bottom of the slope with a speed of 5.3 m/s. What is their speed at the top of the slope?

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Example 28: A food shipper pushes a wood crate of cabbage heads (m = 14 kg) across a concrete floor with a constant horizontal force of 40.0 N. In a straight-line displacement of 0.50 m [right] the speed of the crate decreases from 0.60 m/s to 0.20 m/s. What is the increase in thermal energy of the crate and the floor? What is the coefficient of friction between the crate and the floor?

Example 29: A disabled robot of mass 40.0 kg is dragged by a cable up a 30.0° inclined wall inside a volcano crater. The force on the cable is 332 N and the kinetic friction on the robot from the crater wall is 136 N. If the robot moves 0.50 m along the wall what is the change in kinetic energy of the robot? Explain your result.

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Example 30: A steel ball of mass 5.2 g is fired vertically down from a height of 18 m with a speed of 14 m/s. It penetrates a layer of sand below it to a depth of 21 cm. Assume that the force from the sand that slows and stops the ball is a constant kinetic frictional force. What is the magnitude of the frictional force on the ball from the sand as the ball is being slowed to a stop?

Example 31: A modified Atwood machine is set up such that a block of mass 2.40 kg rests on a table of height h and a block of mass 1.80 kg hangs from a rope over a frictionless pulley. When the blocks are released from rest, they move through a distance of 0.500 m at which point the 1.80 kg block hits the floor. If the coefficient of friction between the 2.40 kg block and the table is 0.450, find the speed of the blocks just before the 1.80 kg block hits the table.

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Section 8-5 Potential Energy Curves and Equipotentials

In a collision between objects, linear momentum is conserved (next unit). In an elastic collision, kinetic energy is the same before and after the collision. In an inelastic collision, kinetic energy is not the same before and after the collision.

In a closed system, the linear momentum is constant throughout the collision In a closed system, the kinetic energy after an elastic collision is the same as the kinetic energy before the

collision. In a closed system the kinetic energy after an inelastic collision is different from the kinetic energy before the

collision

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