Chapter 5: Forces in Equilibrium

“I don't know what I may seem to the world, but, as to myself, I seem to have been only like a

boy playing on the sea shore, and diverting myself in now and then finding a smoother

pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered

before me” – Isaac Newton, circa1727

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5.1 Vectors

• Draw vectors to scale to represent a quantity’s magnitude and direction.

• Solve vector problems.

• Find a vector’s components.

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Scalars vs. Vectors

• Quantities that can be represented by a single value are called Scalars. Scalars have no direction associated with them. Examples are: – Temperature

– Time

– Speed

– Mass

• Vectors are quantities that require more than one value to describe them. Vectors have a size and direction associated with them. Examples are:

– Velocity

– Acceleration

– Force

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Vectors

• Graphically vectors are represented by arrows.

• The length of the arrow represents the size (or magnitude) of the vector.

• The way the arrow is pointing shows the direction.

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Vector Components

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Quiz! What are the “x” and “y” components?

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+7

+3

Think of the components as “coordinates”; some vectors may not have a component in a certain direction, for example G!

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– 5

Free Body Diagrams

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A free-body diagram shows only the forces acting on an object, and does not include the forces an object exerts on other things. When making a free-body diagram, draw only the object you are studying, not any other objects around it.

5.2 Forces and Equilibrium

• Explain what it means to say an object is in equilibrium.

• Use free-body diagrams to find unknown forces.

• Explain how springs exert forces.

• Add force vectors.

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Forces and Equilibrium

• When the net force on an object is zero, we say the object is in equilibrium.

• In which diagram, A or B, is the box in equilibrium?

• Newton’s third law explains why normal forces exist. The book pushes down on the table, so the table pushes up on the book.

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Adding Vectors Graphically

Vectors are added “head to tail”. The parallelogram rule is shown in the figure to the right. Vectors obey the commutative law of addition. The resultant vector, R, is equivalent to A + B.

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Vector Addition

• We are familiar with Scalar Addition: 3.0 grams + 4.0 grams = 7.0 grams.

• Vectors do not add like this! For example, if the left vector is 3 cm and the right vector is 4 cm, when you add them is the resultant 7 cm?

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Vector Addition Continued

If two vectors line up in the same direction (or exactly opposite directions) then you may be able to add them like normal numbers, however, if they don’t…

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What is the unknown force if the polar bear is in equilibrium?

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Solving Equilibrium Problems

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Left diagram: ? = 900 N Top diagram: ? = 450 N

Springs

• Springs have a variety of uses: keeping objects in equilibrium, storing potential energy, accelerate objects.

• A spring always acts to return the spring to its natural length (the reaction in the diagrams to the right).

• A object pressing on a table top causes the atoms in the table surface to compress like a spring and push back!

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Hooke’s Law

• Hooke’s law states that the force exerted by a spring is proportional to its change in length.

• Different springs have different spring constants depending on their uses.

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Coil springs designed for tension

Compression springs store energy when compressed

Hooke’s Law, F = kx

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Remember the force is proportional to only the change in length, not the total length. If you apply 3 N what was the change in length, x? If you double the force to 6 N, what is x? Therefore, what is L in figure (c)?

Springs are elastic, like all elastic materials, springs try to return to their original shape once an applied force is removed. But if you stretch a spring past its elastic limit you may permanently damage the spring.

5.3 Friction

• Distinguish between sliding and static friction.

• Explain the cause of friction.

• Discuss reasons to increase or decrease friction.

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Friction

• Friction is the force opposing motion along a surface.

• Friction is due to the electrical forces between the atoms on two surfaces in contact with each other.

• Friction is all around us in everyday life.

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Types of Friction

• Sliding friction is present when two objects or surfaces slide across each other.

• Static friction exists when forces are acting to cause an object to move but friction is keeping the object from moving.

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Static and Sliding Friction

• Static friction builds up to the point where object moves.

• In general, it takes a little more force to get something moving than to keep it moving.

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Coefficients of friction

• From textbook, page 121: “The greater the force squeezing two surfaces together, the greater the friction force.”

• The amount of friction also depends on the type of material.

• So, we have an equation relating the frictional force, f, with the normal force, N.

• f = mN where m is called the coefficient of friction and depends on the surface material.

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box is moving to the right

Coefficients of friction

Surfaces

Coefficient of Static Friction, ms

Coefficient of Kinetic Friction, mk

Steel on Steel

0.74

0.57

Rubber on Concrete

1.0

0.8

Wood on Wood

0.25-0.5

0.2

Teflon on Teflon

0.04

0.04

Synovial joints in humans

0.01

0.003

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Reducing the Force of Friction

• The friction between a shaft (the long pole in the picture) and an outer part of a machine produces a lot of heat. Friction can be reduced by placing ball bearings between the shaft and the outer part.

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5.4 Torque and Rotational Equilibrium

• Explain how torque is created.

• Calculate the torque on an object.

• Define rotational equilibrium.

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What is Torque?

• A torque causes objects to rotate or spin.

• Torque is the “rotational” equivalent of force.

• The boat in the above diagram is not truly in equilibrium even though the forces balance.

• The door in the bottom diagram rotates around its hinges

• A force creates the greatest torque when the force is applied far from the hinges.

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Torques are created by a force acting at a distance from the axis of rotation

• Forces are vectors and have directions: up, down, left, right.

• Torques also have a direction, but in this case: clockwise and counter-clockwise.

• Only forces acting perpendicular to the “lever arm” produce torques!

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Class Problems

1. How much torque are you producing with the 50 N force on the bolt you are trying to tighten with the wrench?

2. What are the units of torque?

3. Is the torque CW or CCW?

4. In the bottom diagram, which force(s) produce the greatest torque?

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Rotational Equilibrium

• An object is in rotational equilibrium when the net torque is zero.

• The “clockwise” and “counter-clockwise” torques balance about some pivot point. You can use the equation:

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ccwccwcwcw rFrF

(1 lbs)(5 in) = (2 lbs)(2.5 in)

Class Problems

• How far must the boy sit from the center of the seesaw in order to balance?

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mN

Nmr

NmrN

mNrN

rFrF

cw

cw

cw

ccwccwcwcw

2.0500

100

100))(500(

)2)(50())(500(

Class Problems

• In the diagram to the right, which girl is heavier?

• If rKelly = 1.3 m, rAmy = 1.8 m, and the mass of Kelly is 50 kg, what is the mass of Amy?

• In the below diagram, find the unknown lever arm 2 distance given: – Mass 1 = 25 g

– Mass 2 = 40 g

– Mass 3 = 12 g

– Lever arm 1 = 40 cm

– Lever arm 3 = 50 cm

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Chapter 5 Review

1) What is the difference between a scalar and a vector?

2) Is each of these a scalar or a vector: time, mass, velocity, speed, temperature, force?

3) You make a scale drawing where 2 cm = 10 Newtons of Force. If your force vector is 2 cm how much force is this? If it is 4 cm? If it is 3 cm?

4) Can a moving object be in equilibrium? Explain.

5) A spring stretches 10 inches when 5 lbs force is pulling it. How much will it stretch when a 10 lbs forces pulls on it?

6) What factors affect the friction force between two surfaces? Remember f = mN.

7) Explain how the same force can create different amounts of torque on an object.

8) What is the net torque on an object in rotational equilibrium?

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