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Chapter Review

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In this chapter we continue the study of work and energy and encounter another form of energy called potential energy. Most importantly, in this chapter we introduce the concept of the conservation of energy, which is one of the most important concepts in science.

8-1 Conservative and Nonconservative Forces

Forces are generally divided into two catagories depending on properties of the work that a force does. A force can either be a conservative force or a nonconservative force. A force is called conservative if the work it does on any object is independent of the path that object takes during its displacement. Equivalently, a force is conservative if the work it does around any closed path is zero. Any force that does not meet the above condition is a nonconservative force.

The primary reason for the two classifications is that when work is done by a conservative force, that work is in some sense stored within the system and can be easily recovered, usually as kinetic energy. Gravity is a conservative force. When an object is lifted against gravity a certain amount of negative work is done by gravity on the object. When that object is released, or lowered back down, gravity does an equal amount of positive work. Kinetic friction is a nonconservative force. If you slide a block across a table friction does a certain amount of negative work on it. When you slide the block back to its original place, friction does even more negative work on it - the work is not recovered in this case.

Physlet Illustration: Conservative and Non-Conservative Forces
Fx = 0  Fy=y Fx = x  Fy=y Fx = y  Fy=y*x Fx = x*x  Fy=y*y
Which of the above forces are conservative? Drag the object around using the mouse. The bar graph on the right displays the work done along the path you chose, and the table below shows the x and y coordinates and the net work done. Start Reinitialize
Hints A force is considered conservative if the work done is path independent. A force is considered conservative if the total work done around a closed loop is zero.

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

In chapter 7 you were introduced to the idea that when work is done it goes into changing the kinetic energy of the object or system on which the work is done. In that case it was the total work done by all forces acting. A similar concept can be applied to specific, or individual, forces as well. When work is done by a conservative force, this work goes into changing the potential energy, U, of the system. You can think of the potential energy as representing the "storage" of this work within the system.

The mathematical definition of potential energy is that the change in potential energy equals the negative of the work done by conservative forces

DU = -W_c.

Notice that it is only the change in potential energy that has physical meaning. Particular values of potential energy are defined relative to a chosen reference. This reference can be chosen arbitrarily although in some cases certain choices are more convienient and usually adopted by convention. Based on the above definition, it is clear that the SI unit of potential energy is the same as for work and kinetic energy - the joule.

(A) Gravity

Because of the connection with conservative forces, potential energies are always associated with specific forces. Since gravity is a conservative force we can define gravitational potential energy. Near Earth's surface, where we can treat the force of gravity as constant, the gravitational potential energy is

DU = mg(Dy),

where Dy is the change in vertical position. The height at which U = 0 can always be chosen to be the initial height. Therefore, we often write the gravitational potential energy as just

U = mgy.

The above equation is written with the understanding that it is only used this way when the reference height has been specified.

Physlet Illustration: Energy of a Bouncing Ball
Near the surface of the earth, a 100-gm bouncy ball is dropped from rest. The display grid is in meters. The velocity is shown in m/s. Study the graphs of kinetic energy, potential energy, and total energy as functions of time. Can you verify that these energies are calculated correctly? What can you conclude? Start
Hints When (where) does the ball's potential energy have its maximum value? Its minimum value? When (where) does the ball's kinetic energy have its maximum value? Its minimum value? How (why) do these two types of energy change? What should be true, according to the Law of Energy Conservation?

Physlet Illustration: Energy In Projectile Motion
The blue ball (mass = 250 grams) is thrown into the air and follows a parabolic path.  How does its energy change with time?  Start
Hints Choose any time to pause the animation. Use the cursor to "measure" the kinetic energy on the graph.  Is the graph correct? Use the cursor to "measure" the potential energy on the graph.  Is the graph correct? What is the sum of the potential and kinetic energy at any time?

(B) Springs

Besides gravity, a second conservative force that we have encountered is that of Hooke's law. Therefore, we can define a potential energy to be associated with this force as well. We call this the spring potential energy, also often called the elastic potential energy. We have seen in chapter 7 that the work done by a spring while being deformed by an amount x from equilibrium is , so the DU associated with this force will be of opposite sign

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In the case of Hooke's law, it is almost always most convenient to take the equilibrium configuration (x = 0) as the reference level for potential energy (U = 0). With this understanding the spring potential energy will be written as

.

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Example 8.1 Rolling Down: A 0.25 kg ball rolls 2.1 m down a ramp that is inclined at 32 degrees to the horizontal. Determine the change in gravitational potential energy.

Picture the Problem The picture shows the ball rolling down the ramp.

Strategy To determine the change in potential energy we need to find the change in height and use it in the expression for gravitational potential energy.

Solution

1. From trigonometry we see that:
2. Since the ball rolls downhill:
3. The change in potential energy is

Insight Notice that we had to be careful to notice that Dy should be negative. The mathematics alone would not have given the minus sign.

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Physlet Illustration: Energy of a Vertical Mass/Spring System
A 2-kg mass hangs from a massless spring of spring constant k = 5 N/m. The spring has an equilibrium (unstretched) length of 13 m. The displacement of the spring from equilibrium is shown in meters, and the velocity of the mass is shown in m/s. Pull down or push up on the end of the spring to displace the mass from its initial position, and then push "Play" to watch it move up and down. Study the graphs of kinetic energy, gravitational potential energy, spring potential energy, and total energy as functions of time. Can you verify that these energies are calculated correctly? What can you conclude? Start
Hints When (where) does the gravitational potential energy have its maximum value? Its minimum value? When (where) does the spring's potential energy have its maximum value? Its minimum value? When (where) does the mass's kinetic energy have its maximum value? Its minimum value? How (why) do each of  these types of energy change? What should be true, according to the Law of Energy Conservation?

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Exercise 8.2 Hanging from a Spring: A spring of force constant 21 N/cm hangs vertically from a ceiling. Then a 1.2 kg mass is attached to its end and lowered until it hangs motionless from the end of the spring. If we take the potential energy of the Earth-spring-mass system to be zero initially, what is it in the final configuration?

Solution: The following information is given in the problem:
Given: m = 1.2 kg, k = 21 N/cm, U_i = 0; Find: U_f

Here we have changes in both the gravitational potential energy and that of the spring. The change in each depends on how much the spring stretches under the weight of the mass; i.e., x = |Dy|. As we have seen in previous problems, equilibrium between the gravitational force and the Hooke's law force is reached when kx = mg. This implies that

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The change in potential energy is then

.

Inserting the expressions for x and Dy (noting that Dy is negative) we get

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Since DU = U_f - U_i = U_f we can conclude that

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The important thing here was to remember to account for both forms of potential energy.

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Practice Quiz

When an object is thrown up into the air, the gravitational potential energy... increases on the way up and decreases on the way down. decreases on the way up and increases on the way down. changes the same way going up as it does going down. remains constant throughout its motion. [none of the above]

In comparing a spring that is stretched an amount x from equilibrium versus one that is compressed an amount x from equilibrium, the spring potential energy... increases as it is stretched and decreases as it is compressed. decreases as it is stretched and increases as it is compressed. changes the same way being stretched or compressed. remains constant throughout its motion [none of the above]

A 2.0 kg object is lifted 2.0 m off the floor, carried horizontally across the room, and finally placed down on a table that is 1.5 m high. What is the overall change in gravitational potential energy for the Earth-object system? 6.0 J 20 J 29 J 9.8 J 59 J

If a 0.75 kg block is rested on top of a vertical spring of force constant 45 N/m, how much potential energy is stored in the spring relative to its equilibrium configuration? 0 J 14 J 60 J 34 J 0.60 J

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