### The Work–Energy Theorem

In physics, the term work has a very specific definition. Work is application of force, $f$, to move an object over a distance, *d*, in the direction that the force is applied. Work, *W*, is described by the equation

$$W=fd\text{.}$$

Some things that we typically consider to be work are not work in the scientific sense of the term. Let’s consider a few examples. Think about why each of the following statements is true.

- Homework
*is not* work.
- Lifting a rock upwards off the ground
*is* work.
- Carrying a rock in a straight path across the lawn at a constant speed
*is not* work.

The first two examples are fairly simple. Homework is not work because objects are not being moved over a distance. Lifting a rock up off the ground is work because the rock is moving in the direction that force is applied. The last example is less obvious. Recall from the laws of motion that force is *not *required to move an object at constant velocity. Therefore, while some force may be applied to keep the rock up off the ground, no net force is applied to keep the rock moving forward at constant velocity.

Work and energy are closely related. When you do work to move an object, you change the object’s energy. You (or an object) also expend energy to do work. In fact, energy can be defined as the ability to do work. Energy can take a variety of different forms, and one form of energy can transform to another. In this chapter we will be concerned with mechanical energy, which comes in two forms: kinetic energy and potential energy.

- Kinetic energy is also called energy of motion. A moving object has kinetic energy.
- Potential energy, sometimes called stored energy, comes in several forms. Gravitational potential energy is the stored energy an object has as a result of its position above Earth’s surface (or another object in space). A roller coaster car at the top of a hill has gravitational potential energy.

Let’s examine how doing work on an object changes the object’s energy. If we apply force to lift a rock off the ground, we increase the rock’s potential energy, *PE*. If we drop the rock, the force of gravity increases the rock’s kinetic energy as the rock moves downward until it hits the ground.

The force we exert to lift the rock is equal to its weight, *w*, which is equal to its mass, *m*, multiplied by acceleration due to gravity, **g**.

$$f=w=mg$$

The work we do on the rock equals the force we exert multiplied by the distance, *d*, that we lift the rock. The work we do on the rock also equals the rock’s gain in gravitational potential energy, *PE*_{e}.

$$W=P{E}_{e}=fmg$$

Kinetic energy depends on the mass of an object and its velocity, **v**.

$$KE=\frac{1}{2}m{v}^{2}$$

When we drop the rock the force of gravity causes the rock to fall, giving the rock kinetic energy. When work done on an object increases only its kinetic energy, then the net work equals the change in the value of the quantity$\frac{1}{2}m{v}^{2}$. This is a statement of the work–energy theorem, which is expressed mathematically as

$$W=\text{\Delta}KE\text{=}\frac{1}{2}m{v}_{2}^{2}-\frac{1}{2}m{v}_{1}^{2}\text{.}$$

The subscripts _{2} and _{1} indicate the final and initial velocity, respectively. This theorem was proposed and successfully tested by James Joule, shown in Figure 9.2.

Does the name Joule sound familiar? The joule (J) is the metric unit of measurement for both work and energy. The measurement of work and energy with the same unit reinforces the idea that work and energy are related and can be converted into one another. 1.0 J = 1.0 N∙m, the units of force multiplied by distance. 1.0 N = 1.0 k∙m/s^{2}, so 1.0 J = 1.0 k∙m^{2}/s^{2}. Analyzing the units of the term (1/2)*m***v**^{2} will produce the same units for joules.

### Watch Physics

#### Work and Energy

This video explains the work energy theorem and discusses how work done on an object increases the object’s KE.

Grasp Check

True or false—The energy increase of an object acted on only by a gravitational force is equal to the product of the object's weight and the distance the object falls.

- True
- False