Work, Power, and the Work–Energy Theorem, Work, Energy, and Simple Machines | Physics

Learning objectives

Teacher Support

The learning objectives in this section will help your students master the following standards:

(6) Science concepts. The student knows that changes occur within a physical system and applies the laws of conservation of energy and momentum. The student is expected to:
- (A) describe and apply the work–energy theorem;
- (C) describe and calculate work and power.

In addition, the High School Physics Laboratory Manual addresses the following standards:

(6) Science concepts. The student knows that changes occur within a physical system and applies the laws of conservation of energy and momentum. The student is expected to:
- (C) calculate the mechanical energy of, power generated within, impulse applied to, and momentum of a physical system.

Use the lab titled Work and Energy as a supplement to address content in this section.

Section Key Terms

energy	gravitational potential energy	joule	kinetic energy	mechanical energy
potential energy	power	watt	work	work–energy theorem

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 = f d .

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 = m g

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} = m g d

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

K E = \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 = Δ K E = \frac{1}{2} m v_{2}^{2} - \frac{1}{2} m v_{1}^{2} .

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_09_01_Joule.

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 kg∙m/s², so 1.0 J = 1.0 kg∙m²/s². Analyzing the units of the term (1/2)mv² will produce the same units for joules.

View original figure (external; image not reproduced in this text edition)

Original figure description: A black and white photograph of James Joule is shown.

The joule is named after physicist James Joule (1818–1889). (C. H. Jeens, Wikimedia Commons)

Calculations Involving Work and Power

In applications that involve work, we are often interested in how fast the work is done. For example, in roller coaster design, the amount of time it takes to lift a roller coaster car to the top of the first hill is an important consideration. Taking a half hour on the ascent will surely irritate riders and decrease ticket sales. Let’s take a look at how to calculate the time it takes to do work.

Recall that a rate can be used to describe a quantity, such as work, over a period of time. Power is the rate at which work is done. In this case, rate means per unit of time. Power is calculated by dividing the work done by the time it took to do the work.

P = \frac{W}{t}

Let’s consider an example that can help illustrate the differences among work, force, and power. Suppose the woman in Figure_09_01_Lifting lifting the TV with a pulley gets the TV to the fourth floor in two minutes, and the man carrying the TV up the stairs takes five minutes to arrive at the same place. They have done the same amount of work $(f d)$ on the TV, because they have moved the same mass over the same vertical distance, which requires the same amount of upward force. However, the woman using the pulley has generated more power. This is because she did the work in a shorter amount of time, so the denominator of the power formula, t, is smaller. (For simplicity’s sake, we will leave aside for now the fact that the man climbing the stairs has also done work on himself.)

View original figure (external; image not reproduced in this text edition)

Original figure description: A cross-section view of a four-story elevator is shown. The first three floors are labeled with h. A person is carrying a television up a flight of stairs on the second floor. Another person is standing on the ground floor, using a pulley to move a television up the floors.

No matter how you move a TV to the fourth floor, the amount of work performed and the potential energy gain are the same.

Power can be expressed in units of watts (W). This unit can be used to measure power related to any form of energy or work. You have most likely heard the term used in relation to electrical devices, especially light bulbs. Multiplying power by time gives the amount of energy. Electricity is sold in kilowatt-hours because that equals the amount of electrical energy consumed.

The watt unit was named after James Watt (1736–1819) (see Figure_09_01_Watt). He was a Scottish engineer and inventor who discovered how to coax more power out of steam engines.

View original figure (external; image not reproduced in this text edition)

Original figure description: A painting of James Watt is shown.

Is James Watt thinking about watts? (Carl Frederik von Breda, Wikimedia Commons)

Teacher Support

[BL][OL] Review the concept that work changes the energy of an object or system. Review the units of work, energy, force, and distance. Use the equations for mechanical energy and work to show what is work and what is not. Make it clear why holding something off the ground or carrying something over a level surface is not work in the scientific sense.

[OL] Ask the students to use the mechanical energy equations to explain why each of these is or is not work. Ask them to provide more examples until they understand the difference between the scientific term work and a task that is simply difficult but not literally work (in the scientific sense).

[BL][OL] Stress that power is a rate and that rate means "per unit of time." In the metric system this unit is usually seconds. End the section by clearing up any misconceptions about the distinctions between force, work, and power.

[AL] Explain relationships between the units for force, work, and power. If $W = f d$ and work can be expressed in J, then $P = \frac{W}{t} = \frac{f d}{t}$ so power can be expressed in units of $\frac{N \cdot m}{s}$

Also explain that we buy electricity in kilowatt-hours because, when power is multiplied by time, the time units cancel, which leaves work or energy.

Watt’s Steam Engine

James Watt did not invent the steam engine, but by the time he was finished tinkering with it, it was more useful. The first steam engines were not only inefficient, they only produced a back and forth, or reciprocal, motion. This was natural because pistons move in and out as the pressure in the chamber changes. This limitation was okay for simple tasks like pumping water or mashing potatoes, but did not work so well for moving a train. Watt was able build a steam engine that converted reciprocal motion to circular motion. With that one innovation, the industrial revolution was off and running. The world would never be the same. One of Watt's steam engines is shown in Figure_09_01_engine. The video that follows the figure explains the importance of the steam engine in the industrial revolution.

View original figure (external; image not reproduced in this text edition)

Original figure description: A black and white illustration of an early steam engine is shown.

A late version of the Watt steam engine. (Nehemiah Hawkins, Wikimedia Commons)

Before proceeding, be sure you understand the distinctions among force, work, energy, and power. Force exerted on an object over a distance does work. Work can increase energy, and energy can do work. Power is the rate at which work is done.

Applying the Work–Energy Theorem

An ice skater with a mass of 50 kg is gliding across the ice at a speed of 8 m/s when her friend comes up from behind and gives her a push, causing her speed to increase to 12 m/s. How much work did the friend do on the skater?

Solution

Identify the variables. m = 50 kg,

v_{2} = 12 \frac{m}{s}, and v_{1} = 8 \frac{m}{s}

Substitute. $W = \frac{1}{2} 50 (12^{2} - 8^{2}) = 2, 000 J$

Discussion

Work done on an object or system increases its energy. In this case, the increase is to the skater’s kinetic energy. It follows that the increase in energy must be the difference in KE before and after the push.

Practice Problems

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Check Your Understanding

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When a coconut falls from a tree, work W is done on it as it falls to the beach. This work is described by the equation

W = F d = \frac{1}{2} m v_{2}^{2} - \frac{1}{2} m v_{1}^{2} .

Identify the quantities F, d, m, v₁, and v₂ in this event.

F is the force of gravity, which is equal to the weight of the coconut, d is the distance the nut falls, m is the mass of the earth, v₁ is the initial velocity, and v₂ is the velocity with which it hits the beach.
F is the force of gravity, which is equal to the weight of the coconut, d is the distance the nut falls, m is the mass of the coconut, v₁ is the initial velocity, and v₂ is the velocity with which it hits the beach.
F is the force of gravity, which is equal to the weight of the coconut, d is the distance the nut falls, m is the mass of the earth, v₁ is the velocity with which it hits the beach, and v₂ is the initial velocity.
F is the force of gravity, which is equal to the weight of the coconut, d is the distance the nut falls, m is the mass of the coconut, v₁ is the velocity with which it hits the beach, and v₂ is the initial velocity.

Section Summary

Doing work on a system or object changes its energy.
The work–energy theorem states that an amount of work that changes the velocity of an object is equal to the change in kinetic energy of that object.The work–energy theorem states that an amount of work that changes the velocity of an object is equal to the change in kinetic energy of that object.
Power is the rate at which work is done.

Key Equations

equation for work	$W = f d$
force	$f = w = m g$
work equivalencies	$W = P E_{e} = f m g$
kinetic energy	$K E = \frac{1}{2} m v^{2}$
work–energy theorem	$W = Δ KE = \frac{1}{2} m v_{2}^{2} - \frac{1}{2} m v_{1}^{2}$
power	$P = \frac{W}{t}$

Concept Items

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A book with a mass of 0.30 kg falls 2 m from a shelf to the floor. This event is described by the workenergy theorem: $W = f d = \frac{1}{2} m v_{2}^{2} - \frac{1}{2} m v_{1}^{2}$ Explain why this is enough information to calculate the speed with which the book hits the floor.

The mass of the book, m, and distance, d, are stated. F is the weight of the book mg . v₁ is the initial velocity and v₂ is the final velocity. The final velocity is the only unknown quantity.
The mass of the book, m, and distance, d, are stated. F is the weight of the book mg . v₁ is the final velocity and v₂ is the initial velocity. The final velocity is the only unknown quantity.
The mass of the book, m, and distance, d, are stated. F is the weight of the book mg . v₁ is the initial velocity and v₂ is the final velocity. The final velocity and the initial velocities are the only unknown quantities.
The mass of the book, m, and distance, d, are stated. F is the weight of the book mg . v₁ is the final velocity and v₂ is the initial velocity. The final velocity and the initial velocities are the only unknown quantities.

Critical Thinking

Which activity requires a person to exert force on an object that causes the object to move but does not change the kinetic or potential energy of the object?

Moving an object to a greater height with acceleration
Moving an object to a greater height without acceleration
Carrying an object with acceleration at the same height
Carrying an object without acceleration at the same height

Which statement explains how it is possible to carry books to school without changing the kinetic or potential energy of the books or doing any work?

By moving the book without acceleration and keeping the height of the book constant
By moving the book with acceleration and keeping the height of the book constant
By moving the book without acceleration and changing the height of the book
By moving the book with acceleration and changing the height of the book

Problems

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A boy pushes his little sister on a sled. The sled accelerates from 0 to 3.2 m/s . If the combined mass of his sister and the sled is 40.0 kg and 18 W of power were generated, how long did the boy push the sled?

205 s
128 s
23 s
11 s

Test Prep Multiple Choice

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The workenergy theorem states that the change in the kinetic energy of an object is equal to what?

The work done on the object
The force applied to the object
The loss of the objects potential energy
The objects total mechanical energy minus its kinetic energy

A runner at the start of a race generates 250 W of power as he accelerates to 5 m/s . If the runner has a mass of 60 kg, how long did it take him to reach that speed?

0.33 s
0.83 s
1.2 s
3.0 s

A cars engine generates 100,000 W of power as it exerts a force of 10,000 N. How long does it take the car to travel 100 m?

0.001 s
0.01 s
10 s
1,000 s

Test Prep Short Answer

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True or falseWhile riding a bicycle up a gentle hill, it is fairly easy to increase your potential energy, but to increase your kinetic energy would make you feel exhausted.

True
False

Which statement best explains why running on a track with constant speed at 3 m/s is not work, but climbing a mountain at 1 m/s is work?

At constant speed, change in the kinetic energy is zero but climbing a mountain produces change in the potential energy.
At constant speed, change in the potential energy is zero, but climbing a mountain produces change in the kinetic energy.
At constant speed, change in the kinetic energy is finite, but climbing a mountain produces no change in the potential energy.
At constant speed, change in the potential energy is finite, but climbing a mountain produces no change in the kinetic energy.

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Test Prep Extended Response

Work can be negative as well as positive because an object or system can do work on its surroundings as well as have work done on it. Which of the following statements describes:
a situation in which an object does work on its surroundings by decreasing its velocity and
a situation in which an object can do work on its surroundings by decreasing its altitude?

A gasoline engine burns less fuel at a slower speed.
Solar cells capture sunlight to generate electricity.
A hybrid car charges its batteries as it decelerates.
Falling water turns a turbine to generate electricity.
Airplane flaps use air resistance to slow down for landing.
Rising steam turns a turbine to generate electricity.
An electric train requires less electrical energy as it decelerates.
A parachute captures air to slow a skydivers fall.

A boy is pulling a girl in a childs wagon at a constant speed. He begins to pull harder, which increases the speed of the wagon. Which of the following describes two ways you could calculate the change in energy of the wagon and girl if you had all the information you needed?

Calculate work done from the force and the velocity.
Calculate work done from the change in the potential energy of the system.
Calculate work done from the force and the displacement.
Calculate work done from the change in the potential energy of the system.
Calculate work done from the force and the velocity.
Calculate work done from the change in the kinetic energy of the system.
Calculate work done from the force and the displacement.
Calculate work done from the change in the kinetic energy of the system.

Glossary

energy: the ability to do work
gravitational potential energy: energy acquired by doing work against gravity
joule: the metric unit for work and energy; equal to 1 newton meter (N∙m)
kinetic energy: energy of motion
mechanical energy: kinetic or potential energy
power: the rate at which work is done
potential energy: stored energy
watt: the metric unit of power; equivalent to joules per second
work: force multiplied by distance
work–energy theorem: states that the net work done on a system equals the change in kinetic energy

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