# 7.1: Work

- Page ID
- 2821

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For some, the exciting part of a roller coaster is speeding down; for others it is the anticipation of climbing up. While the coaster is being towed up, it is having work done on it. The work done towing it to the top of the hill becomes potential energy stored in the coaster and that potential energy is converted to kinetic energy as the coaster runs down from the top of the hill to the bottom.

## Work

The word **work** has both an everyday meaning and a specific scientific meaning. In the everyday use of the word, work would refer to anything which required a person to make an effort. In physics, however, work is defined as the force exerted on an object multiplied by the distance the object moves due to that force.

**W=Fd**

In the scientific definition of the word, if you push against an automobile with a force of 200 N for 3 minutes but the automobile does not move, then you have done no work. Multiplying 200 N times 0 meters yields zero work. If you are holding an object in your arms, the upward force you are exerting is equal to the object’s weight. If you hold the object until your arms become very tired, you have still done no work because you did not move the object in the direction of the force. When you lift an object, you exert a force equal to the object’s weight and the object moves due to that lifting force. If an object weighs 200. N and you lift it 1.50 meters, then your work is W=Fd=(200. N)(1.50 m)=300. N m.

One of the units you will see for work is shown above: the Newton meter (Nm). More often, however, units of work are given as the **Joule** (pronounced "jool") in honor of James Prescott Joule, a nineteenth century English physicist. A Joule is a kg·m^{2}/s^{2}.

Example 7.1.1

A boy lifts a box of apples that weighs 185 N. The box is lifted a height of 0.800 m. How much work did the boy do?

**Solution**

W=Fd=(185 N)(0.800 m)=148 N m=148 Joules

Work is done only if a force is exerted in the direction of motion. If the motion is perpendicular to the force, no work has been done. If the force is at an angle to the motion, then the component of the force in the direction of the motion is used to determine the work done.

Example 7.1.2

Suppose a 125 N force is applied to a lawnmower handle at an angle of 25° with the ground and the lawnmower moves along the surface of the ground. If the lawnmower moves 56 m, how much work was done?

**Solution**

The solution requires that we determine the component of the force that was in the direction of the motion of the lawnmower because the component of the force that was pushing down on the ground does not contribute to the work done.

F_{parallel}=(Force)(cos25∘)=(125 N)(0.906)=113 N

W=F_{parallel}d=(113 N)(56 m)=630 J

Getting a piano into a truck isn't easy, so we often use a ramp to lower the amount of required force. Although the force is smaller, the distance we have to apply the force is greater. So in a way, we exchange force for distance. However, the total amount of work done remains the same. Play around with the Ramp and Piano SIM below to learn more:

Interactive Element

## Summary

- Work is the force exerted on an object multiplied by the distance the object moves due to that force.
- The unit for work is called the joule, which is a kg m
^{2}/s^{2}. - If the force is at an angle to the motion, then the component of the force in the direction of the motion is used to determine the work done.

## Review

- How much work is done by the force of gravity when a 45 N object falls to the ground from a height of 4.6 m?
- A workman carries some lumber up a staircase. The workman moves 9.6 m vertically and 22 m horizontally. If the lumber weighs 45 N, how much work was done by the workman?
- A barge is pulled down a canal by a horse walking beside the canal. If the angle of the rope is 60.0°, the force exerted is 400. N, and the barge is pulled 100. m, how much work did the horse do?

## Explore More

The following video introduces energy and work. Use this resource to answer the questions that follow.

- How can mechanical energy be added to or removed from a system?
- How can you use a Force vs Displacement Graph to derive the work done on a system?

## Additional Resources

Study Guide: Energy Study Guide

Videos: Work - Overview, The Work-Energy Theorem - Overview

Real World Applications: The 1015-pound Deadlift, The Pillars of Hercules

Interactive: Roller Coaster