\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\)

Work

Energy can be transferred from one object to another through two mechanisms. One method is heat, which is when energy flows from a hotter object to a colder object. We’ll talk about that in Heat.

The other mechanism is work, which is the exchange of energy by means of a force. When you give energy to an object, such as when you make a box go faster, then you are doing positive work on the object. Negative work is when you steal energy from an object, such as when you slow down a runaway shopping cart. Work is reciprocal, so if you do positive work on a box (by running into it, for example), the box does negative work on you (by slowing you down); thus it's important to specify which object is receiving the work (the target) and which object is the source of the work.

Work can be calculated using the formula

where

• $F$ is the magnitude of the force,
• $d$ is the distance (not displacement!) the object moves while the force is being applied: If an object doesn't move then you don't do any work on it, even if you are exerting a great effort. (You may be doing work on your muscles however, making them sore.)
• $\theta$ is the angle between the force and the motion. Remember from Trigonometric Functions that the cosine is a measure of how parallel two vectors are.

Consider the three forces on the car in this picture; the car is moving to the right.

• If the force points in the direction of the motion, then $\cos\theta=1$, and the work is simply $W=+Fd$: energy is given to the target. If it points in the same-ish direction, then the cosine is positive, and the work done by the force is positive as well, although smaller than $Fd$.
• If the force points in the opposite direction of the motion, then the work is $W=-Fd$: energy is stolen from the target. If it points in the opposite-ish direction, then the work done by the force still negative, but smaller than $Fd$.
• If the force is perpendicular to the motion then $\cos\theta=0$ and no work is done on the object.

Because work is the only change in energy we will study in this chapter, we can write energy conservation as