# Work

So far we have thought about decreasing during collisions by increasing the time over which the force is applied. Instead of using the , which combines average force and time over which it is applied:

(1)

We can instead combine average force with the distance over which it is applied to define a quantity known as the .

(2)

The tells us whether the work is transferring into or out of a particular object:

- A applied to an object in the opposite direction to its motion will tend to slow it down, and thus would transfer out of the object. With energy leaving the object, the done on the object should be negative. The angle between the object’s motion and the force in such a case is 180° and , so that checks out.
- A applied to an object in the same direction to its motion will tend to cause it to speed up, and thus would transfer in to the object. With energy entering the object, the done on the object should be positive. The angle between the object’s motion and the force in such a case is 0° and so that also checks out.
- Finally, if a force acts to an objects motion it can only change its direction of motion, but won’t cause it to speed up or slow down, so the kinetic energy doesn’t change. That type of force should do zero . The angle between the object’s motion and the force in such a case is 90° and so once again, the in the work equation gives the required result. For more on this particular type of situation read the chapter on weightlessness at the end of this unit.

We see that and both have units of **Nm** (or **J**). Work is a quantity of , however is not a type of energy. Rather, work is an amount of energy transferred by the application of a force over a distance. *Doing work is the act of transferring energy from one form to another and/or one object to another*. The sign of the work indicates if energy is coming in or going out, rather than indicating a direction in space like the signs on a vector such as and . Therefore, we have not made work (*W*) **bold** in the previous equation because it is not a vector. Also, when calculating work the accounts for the force direction so we only use the size of the force (*F*) in the equation, which is why we have not made force **bold** either.

### Everyday Example

Let’s apply this concept to our crumple zone example. The crumple zone increases the distance over which the on the car, from the wall, is applied. That force on the car points in the opposite direction of its motion, thus it will be a negative that transfers out of the car (no work is “absorbed” here, just transferred from to another type). In order to stop, the *KE* of the car must change from its initial value to zero, therefore the amount of work that needs to be done is equal to the total amount of *KE* the car had to begin with. The initial *KE* is determined by the and of the car, but that * can be done by a larger force over a shorter distance or a smaller force over a longer distance*. The crumple zone ensures the force is applied over a longer distance thus smaller force. (If the car didn’t crumple at all, but only received a small dent, then would only be a few centimeters).

The equation gives the correct work done by a force, no matter the angle between the direction of force and the direction of motion, even if the force points off at some angle other than 0°, 90°, or 180°. In such a case, some part of the force will be doing work and some part won’t, but the tells us just how much of the force vector is contributing to work.

### Reinforcement Exercises

How much is done by the larger child pulling the smaller one for a distance of 30.0 **m** in a wagon as shown?

- Adapted from Insuknawr (Rod Pushing Sport by H. Thangchungnunga [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], from Wikimedia Commons ↵
- "This work" by BC Open Textbooks is licensed under CC BY 4.0 ↵

any interaction that causes objects with mass to change speed and/or direction of motion, except when balanced by other forces. We experience forces as pushes and pulls.

the average force applied during a time interval multiplied by the time interval

A quantity representing the effect of applying a force to an object or system while it moves some distance.

A quantity representing the capacity of an object or system to do work.

energy which a body possesses by virtue of being in motion, energy stored by an object in motion

at an angle of 90° to a given line, plane, or surface

a quantity of speed with a defined direction, the change in speed per unit time, the slope of the position vs. time graph

distance traveled per unit time

a measurement of the amount of matter in an object made by determining its resistance to changes in motion (inertial mass) or the force of gravity applied to it by another known mass from a known distance (gravitational mass). The gravitational mass and an inertial mass appear equal.