82 Doing 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)   \begin{equation*} \bold{Impulse} = \Delta t \bold{F_{ave}} \end{equation*}

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

(2)   \begin{equation*} W = Fdcos\theta \end{equation*}

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

  1. 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 cos(180^{\circ}) = -1, so that checks out.
  2. 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 cos(0^{\circ}) = 1 so that also checks out.
  3. 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 cos (90^{\circ}) = 0 so once again, the cos\theta 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.
Two people push inwards on opposite ends of a pole, from the right and from the left. The direction of motion is indicated to the left. Therefore the person on the right, applying a leftward force, is doing positive work. The person on the right is doing negative work.
Insuknawr, or Rod Pushing Sport is an indigenous game of Mizoram, one of the North Eastern States of India. A force applied in the same direction as an objects motion does positive work. A force applied in the opposite direction to motion does negative work. Image adapted from  <a href=”https://commons.wikimedia.org/wiki/File:Insuknawr(Rod_Pushing_Sport).JPG”>Insuknawr (Rod Pushing Sport by H. Thangchungnunga via  Wikimedia Commons


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 costheta 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 cos\theta 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?

A child is sitting inside a wagon and being pulled by a boy with a force F at an angle thirty degrees upward from the horizontal. F is equal to fifty newtons, the displacement vector d is horizontal in the direction of motion. The magnitude of d is thirty meters.
A child pulls another in a wagon, exerting a force at an angle relative to the direction of motion.  “This work”  by BC Open Textbooks



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