The Work-Energy Theorem

What is energy?

How is energy different from force?

Copyright  © 2009-2012, by Marcus Milling

edited by Julia Pian

A girl, Sally, pushes a 50kg TV set across a gym floor with a force of 150N.  The TV set is on wheels that are well greased, so it experiences almost no friction.  Her identical twin, Bosh, pushes an identical TV with the same wheels across the floor with an identical force of 150N. 

So far it sounds like two identical situations.  The acceleration of each TV is 3 m/s/s (you should be able to calculate this in your head now). 

However, Bosh pushes the TV this way for 6 seconds, while Sally only pushed the TV this way for 3 seconds.  Now, what is different between the two TV’s motion? 

If they both accelerated at 3 m/s/s, but Bosh’s TV accelerated for twice the time, it should be going twice as fast in the end.  Bosh’s TV ends up going 18 m/s while Sally’s TV ends up going 9 m/s (you should be able to calculate this in your head too). 

Who ‘worked’ up more of a sweat in pushing their respective TVs?  Which TV has more energy in the end?

Figure 1, Bosh and Sally push identical TVs with identical forces, but Bosh pushes the TV for 6 seconds and Sally pushes the TV for only 3 seconds.

Example 5.1.1

Sally’s TV

Bosh’s TV has 4 times the energy of Sally’s TV.

Clearly Bosh’s TV has more energy if it is going faster, since it would be harder for you to stop it.  Bosh imparts more energy to her TV with an identical push, compared to Sally’s, because she pushed her TV for a longer period of time, or we could also say she pushed her TV over a greater distance. 

Bosh could impart even more energy to her TV over the 6 seconds if she pushed with more force than 150N.  The amount of energy gained by the TV depends on two factors: (1)  he amount of force used and (2) the distance over which the force is applied.  We can use this information to build an equation for energy.

                                      Energy = Force  x  distance

If you push with twice as much force, you give the TV twice as much energy, or if you push over twice the distance, you give the TV twice as much energy.  This can be written as

                                                        ∆E = F.d

The ∆ (delta) symbol signifies a change in energy.  ∆E is the energy gained by the TV, but it is also the energy lost by the person.  Bosh must be tired after pushing the TV!  The change in energy (∆E) is also called the work.  Work is energy and a transfer of energy from one place to another.  Usually, the situation is described by drawing an imaginary line around the object or objects of interest and calling them the system.  Everything else (outside our imaginary line) is called the surroundings.  Work is a transfer of energy between the system and the surroundings.  We can also write

                                                        W = F.d

You can calculate the distance each TV traveled during the time it is pushed (it is OK if you can not do this calculation in your head, but check my calculations).  During Bosh’s 6 second push, her TV travels 54.0 meters.  During Sally’s 3 second push, her TV travels 13.5 meters.  We can also calculate the energy gained by each TV.

50 kg

50 kg

13.5 m

54.0 m

Bosh’s TV

Let’s look a bit more closely at the units of energy (N.m).

Remember what a N (Newton) represents.

We replace N in N.m with kg.m/s2.

This is the unit of energy and equivalent to N.m.

These units are often represented with just a J, standing for a Joule, so we have three identical energy units.

Bosh

Sally

Example 5.1.2

Bosh is back, but now she is pushing a new statue of Edger Allen Poe.  She pushes with a force of 230 N, but because the base of the statue has significant friction with the ground, it does not move.  What is the work that Bosh does on the statue?

We will use the work equation,

In this case, the force is 230 N, but the distance is ZERO, because the statue does not move.  This means

Remember the work done is the amount of energy transferred from the system to the surroundings.  The calculation suggests that no energy was transferred from the system (Bosh) to the statue (which is just part of the surroundings).  This result makes sense.  If the statue does not move, it cannot gain kinetic energy.  If the statue does not end up higher off the ground, it did not gain potential energy.  If the statue did not gain energy, Bosh did no work on the statue.

Does this mean Bosh did not lose energy?  No!  Clearly, if you push for a period of time, you will get tired.  Your heart pumps your blood around to your muscles. You may heat up and lose some of this heat to the surroundings.  Since you lost energy, you did do work, just not on the statue.

F

Example 5.1.3

In this example, Bosh and Sally are skating on a frozen pond (uncommon in San Diego, but take my word, they do exist!).  A box was pushed by a friend of theirs and the box is moving across the pond at constant velocity (there is no friction on this ice).  As the box moves across the ice, Bosh and Sally skate up beside it and push from either side with equal but opposite forces.  A top view of this situation is shown below.

If the forces applied to the box are equal and opposite, then the net force is zero, and the box will continue to slide at the constant velocity the box had just before the forces were applied.

The box will not speed up and since they are not lifting the box, it will not gain kinetic energy or potential energy.  No work is done on the box. 

Thus, in the work equation, the distance (d) must be in the same direction as the applied force or no work is done on the object.

As in the previous example, Bosh and Sally do work because they lose energy, but they do no work ON the box, because the box does not gain that energy.

F

F

v

The distance the box traveled in the direction of the force is zero.

No work is done on the box, and no energy is transferred to the box.

the system