Example: Moving a Box

Question:

How much work is done in the following three actions, all performed very slowly and at constant speed?

I lift a (heavy!) box of mass 20 kg from the floor to a height of h = 1 m.
I carry the box all the way across my apartment, a distance of 15 m.
I lower the box from 1 m height back down on the floor.

Answer:

I have to do work against the force of gravity. Since the box moves with constant speed, the force I use to lift it is equal and opposite to the force of gravity. The force of gravity points straight down and is F=-mg; therefore my lifting force must point straight up, F=mg. The displacement vector points straight up, too, because the box is gaining height. This means that the angle between force and displacement vectors is 0°. The work therefore is: W = F h cos(0°) = F h = mg h = (20 kg)(9.81 m/s²)(1 m) = 196.2 J
We might now think that it sure takes more work to lug this heavy box all the way across the apartment than simply lifting it. However, this is not what our equations say. Let's have a closer look: The displacement vector is pointing in horizontal direction; the force I use to hold the box in the air is still pointing straight up; thus the angle between the two is 90°. The work therefore is in this case: W = F d cos(90°) = 0
Now I lower the box. As anybody who is lifting weights can tell you: lowering the weight can be almost as tough as lifting it - in particular if you are doing it slowly. So does this mean that the answer here is the same as in part 1? No. Since I lower the box very slowly, my force still must balance that of gravity and is pointing straight up. The displacement vector now points straight down, and thus the angle between my force and the displacement vector is now 180°. The work therefore is in this case: W = F h cos(180°) = - F h = - mg h = -(20 kg)(9.81 m/s²)(1 m) = -196.2 J

What is going on here? Zero work done carrying a box around? -- even negative work done? Before you loose your faith in the predictive power of physics, a few explanations are in order. First of all and most importantly, we are only dealing with mechanical work in this chapter. There are other forms of energy and work. (In fact, what makes us use up most of our calories in lugging the box around is not the mechanical work, but an increase in our metabolic rate. This will be dealt with in the chapters on temperature, heat, and thermodynamics.)

Negative mechanical work is perfectly fine. In fact, if you add up all three contributions above, you end up with a sum of exactly 0. This is because the box is in the end at the same height that is was in the beginning. If we had put the box on a 1 m high table, then we could have stored the ability to do work by lowering the box down. We will call this stored ability to do work potential energy, and later in this chapter we will learn all about it.