Net Work Done When Lifting an Object at a constant speed

Question

I am confused about the amount of work done when lifting an object at a constant speed. If you find the work done by you on the object and the work done by gravity on the object and add them the net work would be 0. How is there an increase in Potential Energy if the net work done on the object is 0? I was told that 0 was the incorrect answer and the net work should be equal to the potential energy. Which answer is correct?

Answer 1

If you lift a heavy box off of the floor and place it on a high shelf you have done work to increase the box's gravitational potential energy. The net work you have done against gravity is not zero. If the box later falls off of the shelf this potential energy is converted to kinetic energy as gravity accelerates it downwards.

Answer 2

I will begin from a mathematical perspective. Perhaps this will clear the confusion: the Net Work, $W_{\rm net}$, is defined as the sum of all works, and is equal to the change in KE, as follows:

$$W_{\rm net} = \sum_iW_i = \Delta \rm KE$$

Now in your case, you have 2 forces: the force of gravity $\vec{F}_g$ and the force you apply $\vec{F}_{\rm app}$. Each of these forces will do some work, which I will denote $W_{\rm gravity}$ and $W_{\rm you}$ respectively. These two works, by our above formula, will sum to the Net work:

$$W_{\rm net} = W_{\rm gravity} +W_{\rm you} = \Delta \rm KE.$$

Since the speed in constant, the KE does not change. Thus, $\Delta \rm KE$ is zero; then we know that the Net Work is zero. (why? because net work = change in KE). We then have:

$$W_{\rm net} = W_{\rm gravity} +W_{\rm you} = 0.$$

From there, it is obvious that

$$-W_{\rm gravity} =W_{\rm you}.$$

Since for any (conservative) force $\Delta {\rm PE_{force}}=-W_{\rm force}$ so then

$$\Delta {\rm PE}_{\rm gravity} =-W_{\rm gravity}=W_{\rm you}.$$

Therefore, the work you put into the system increases the object's gravitational PE.

How is there an increase in Potential Energy if the net work done on the object is 0?

The net work is zero. The work you do is non zero. And the work gravity does is also non zero. Together, the work you do and the work gravity does SUM to 0 (the NET work).

The work you do by bringing the block up is positive, while gravity does negative work while the block is going up. As shown above, $\Delta {\rm PE}_{\rm gravity} =-W_{\rm gravity}=W_{\rm you}.$. Therefore, there is an increase in gravitational PE, which corresponds exactly to the work you put in.

Key is to recognize the following:

• Forces in your system will do work; each force that does work, does a specific amount of work, that is due to that specific force
• The sum of the work by all the forces is equal to the net work
• The net work is equal to the change in KE