# How to understand the work-energy theorem?

How to understand the work-energy theorem?

I took a short lecture on physics for engineering last week. The lecturer emphasized that the work done on an object will cause the kinetic energy change as

$$W = \Delta \text{KE}.$$

I know this concept might be so common to you but to me, as a beginner, it is pretty hard to understand the reason. My understanding is that 'work' is the energy an external object 'injects into' the object or is the energy an external object 'takes away' from the object. I think the work done by the object should equal to the total energy changed on that object, which could be in any form (heat, potential or kinetic energy.) Why does the theorem only explicitly refer to kinetic energy? Will this theorem work in some cases or in all cases?

The total work can be split up into two parts:

$$W_{net} = W_{conservative}+W_{non-conservative}.$$

With the conservative part you can associate a potential energy:

$$W_{conservative}=-\Delta PE$$

(this is in fact the definition of a conservative force) so that the Work-Energy theorem becomes

$$W_{non-conservative}=\Delta KE + \Delta PE = \Delta E.$$

This is another way of writing the Work-Energy theorem and in my mind it's a little bit clearer. Restated, the work done by non-conservative forces is equal to the overall change in energy of the system.

For example, work done by friction is negative, so it dissipates energy away from a system.

On the other hand, gravity is a conservative force. Imagine the motion of a falling ball. Unless something doing work on the ball to slow it down (for example, air) the ball will speed up as it falls. In this case, the equation

$$W_{gravity} = -\Delta PE = \Delta KE$$

is equivalent to that statement. (As the potential energy becomes more negative, kinetic energy becomes more positive.)

• thanks Alec. I am still a bit confusing on that. From your statement, should I say the conservative force will only change the potential energy but not the kinetic energy? I know that gravity is a conservative force, so if I freely drop an object from high place (ignore the friction), the gravity will do the work on the object so to lower the potential energy but its speed will change too (so does the kinetic energy). So how to understand this? Commented Mar 26, 2013 at 20:42
• I've been editing this to make it more clear. I have added an equation to the end illustrating the connection between gravitation work, potential energy and kinetic energy. Commented Mar 26, 2013 at 20:46
• by the way, in some books, some cases, I find that they will put the internal energy change in the work-energy theorem too. So in what situation I should consider the internal energy and what's the reason to cause the change of internal energy? If some case it said the internal energy of the system changed, can I said there must be some non-conservative force applied on the system? thanks Commented Mar 26, 2013 at 20:46
• @user1285419 Ok sure. So imagine you throw a ball up in the air. You did work on it, so you increase its kinetic energy. But something happens as it rises -- it slows down! That's because gravity is also doing work on the ball -- negative work, in fact, so that the ball's kinetic energy goes down. But the difference between a force like gravity and one like friction is that you can easily get the energy back from gravity. That's why we call it a conservative force. Commented Mar 28, 2013 at 17:08
• This answer has problems. It does not address the issue of system and environment and does not distinguish between internal and external forces. This accounts for the confusion expressed in many comments. Commented Jan 24, 2021 at 14:58

I think the problem with the Work-Energy theorem is that so few books properly cover the concept of defining a system. If we take the example of Earth and a ball near the surface of Earth as an example, we can ask "What is the total energy of the system?". Obviously, this is the sum of the kinetic energy of Earth, the kinetic energy of the ball, and the gravitational potential energy of the earth and ball. $$E_\mathrm{tot} = K_{\mathrm{earth}} + K_\mathrm{ball} + U_\mathrm{earth-ball}$$

When we apply the Work-Energy theorem to the ball, the earth is no longer part of our system, and therefore the gravitational potential energy is also not part of the definition of the total energy of the system. $$E_\mathrm{tot} = K_\mathrm{ball}$$ Therefore, any change in the gravitational potential energy between Earth and the ball must be considered work being done on the system (which for the Work-Energy theorem is just the ball). Since the only energy term in the total energy in this case is the kinetic energy of the ball, the work done by a change in the gravitational potential energy between Earth and the ball must change the ball's kinetic energy.