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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?

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1 Answer 1

up vote 7 down vote accepted

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.)

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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? – user1285419 Mar 26 '13 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. – Pricklebush Tickletush Mar 26 '13 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 – user1285419 Mar 26 '13 at 20:46
In the equation $W_{non-conservative}=\Delta KE + \Delta PE = \Delta E$ you can divide $\Delta E$ into as many parts as you like. It really depends on the context. If you're talking about a chemical reaction, for example, you may want to have an internal energy term, but not if you're talking about pushing blocks on inclined planes. – Pricklebush Tickletush Mar 26 '13 at 20:50
Thanks Alec, I think your explanation sort of giving me some points to start. Thanks again. – user1285419 Mar 26 '13 at 20:52

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