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The statement of the law of conservation of mechanical energy:

"The total mechanical energy of a system is constant, if the internal forces doing work on it are conservative and the external forces do no work."

And this statement is verified by the case of a freely falling body where the energy is always mgh.

How is that possible? Doesn't gravity do work and isn't it an external force?

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2 Answers 2

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One must consider the system of the planet and the falling body, not just the falling body. This is a system with no external forces doing work (more or less), and all internal forces (gravity) are conservative.

Interestingly, if you just consider the falling body and try to integrate the work done by gravity, you'll come up a little shy on energy. That's because as the body falls towards the planet, the planet also falls towards the body. The planet gains a trivial amount of velocity, which accounts for the missing energy from the body on its own. Of course, with the tremendous masses of planets, we hardly notice.

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When you consider the earth (with its gravitational field) and the freely falling body together as the system, then the force of gravity is an internal force and not external force to the system. Then if a body freely falls from a height of $h$ near the surface of the earth where it initially has potential energy of $mgh$, gravity does positive work taking away its gravitational potential energy and giving it kinetic energy equal to $\frac{mv^2}{2}$. Energy is conserved because

$$\frac{mv^2}{2} = mgh$$.

Hope this helps.

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