Why do you integrate to find the potential energy?

I'm looking at the following question:

ImageUploadedByStudent Room1429694174.119062.jpg

I've looked at the solution to it and it says that you need to integrate the force F(x) to find the potential energy. Why is this?

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Reply 1

atsruser

Original post by A Confused Guy

I'm looking at the following question:

ImageUploadedByStudent Room1429694174.119062.jpg

I've looked at the solution to it and it says that you need to integrate the force F(x) to find the potential energy. Why is this?

Posted from TSR Mobile

If you have a mass in, say, a gravitational field, you have to do work to lift it in that field. If you lift a mass by 10 m with your arms, your arms get tired by losing energy, and the lost energy is stored in the gravitational field by virtue of the position of the mass - this is called its potential energy. Your arms get tired as they are pulling a force (in this case weight) through a distance.

If you pull the force through twice the distance, your arms lose twice as much energy; if you pull twice the force through the same distance, your arms lose twice as much energy. So the potential energy, U, that you transfer has the property that

U \propto F, U \propto x \Rightarrow U \propto Fx

which gives

U=Fx

in SI units.

If you move a force,

F(x)

, which can vary with position, through an infinitesimal distance

dx

you do infinitesimal work

dU = F(x)dx

where we assume that

F(x)

is constant over the very short distance. If you move the force

F(x)

through a large distance you can find the total work done by adding up all of the infinitesimal forces:

U= \sum dU = \sum F(x) dx

. Mathematically, it turns out that if you consider the value of that sum as

dx

gets smaller and smaller, then it is precisely equivalent to the value of the integral

U = \int F(x) dx

The result is therefore ultimately due to the mathematical equivalence between the limit of a sum and a related integral.

Reply 2

A Confused Guy

Okay, think I got that. Thanks very much for the help!

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Why do you integrate to find the potential energy?

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