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August 14th, 2018, 04:27 AM   #1
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Calculating Pendulum oscillation Work (Do I have errors in my calculations?)

Please consider the following:


In physics, if all the forces are constant then work is equal to the dot product:
$\displaystyle
\vec{F_{net}} \cdot \vec{d}
$

where Fnet is the sum of all forces, and d the distance that the ball has traveled.

But here the force T from the rope to the ball always change (in direction only) while the ball is changing positions.

So in order to calculate T, I had to calculate its direction every time the position of the ball changes.

This is easy if you know the position you need to look at. And this is where the rope is hanged on the wall (0,h).So if you take the difference (0,h) - (x,y) where (x,y) is the current position of the ball, then you have a vector that always points from the position of the ball towards the wall where the rope is being hanged. Now you can normalize to get only the direction and multiply it with the magnitude of the T force.

So now, in order to find the Work, you need to apply:
$\displaystyle
\vec{F_{net}} \cdot \vec{d}
$

In small segments that they are close to 0 (distance between them) and then sum all these works to get the final total work. This leads to a double integration as you can see on the image.

Am I correct or do my calculations have errors?

Last edited by skipjack; August 16th, 2018 at 03:53 PM.
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August 16th, 2018, 08:51 AM   #2
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Well except the final result, pretend you didn't see how i took -mg outside the integration
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