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A ball of mass $m$ is moving inside a vertical hollow hoop with radius $L$ and with angular velocity $\omega$.

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I was asked to find a term for the total potential energy of the ball.

Intuitively, I thought of the term $u=mgL(1-cos\theta )$ since when $\theta=0$ I expect it to be $0$ and when $\theta=180$ to be $2mgL$.

Can someone help me understand if and why this term works? how can I get to it with looking on the energy in the system?

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  • $\begingroup$ Hint: the gravitational potential energy near Earth's surface is $U = mg(y-y_0)$, where the constant $y_0$ is chosen arbitrarily. Set up both a Cartesian coordinate grid and a polar coordinate grid that share their origin at the center of the loop. What is $y$ for a particle on the loop expressed in polar coordinates? $\endgroup$ – JM1 Dec 24 '17 at 7:42
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Gravitational potential energy = meh

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