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The picture is showing the swinging atwood and a hoop and pulley.

I know the lagrangian for both two, I have no problem with the kinetic energy of both but i couldn't convince myself that for the swinging atwood, the potential energy is : $$V = Mgr - mgr \cos\theta$$ and for the hoop and pulley, the potential energy is : $$V = -Mgr \cos\theta - mgr\theta$$

where $\theta$ is the angle mass $m$ makes with vertical

I'm having the sign problem here. I don't understand why for the atwood we use $+Mgr$ and the hoop we use $-mgr\theta$

When and how do we use the $+$ and $-$ sign?

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In the first case we have $Mgr$ because as $r$ increases the potential energy increases for the block of mass $M$. In the second case we have $-mgR\theta$ because when $\theta$ increases the potential energy decreases for the block of mass $m$.

In general you pick the sign so that the potential energy function behaves how it should given the variables it depends on.

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