Problem 1 - Sliding chain on a sphere

Question:

A chain of length l is placed on a smooth spherical surface of radius $R$ with one of its ends fixed at the top of the sphere. What will be the acceleration of each element of the chain when the upper end is released? It is assumed that the length of the chain $l<\frac{1}{2}\pi R$

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Solution

Each element $dm$ experiences an acceleration $g\sin \theta$ along the tangent of the circle at that point.

The net acceleration would be the total force experienced by the body divided by the total mass of the body.

Therefore:

$$a=\frac{{\int }_0^{\frac{l}{R}}g\sin \theta dm}{M}$$

Substituting $dm=\frac{M}{\frac{l}{R}}d\theta$ in the above equation,

$$a=\frac{Rg}{l}{\int }_0^{\frac{l}{R}}\sin \theta$$ $$a=\frac{Rg}{l}(1-\cos (\frac{l}{R}))$$