# Challenges in Mechanics by Ronak Agarwal (Part 4)

On a smooth ground a rough sphere of mass $${m}_{1}$$ and radius $${r}_{1}$$ is placed. On this big sphere a small sphere of mass $${m}_{2}$$ and radius $${r}_{2}$$ is placed right at the top as shown in the figure. The system is in unstable equilibrium. Now the equilibrium is disturbed by giving a slight push to the upper sphere.

Now if the upper sphere makes an angle $$\theta$$ with the vertical when it leaves contact with the lower sphere then $$\cos(\theta) = \dfrac{a}{b}$$, find $$a+b$$

Details and Assumptions:

1) There is no friction between ground and the lower sphere.Assume sufficient friction between the two sphere's at all times. ( This assumption may seem a little incorrect since one may argue that as normal is tending to zero there must come a point where friction is insufficient for a finite co-efficient of friction, so you can assume infinite co-efficient of friction)

2) $${m}_{1} = 5 \text{Kg} , {m}_{2} = 7 \text{Kg}, {r}_{1} = 3 \text{m} , {r}_{2} = 1 \text{m}, g=10 m/{s}^{2}$$

3) The sphere's are solid spheres.

4) $$a,b$$ are positive co-prime integers less than $$20$$.

My series of problem Challenges in Mechanics( although only three problems) got quite famous hence I decided it to extend it. hence the fourth part of this series.

Part 1

Part 2

Part 3

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