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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