Four identical rods, each of mass m are welded at their ends to form a square, and the corners are then welded to a light metal hoop of radius r. The rigid assembly of rods and hoop is allowed to roll down the inclined rough surface. If the minimum value of the coefficient of static friction which will prevent slipping is $\frac{k}{10}$ , then find the value of k.

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Detailed Solution

Consider rotational motion about instantaneous point of rotation O.
$4 mgsin θ \times r = [\frac{8}{3} {mr}^{2} + 4 {mr}^{2}] α \Rightarrow 4 mgrsin θ = \frac{20}{3} {mr}^{2} α$

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