A sphere of mass M and radius R is on a smooth fixed inclined plane in equilibrium as shown in the figure. If now the sphere is displaced through a small distance along the plane, what will be the angular frequency (in radian/sec) of the resulting SHM? (Given k = 4M/3).

Engineering

Physics

Simple Harmonic Motion

Question

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Say the sphere is moved down through a distance x. Then elongation in the spring = x cos 30°
⇒ Restoring force = kx cos² 30° = 3kx/4

$\Rightarrow M \frac{d^{2} x}{{dt}^{2}} = \frac{- 3}{4} kx$

$\Rightarrow ω = \sqrt{\frac{3 k}{4 M}} = 1$

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