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$$ $$rads-1) of the resulting SHM? Given, $$k=4M3$$

Question

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$$ $$rads-1) of the resulting SHM?  Given, $$k=4M3$$

Infinity Learn · Accepted Answer

If the sphere moves down along the plane by a distance x, extension in the spring = x cos 30^o
$∴$ Restoring force = F cos 30^o

$= (k x \cos 30^{\circ}) \cos 30^{\circ} = k x {(\frac{\sqrt{3}}{2})}^{2} = \frac{3}{4} k x$

As it is restoring in nature $F_{R} = - \frac{3}{4} k x$

$\begin{array}{l} a = - \frac{3}{4} \cdot \frac{k}{M} x \\ ω = \sqrt{\frac{3}{4} \cdot \frac{k}{M}} = \sqrt{\frac{3}{4 M} \cdot \frac{4 M}{3}} = 1 {rads}^{- 1} \end{array}$

Applications of SHM

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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 rads^-1) of the resulting SHM? $(Given, k = \frac{4 M}{3})$

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Applications of SHM

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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 rads-1) of the resulting SHM? Given, k=4M3

If the sphere moves down along the plane by a distance x, extension in the spring = x cos 30o∴ Restoring force = F cos 30o=kxcos⁡30∘cos⁡30∘=kx322=34kxAs it is restoring in nature FR=−34kxa=−34⋅kMxω=34⋅kM=34M⋅4M3=1rads−1

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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 rads^-1) of the resulting SHM? $(Given, k = \frac{4 M}{3})$