Question

A uniform cylinder of mass $m$ and radius $R$ is in equilibrium on an inclined plane by the action of a light spring of stiffness $k,$ gravity and reaction force acting on it. If the angle of inclination of the plane is $ϕ,$ find the angular frequency of small oscillation of the cylinder.

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Accepted Answer

Here point of contact $P,$ will be instantaneous center of rotation. Let the cylinder rotates a small angle $k,$ beyond its equilibrium position about an axis passing through $P .$ Here the increase in length of spring will be $2 R θ .$ Thus the increased spring force provides a restoring torque about $P .$

Now writing torque equation about $P$ as: $τ_{P} = I_{p} α$

$(K \times 2 R θ) \times 2 R = [\frac{m R^{2}}{2} + m R^{2}] α \Rightarrow 4 K R^{2} α = \frac{3}{2} m R^{2} α$

or $α = (\frac{8 K}{3 m}) θ$ $α$ is the restoring angular acceleration $\Rightarrow α = - (\frac{8 K}{3 m}) θ$

$α = - ω^{2} θ$ , Therefore, $ω^{2} = \frac{8 K}{3 m}$ $\Rightarrow ω = 2 \sqrt{\frac{2 K}{3 m}}$

B M Sharma Solutions for Chapter: Linear and Angular Simple Harmonic Motion, Exercise 1: ILLUSTRATIONS

B M Sharma Physics Solutions for Exercise - B M Sharma Solutions for Chapter: Linear and Angular Simple Harmonic Motion, Exercise 1: ILLUSTRATIONS

Questions from B M Sharma Solutions for Chapter: Linear and Angular Simple Harmonic Motion, Exercise 1: ILLUSTRATIONS with Hints & Solutions

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