A uniform solid sphere of mass M and radius R rolls without slipping on a rough horizontal surface. Its centre is attached to an ideal horizontal spring of force constant k. Static friction is sufficient to maintain pure rolling. What is the angular frequency of small oscillations?
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Question

A uniform solid sphere of mass M and radius R rolls without slipping on a rough horizontal surface. Its centre is attached to an ideal horizontal spring of force constant k. Static friction is sufficient to maintain pure rolling. What is the angular frequency of small oscillations?

Options

G1 Physics · Accepted Answer

For pure rolling, v = RΩ. Total kinetic energy is K = ½Mv²+½IΩ². For a solid sphere, I = 2MR²/5. Thus, K = ½Mv² +½(2MR²/5)(v²/R²) = ½(7M/5)v². Hence the effective inertial mass is Mₑ = 7M/5. The spring potential energy is U = ½kx². Therefore, ω² = k/Mₑ = 5k/(7M). Thus, ω = √[5k/(7M)]. Correct option: B.