Rotational Motion

Olympiad (NSEP) level, single correct, hard

3 of 280

Question 3 of 280

A uniform solid cylinder of mass mm and radius RR rolls without slipping inside a concave parabolic track y=αx2y = \alpha x^2 about the vertex. The time period for small amplitude oscillations is:

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Answer: (A) 2π34αg2\pi \sqrt{\frac{3}{4 \alpha g}}

Radius of curvature at vertex is ρ=1y′′(0)=12α\rho = \frac{1}{y''(0)} = \frac{1}{2\alpha}. For pure rolling of cylinder on curved surface of radius ρ\rho, effective length Leff=32(ρ−R)≈34αL_{eff} = \frac{3}{2}(\rho - R) \approx \frac{3}{4\alpha}. Time period T=2π34αgT = 2\pi \sqrt{\frac{3}{4\alpha g}}.
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