A plank of mass m is kept on rest on a solid cylinder of mass m and radius R. The plank is connected to two identical springs of spring constant 'k' which in turn are attached to the walls as shown. The time period of small oscillations of the plank is \pi \sqrt{\frac{11 m}{Nk}}. Assume pure rolling of cylinder and no slipping condition between plank and the cylinder, suggest value of N.

Engineering

Physics

Simple Harmonic Motion

Question

A plank of mass m is kept on rest on a solid cylinder of mass m and radius R. The plank is connected to two identical springs of spring constant 'k' which in turn are attached to the walls as shown. The time period of small oscillations of the plank is $π \sqrt{\frac{11 m}{Nk}}$ . Assume pure rolling of cylinder and no slipping condition between plank and the cylinder, suggest value of N.

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$f_{1} \times 2 R = \frac{- 3}{2} {mR}^{2} α$

$f_{1} - 2 kx = m \times 2 Rα$

$f_{1} = \frac{- 3}{4} mRα = \frac{- 3}{4} ma$

$- 2 kx = \frac{11 ma}{4}$

$a = \frac{- 8 kx}{11 m}$

$\frac{2 π}{T} = ω = \sqrt{\frac{8 k}{11 m}}$

$\frac{π}{T} = \sqrt{\frac{2 k}{11 m}}$

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