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A uniform plank of mass $m = 1$ kg, free to move in the horizontal direction only, is placed at the top of a solid cylinder of mass 2m and radius R. The plank is attached to a fixed wall by means of a light spring of spring constant $k = 7 N / m$ . There is no slipping between the cylinder and the plank, and between the cylinder and the ground. Find the angular frequency of small oscillations of the system in $r a d / s e c$ .

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answer is 2.

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Detailed Solution

Suppose that the plank is displaced from its equilibrium position by $x$ at time $t$ , the centre of the cylinder is, therefore, displaced by $\frac{x}{2}$ .

the mechanical energy of the system is given by,

$E=K.E (Plank) + P.E.(spring) + K.E. (cylinder)$

$E = \frac{1}{2} m {(\frac{d x}{d t})}^{2} + \frac{1}{2} k x^{2} + \frac{1}{2} 2 m {\frac{d}{d t} (\frac{x}{2})}^{2}$

$= \frac{1}{2} (\frac{7}{4} m) {(\frac{d x}{d t})}^{2} + \frac{1}{2} k x^{2}$

After differentiating the total energy and equating it to zero, one finds acceleration = $- ω^{2} x$

The angular frequency $ω = \sqrt{\frac{4 k}{7 m}} = 2$

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