A highly rigid cubical block A of small mass M and slide L is fixed rigidly on to another cubical block B of the same dimensions…

Q: A highly rigid cubical block A of small mass M and slide L is fixed rigidly on to another cubical block B of the same dimensions and of low modulus of rigidly η such that the lower face of A completely covers the upper face of B. The lower face of B is rigidly held on a horizontal surface. A small force F is applied perpendicular to one of the side faces of A. After the force is withdrawn, block A executes small oscillations, the time period of which is given by

(a) $\large 2\pi\sqrt{M \eta L}$

(b) $\large 2\pi\sqrt{\frac{M \eta}{L}}$

(c) $\large 2\pi\sqrt{\frac{M L}{\eta}}$

(d) $\large 2\pi\sqrt{\frac{M }{\eta L}}$

Ans: (d)

Sol: Modulus of Rigidity $\large \eta = \frac{F}{A \theta}$

Where $\large \theta = \frac{x}{L}$ and A = L2

Restoring force $\large F = – \eta A \theta$

$\large F = – \eta L x$

Acceleration $\large a = \frac{F}{M} = -\frac{\eta L}{M} x$

In S.H.M , a = -ω2 x

$\large \omega = \sqrt{\frac{\eta L}{M}}$

Time Period $\large T = \frac{2\pi}{\omega}$

$\large T = 2\pi\sqrt{\frac{M }{\eta L}}$