A highly rigid cubical block B₁ of small mass m and side L is fixed rigidly onto another cubical block B₂ of same dimensions and of low modulus of rigidity $η$ , such that the lower face of B₁ completely covers the upper face of B₂. The lower face of B₂ is rigidly held on a horizontal surface. A small force F is applied perpendicular to one of the side faces of B₁. After force is withdrawn, block B₁ executes small oscillations, the time period of which is $π \sqrt{\frac{α m}{η L}}$ . The value of $α$ is __________.

Difficult

Solution

$η = \frac{F}{A θ}$
$Here, A = L^{2} and θ = \frac{x}{L} .$
$Restoring force is F = - η A θ = - η L x$
$Acceleration is a = - \frac{η L}{m} x$
$\Rightarrow a \propto - x$
$∴ Oscillations are simple harmonic with ω = \sqrt{\frac{η L}{m}}$
$So, T = \frac{2 π}{ω} = 2 π \sqrt{\frac{m}{η L}} = π \sqrt{\frac{4 m}{η L}}$
$\Rightarrow α = 4$

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