A highly rigid cubical block $A$ of small mass $M$ and side $L$ is fixed rigidly onto another cubical block $B$ of same dimensions and of low modulus of rigidly $η$ such that lower 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 face of $A$ . After the force is withdrawn, block $A$ executes small oscillations, the time period of which is given by.

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$2 π \sqrt{M η L}$

$2 π \sqrt{\frac{M}{η L}}$

$2 π \sqrt{\frac{M L}{η}}$

$\sqrt{\frac{M η}{L}}$

answer is B.

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Detailed Solution

Stress = $F / A$

$s h e a r s t r a i n = θ = \frac{x}{L}; η = \frac{stress}{strain} = \frac{F / A}{x / L} \Rightarrow F = \frac{η L^{2}}{L} x \Rightarrow a = \frac{- η L x}{M} \Rightarrow \vec{a} \propto - \vec{x}$

Motion is S.H.M

Standard equation is $a = - ω^{2} x \Rightarrow ω^{2} = \frac{η L}{M}$

$T = \frac{2 π}{ω} = 2 π \sqrt{\frac{M}{η L}}$

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