Question

A rigid cubical block A of mass M and side L is fixed rigidly on to another cubical block of the same dimensions and of modulus of rigidity $n$ 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 A small F is applied perpendicular to one of the side faces of A.After the force is withdrawn, block A executes small oscillations, the time period of which is

• A. $2\pi \sqrt{Mnl}$
• B. $2\pi \sqrt{\frac{Mn}{L}}$
• C. $2\pi \sqrt{\frac{ML}{n}}$
• D. $2\pi \sqrt{\frac{M}{nL}}$

Answer 1

The correct option is D $2\pi \sqrt{\frac{M}{nL}}$

Let the force F produce a deformation x. When this force is whithdrawn, the force that tries to restore block A to its equilibrium position is proportional to x and is given by
$f=-\eta Lx$
$\therefore$ Acceleration $\frac{d^{2}x}{d{t}^2}=\frac{f}{M}=-\frac{\eta L}{M}X=-{\omega }^{2}X$
where $\omega =\sqrt{\frac{\eta L}{M}}$ The angular frequency of this simple harmonic frequency is $\omega$. Now $T=\frac{2T}{\omega }$
Therefore
$T=2\pi \sqrt{\frac{M}{\eta L}}$