Question

A highly rigid cubical block A of small mass M and side L is fixed rigidly onto another cubical block B of the same dimensions and of low modulus of rigidity $\eta$ 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 A small force F is applied perpendicular to one of the side faces of A. After the force is withdrawn block A executes small oscillations.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. $2\pi\sqrt{\frac{M\eta}{L}}$
• B. $2\pi\sqrt{\frac{L}{M\eta}}$
• C. $2\pi\sqrt{\frac{ML}{\eta}}$
• D. $2\pi\sqrt{\frac{M}{\eta L}}$

Answer 1

Answer: (D)

$2\pi\sqrt{\frac{M}{\eta L}}$

Answer 2

Given m = mass = [M], $\eta$ = coefficient of rigidity =

$\lbrack ML^{- 1}T^{- 2}\rbrack$, L = length = [L]

By substituting the dimension of these quantity we can check the accuracy of the given formulae

$\lbrack T\rbrack = 2\pi\left( \frac{\lbrack M\rbrack}{\lbrack\eta\rbrack\lbrack L\rbrack} \right)^{1/2}$= $\left\lbrack \frac{M}{ML^{- 1}T^{- 2}L} \right\rbrack^{1/2}$= [T].

L.H.S. = R.H.S. i.e., the above formula is Correct**.**