A highly rigid cubical block `A` of small mass `M` and side `L` is fixed rigidly on the other cubical block of same dimensions and of 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 After the force is withdrawn , block `A` executes faces of `A`. After the force is withdrawn , block `A` exceutes small oscillations , the time period of which is given by

Options

A. `2 pi sqrt(M eta L)`
B. `2 pi sqrt((M - eta)/(L))`
C. `2 pi sqrt((M - L)/(eta))`
D. 2 pi sqrt((M - N)/(eta L))`

Answer

D

Explanation

When a force is applied on cubical block `A` in the horizontal direction, then the lower block `B` will get distorted as shown by the dotted lines and `A` will attain a new position (without distortion as `A` is a rigid body) as For cubical block `B`,
`eta = (F//A)/(Delta L//L) = (F)/(A) xx (L)/(Delta L) = (F)/(L^(2)) xx (L)/(Delta L) =(F)/(L xx Delta L)`
`rArr F= eta L Delta L`

`eta L` is a constant
`rArr F prop Delta L` and directed towards the mean position
`rArr` oscilation will be simple harmonic in nature. Here,
`M omega^(2) = eta L`
`rArr omega = sqrt((eta L)/(M)) = (2pi)/(T)`
`rArr T=2pi sqrt((M)/(eta L))`.