Question

A block is kept on a horizontal table. The table is executing simple harmonic motion of time period T in the horizontal plane. The coefficient of static friction between the block and the table is $\mu$. The maximum amplitude of the table for which the block does not slip on the surface of the table is

• A. $\frac{\mu gT}{2\pi }$
• B. $\frac{\mu g{T}^2}{2{\pi }^2}$
• C. $\frac{\mu g{T}^2}{4{\pi }^2}$
• D. $\mu g{T}^2$

Answer 1

The correct option is C

$\frac{\mu g{T}^2}{4{\pi }^2}$

Refer to Fig. Let the mass of the block be m and let, at a certain instant of time, the direction of acceleration a of the table (executing simple harmonic motion) be along the positive x – direction. As a result, the block will experience a force ma directed along the negative x – axis. Consequently, the force of friction 𝜇𝑚𝑔 will act along the positive x – axis. The weight mg of the block will be balanced by the normal reaction R. The block will not slip on the surface of the table, if the acceleration a of the motion of the table is such that $\mu mg\ge maor\mu g\ge a$

Therefore, for no slipping, the table can have a maximum acceleration $a_{max}=\mu g.$ We know that, for a simple harmonic motion, $a_{max}={\omega }^{2}A.$ where $\omega$ is the angular frequency and A the amplitude of the motion of the table. Therefore the maximum alplitude is given by

${\omega }^2{A}_{max}=\mu g$

or $A_{max}=\frac{\mu g}{{\omega }^2}=\frac{\mu g{T}^2}{4{\pi }^2}$