Question

A block is kept on a horizontal table. The table is undergoing simple harmonic motion of frequency 3 Hz in a horizontal plane. The coefficient of static friction between the block and the table is 0.72. The maximum amplitude in cm of the table for which the block does not slip on the surface of the table is ___ . Take $g=10m{s}^{-2}$ .

Answer 1

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 negative x-axis. Consequently, the force of friction $\mu mg$ 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 is such that $\mu gm\ge ma$ or $\mu g\ge a$

Therefore, for no slipping, the table can have a maximum acceleration $a_{max}=\mu g$. Where $a_{max}={\omega }^{2}A$. Therefore, the maximum amplitude can be found as follows,
$w^2{A}_{max}=\mu g$

or $A_{max}=\frac{\mu g}{2^2}=\frac{\mu g}{4{\pi }^2{v}^2}(\because w=2\pi v)$

$=\frac{0.72\times 10}{4{\pi }^2\times (3{)}^2}=0.02m=2cm$