Question

A large horizontal surface is in $SHM$ with an amplitude of $1\text{cm}$. A mass of $10\text{kg}$ is kept gently on it when it is at mean position. For the mass to always remain in contact with the surface, the maximum frequency of $SHM$ will be

• A. $5\text{Hz}$
• B. $6\text{Hz}$
• C. $7\text{Hz}$
• D. $8\text{Hz}$

Answer 1

The correct option is A $5\text{Hz}$

We know that for the mass to always remain in contact with the horizontal surface, the mass should move together with the horizontal surface and with the same velocity and acceleration.
The chance of losing the contact is maximum when the horizontal surface is at top extreme position and have maximum acceleration.
From the frame of reference of the horizontal surface, $FBD$ of the mass,
From equilibrium condition,
$N+m{\omega }^{2}A=mg$
At critical condition, $N=0$
$\Rightarrow {\omega }^{2}A=g$
$\Rightarrow \omega =\sqrt{\frac{g}{A}}$
$\Rightarrow 2\pi f=\sqrt{\frac{g}{A}}$
$\Rightarrow f=\frac{1}{2\pi }\sqrt{\frac{g}{A}}$
$\Rightarrow f=\frac{1}{2\pi }\sqrt{\frac{10}{0.01}}=5\text{Hz}$