Question

In the arrangement shown in the figure, a block of mass $m$ is rigidly attached to two identical springs of stiffness constant $K$ each. The other ends of the springs are connected to fixed walls. The block is initially at rest and can slide on a frictionless horizontal bar $AB$ when displaced. The arrangement rotates with a constant angular velocity $\omega$ about an axis passing through the middle of the bar. Find the time period of small oscillations of the block.

• A. $T=2\pi \sqrt{\frac{m}{K}}$
• B. $T=2\pi \sqrt{\frac{m}{K-m{\omega }^2}}$
• C. $T=2\pi \sqrt{\frac{m}{{\omega }^2}}$
• D. $T=2\pi \sqrt{\frac{m}{2K-m{\omega }^2}}$

Answer 1

The correct option is D $T=2\pi \sqrt{\frac{m}{2K-m{\omega }^2}}$

Let us analyse the problem relative to the rotating bar $AB$. Assume the block is displaced by distance $x$ to the right. Since the angular velocity is constant, the acceleration of the block is centripetal and a pseudo force will act on the block away from the centre and will be of magnitude $m{\omega }^{2}x$.


The net force acting on the block towards the centre
$F=-2Kx+m{\omega }^{2}x=-(2K-m{\omega }^2)x$
Comparing this with $F=-K^{\prime }x$, we get
$K^{\prime }=2K-m{\omega }^2$
$\therefore$ Time period of oscillation of block is given by
$T=2\pi \sqrt{\frac{m}{K^{\prime }}}\Rightarrow T=2\pi \sqrt{\frac{m}{2K-m{\omega }^2}}$
Thus, option (d) is the correct answer.