Question

Find the natural frequency of oscillation of the system as shown in the figure. Pulleys are massless and frictionless. Spring and string are also massless. (Take ${\pi }^2=10$) :

• A. $\frac{\pi }{2}Hz$
• B. $\sqrt{\pi }Hz$
• C. $\frac{10}{\sqrt{\pi }}Hz$
• D. $\pi Hz$

Answer 1

The correct option is D $\pi Hz$

Let the mass be displaced by a distance $x$ and the force exerted on the mass by the lower pulley be $F$.

Since the pulley is massless, net moment on pulley is zero. Thus, tension in each string of the lower pulley is $F/2$

Thus, force experienced by the lower pulley is $F/2$, for which the spring and string contribute half. Thus, force in spring is $F/4$.

But, if the mass goes down by $x$, then lower pulley goes down by $x$.

Thus, the upper pulley goes down by $2x$.

Thus, spring stretches by $4x$.

Thus, $k\times 4x=F/4\Rightarrow F=16kx$

Thus, acceleration of the mass is $-16kx/m=-(16\times 25/1)x=-400x$

Thus, the mass exhibits an SHM with $\omega =\sqrt{400}=20\text{rad/s}$

Thus, frequency $f=\omega /\left(2\pi \right)=20/\left(2\pi \right)=10/\pi =\pi \text{Hz}$