Question

A rod of mass $M$ and Length $L$ is hinged at its one end and carries a block of mass m at its lower end. A spring of force constant $k_1$ is installed at distance $a$ from the hinge and another of force constant $k_2$ at a distance $b$ as shown in the figure. If the whole arrangement rests on a smooth horizontal table top, the frequency of vibration is

• A. $\frac{1}{2\pi }\sqrt{\frac{k_1{a}^2+k_2{b}^2}{L^2(m+\frac{M}{3})}}$
• B. $\frac{1}{2\pi }\sqrt{\frac{k_2+k_1}{M+m}}$
• C. $\frac{1}{2\pi }\sqrt{\frac{k_2+k_1\frac{a^2}{b^2}}{4\frac{M}{3}+m}}$
• D. $\frac{1}{2\pi }\sqrt{\frac{k1+\frac{k_2{b}^2}{a^2}}{\frac{4}{3}m+M}}$

Answer 1

The correct option is A $\frac{1}{2\pi }\sqrt{\frac{k_1{a}^2+k_2{b}^2}{L^2(m+\frac{M}{3})}}$

$A$ is the point where the rod is hinged.

Let $\theta$ be the angular displacement from mean position.

The extension of spring connected at point $B$ is $=b\theta$ ($\theta$ is so small that we take it as linear displacement)

So restoring force at point $B=k_{2}b\theta$

Moment of this force at point $A=k_{2}b\theta .b=k_2{b}^2\theta$ ($b$ is the distance of force from $A$)

The contraction of spring connected at point $c$ is:

$=a\theta$

So, restoring force is $=k_{1}a\theta$

and moment of force about $A$ is $k_{1}a\theta .a=k_1{a}^2\theta$ ($a$ is the distance from point $A$ of restoring force)

So, total restoring force $=(k_1{a}^2\theta +k_2{b}^2\theta )\theta$

Now the total moment of inertia of the system about point $A$ is:

$I=$ Moment of inertia about point $A$ +Mooment of inertial of mass $m$ about $A$

$=\frac{M{L}^2}{3}+M{L}^2$

So, the angular acceleration

$\frac{d^2\theta }{d{t}^2}=-\frac{{(k}_1{a}^2+k_2{b}^2)\theta }{I}=-\frac{{(k}_1{a}^2+k_2{b}^2)\theta }{L^2(\frac{M}{3}+M)}$

$\Rightarrow \frac{d^2\theta }{d{t}^2}+\frac{{(k}_1{a}^2+k_2{b}^2)}{L^2(\frac{M}{3}+M)}\theta =0$

So angular frequency $\omega =\sqrt{\frac{{(k}_1{a}^2+k_2{b}^2)}{L^2(\frac{M}{3}+M)}}$

and frequency, $f=\frac{\omega }{2\pi }=\frac{1}{2\pi }\sqrt{\frac{{(k}_1{a}^2+k_2{b}^2)}{L^2(\frac{M}{3}+M)}}$