Question 348: Oscillations

Question

As shown in the figures, a uniform rod
OO'
of length
l
is hinged at the point
O
and held in place vertically between two walls using two massless springs of same spring constant. The springs are connected at the midpoint and at the top-end (
O'
) of the rod, as shown in Fig. 1 and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is
f₁
. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2 and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is
f₂
. Ignoring gravity and assuming motion only in the plane of the diagram, the value of
$ \frac{f_1}{f_2} $
is:
![](https://app-content.cdn.examgoal.net/fly/@width/image/jaoe38c1maw4p1b6/a8448357-b3f1-41c9-a27b-ee9822214b90/02ed1c20-3543-11f0-b8a5-69aa313ed27c/file-jaoe38c1maw4p1b7.png?format=png)

Accepted Answer

$\begin{aligned} & \frac{\mathrm{ML}^2}{3} \ddot{	heta}+\mathrm{K} \cdot \frac{\ell}{2} 	heta \cdot \frac{\ell}{2}+\mathrm{K} \cdot \ell \cdot 	heta \cdot \ell=0 \ & \ddot{	heta}+\frac{15}{4} \frac{\mathrm{k}}{\mathrm{M}} 	heta=0 \ & \ddot{	heta}=-\left(\frac{15}{4} \frac{\mathrm{k}}{\mathrm{M}}ight) 	heta \ & \omega_1=\sqrt{\frac{15 \mathrm{~K}}{4 \mathrm{M}}}\end{aligned}$
$\frac{1}{3} \mathrm{ML}^2 \ddot{	heta}+2 \mathrm{~K} \frac{\mathrm{~L}}{2} 	heta \cdot \frac{\mathrm{~L}}{2}=0$
$\begin{aligned} & \ddot{	heta}+\frac{3}{2} \frac{K}{M} 	heta=0 \ & \ddot{	heta}=-\frac{3}{2} \frac{K}{M} 	heta \ & \omega_2=\sqrt{\frac{3}{2} \frac{K}{M}} \ & \frac{\omega_1}{\omega_2}=\sqrt{\frac{15}{4} 	imes \frac{2}{3}}=\sqrt{\frac{5}{2}}\end{aligned}$
![](['https://app-content.cdn.examgoal.net/fly/@width/image/jaoe38c1maxzk0ho/2bb11068-61cc-4b7b-bc68-d5f64e9fe0c2/7ae68bc0-3648-11f0-b4c7-3fd17612738b/file-jaoe38c1maxzk0hp.png?format=png', 'https://app-content.cdn.examgoal.net/fly/@width/image/jaoe38c1maxzmdli/2b4a86e1-f733-4766-8d9b-da8241a1dc24/bca4af60-3648-11f0-b4c7-3fd17612738b/file-jaoe38c1maxzmdlj.png?format=png'])

Answer

2

Answer

$\sqrt{2}$

Answer

$\sqrt{\frac{5}{2}}$

Answer

$\sqrt{\frac{2}{5}}$

Oscillations

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