Question 15: 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:

Accepted Answer

$\begin{aligned} & \frac{\mathrm{ML}^2}{3} \ddot{	heta}+\mathrm{K} \cdot \frac{\ell}{2} 	heta \cdot \frac{\ell}{2}+\mathrm{K} \cdot \ell \cdot 	heta \cdot \ell=0 \ & \ddot{	heta}+\frac{15}{4} \frac{\mathrm{k}}{\mathrm{M}} 	heta=0 \ & \ddot{	heta}=-\left(\frac{15}{4} \frac{\mathrm{k}}{\mathrm{M}}ight) 	heta \ & \omega_1=\sqrt{\frac{15 \mathrm{~K}}{4 \mathrm{M}}}\end{aligned}$
$\frac{1}{3} \mathrm{ML}^2 \ddot{	heta}+2 \mathrm{~K} \frac{\mathrm{~L}}{2} 	heta \cdot \frac{\mathrm{~L}}{2}=0$
$\begin{aligned} & \ddot{	heta}+\frac{3}{2} \frac{K}{M} 	heta=0 \ & \ddot{	heta}=-\frac{3}{2} \frac{K}{M} 	heta \ & \omega_2=\sqrt{\frac{3}{2} \frac{K}{M}} \ & \frac{\omega_1}{\omega_2}=\sqrt{\frac{15}{4} 	imes \frac{2}{3}}=\sqrt{\frac{5}{2}}\end{aligned}$

Answer

2

Answer

$\sqrt{2}$

Answer

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

Answer

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

Oscillations

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