# Figure below shows a uniform rod of length 2l and mass m. At one end it is connected to a frictionless axis at A…

Q: Figure below shows a uniform rod of length 2l and mass m. At one end it is connected to a frictionless axis at A. The other end is connected to a spring of force constant k. The rod is displaced by a small angle θ from the end connected to the spring and released. What is the frequency of oscillations of the rod ?

(a) $\displaystyle \frac{1}{2\pi} \sqrt{\frac{3k}{m}}$

(b) $\displaystyle \frac{1}{2\pi} \sqrt{\frac{k}{3 m}}$

(c) $\displaystyle \frac{1}{2\pi} \sqrt{\frac{k}{m}}$

(d) $\displaystyle \frac{1}{2\pi} \sqrt{\frac{4 k}{3 m}}$

Ans: (a)

Sol: Resting force in spring when stretched through length x = k x= k(2lθ)  ; where θ is the small angle through which the rod is rotated

Restoring Torque $\displaystyle \tau = – kx \times 2l$

$\displaystyle \tau = – k(2l\theta) \times 2l$

$\displaystyle \tau = – 4k l^2 \theta$

As $\displaystyle \tau = I \alpha$

$\displaystyle I \alpha = – 4k l^2 \theta$

$\displaystyle (\frac{m(2l)^2}{3})\alpha = – 4k l^2 \theta$

$\displaystyle \alpha = – \frac{3 k}{m} \theta$

$\displaystyle \omega^2 = \frac{3 k}{m}$

$\displaystyle \omega = \sqrt{\frac{3 k}{m}}$

Frequency , $\displaystyle \nu = \frac{2\pi}{\omega}$