Question

Two identical thin rods each with mass m and length L are joined at right angles to form an L-shpaed object This object is balanced on top of a sharp edge as shown in the figure if the L-shaped object is deflected slightly it oscillates The frequency of oscillation is

• A. $\frac{1}{2\pi }\sqrt{\frac{3g}{\sqrt{2}L}}$
• B. $\frac{1}{2\pi }\sqrt{\frac{3g}{4\sqrt{2}L}}$
• C. $\frac{1}{2\pi }\sqrt{\frac{3g}{2\sqrt{2}L}}$
• D. $\frac{1}{2\pi }\sqrt{\frac{3g}{L}}$

Answer 1

Answer: (C)

$\frac{1}{2\pi }\sqrt{\frac{3g}{2\sqrt{2}L}}$

If L- shaped object is slightly displaced by an angle ${}^{\prime }{\theta }^{\prime }$then

${\tau }_0=mg\frac{L}{2}\left\{\sin \right({45}^{\circ }-\theta )-\sin ({45}^{\circ }+\theta \}=mg\frac{L}{2}\{-\frac{2}{\sqrt{2}}\sin \theta \}$

$2\left(\frac{m{L}^2}{3}\right)\alpha =-\frac{mgL}{\sqrt{2}}\theta$

$\alpha =-\frac{3g}{2\sqrt{2L}}\theta$

$\omega =\sqrt{\frac{3g}{2\sqrt{2L}}}$