Question

A uniform rod AB of length $l$ and mass $m$ hangs from point $A$ in a car moving velocity $V_0$ on an inclined plane as shown in the figure. The rod can rotate in vertical plane about the axis at point $A$. If the car suddenly stops , the angular speed with which the rod starts rotating is

• A. $\frac{3}{2}\frac{V_0}{l}\cos \theta$
• B. $\frac{V_0}{2}\frac{\cos \theta }{l}$
• C. $\frac{3}{2}\frac{V_0}{l}\sin \theta$
• D. $\frac{5}{2}\frac{V_0}{l}\sin \theta$

Answer 1

The correct option is A $\frac{3}{2}\frac{V_0}{l}\cos \theta$

In the given situation, no external torque is applied.

initial momentumm of rod $L_i=m{V}_{0}r$
where, $r=\frac{l}{2}cos\theta L_i=m{V}_0\frac{l}{2}cos\theta$
Final momentum of rod $L_f=I\omega$

calculation of moment of inertia $I=\frac{L^2}{12}+m(\frac{l}{2}{)}^{2}I=\frac{m{L}^2}{3}$
$L_f=\frac{m{L}^2}{3}\times \omega$

Using conservation of anguar momentum $L_I=L_F$
$m{V}_0\frac{l}{2}cos\theta =\frac{m{L}^2}{3}\times \omega$
$\omega =\frac{3}{2}\frac{V_0}{l}cos\theta$