Question

A thin uniform rod of length $l$ and mass $m$ suspended vertically is free to rotate about a smooth horizontal axis which passes through its end $A$ . $A$ point mass $m$ moving horizontally with velocity $v_{0}$ hits the lower end $B$ of the rod and sticks to it. After collision the rod just reaches the horizontal position. Which of the following statements is WRONG?

• A. The value of $v_{0}$ is $\sqrt{4 g l}$
• B. The angular acceleration of the rod when it reaches the horizontal position is $\frac{9 g}{8 l}$
• C. The magnitude of reaction force by the hinge on the rod in horizontal position is $\frac{5 mg}{8 l}$
• D. The magnitude of acceleration of centre of mass of the system (rod $+$ point mass) when the rod is in horizontal position is $\frac{27 g}{32}$ .

Answer 1

Answer: (C)

$mv_{0}l=\frac{4}{3}ml^{2}\omega \Rightarrow \omega =\frac{3 v_{0}}{4 l}$
$\frac{1}{2}\frac{4}{3}ml^{2}\left(\frac{3 v_{0}}{4 l}\right)^{2}=3mg\frac{l}{2}\Rightarrow v_{0}=\sqrt{4 g l}$
$3mg\frac{l}{2}=\frac{4}{3}ml^{2}\alpha \Rightarrow \alpha =\frac{9 g}{8 l}$
$a_{cm}=\alpha r=\frac{9 g}{8 \ell }\times \frac{3 \ell }{4}=\frac{27 g}{32}$
$2mg-F=2m\times \frac{27 g}{32}\Rightarrow F=\frac{5 m g}{16}$