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

Answer 1

Answer: (C)

$m{v}_{0}l=\frac{4}{3}m{l}^2\omega \Rightarrow \omega =\frac{3v_0}{4l}$
$\frac{1}{2}\frac{4}{3}m{l}^2{\left(\frac{3v_0}{4l}\right)}^2=3mg\frac{l}{2}\Rightarrow v_0=\sqrt{4gl}$
$3mg\frac{l}{2}=\frac{4}{3}m{l}^2\alpha \Rightarrow \alpha =\frac{9g}{8l}$
$a_{cm}=\alpha r=\frac{9g}{8\ell }\times \frac{3\ell }{4}=\frac{27g}{32}$
$2mg-F=2m\times \frac{27g}{32}\Rightarrow F=\frac{5mg}{16}$