Question

A uniform beam of length $L$ and mass $m$ is supported as shown. If the cable at $B$ suddenly breaks, determine;
(a) the acceleration of end $B$
(b) the reaction at the pin support.

Answer 1

Torque about point $P$-

$\frac{3m}{4}\left(\frac{3L}{8}\right)-\frac{m}{4}\frac{L}{8}=I_p\alpha \left(clockwise\right)$

$I_p=\frac{m{L}^2}{12}+m{\left(\frac{L}{4}\right)}^2=\frac{7m{L}^2}{48}$

$\frac{3m}{4}\frac{3L}{8}-\frac{m}{4}\frac{L}{8}=\frac{7m{L}^2}{48}\alpha$

$\alpha =\frac{12g}{7L}$

${\alpha }_B=\alpha \frac{3L}{4}=\frac{98}{7}(v.downward)$

$\frac{RL}{4}=I_c\alpha \Rightarrow \frac{m{L}^2}{12}\alpha$

$R=\frac{4mg}{7}$