Question

A meter strick is placed vertically at origin on a smooth surface. A gentle push in $+x$ direction is given to top most point, when it has fallen completely, find the x-coordinate of COM of rod.

Answer 1

Conservation of energy $mg\frac{l}{2}=\frac{1}{2}I{\omega }^2$

$mg\frac{l}{2}=\frac{1}{2}\frac{m{l}^2}{3}{\omega }^2$

$\omega =\sqrt{\frac{3g}{l}}$

Velocity $v=\omega \frac{l}{2}$

=$\frac{1}{2}\sqrt{3gl}$

Time taken to turn $\frac{\pi }{2}$ rad.

$T=\frac{\pi }{2\omega }=\frac{\pi }{2\sqrt{\frac{3g}{l}}}$

Now x-distance moved by velocity v,

$x=vt=\frac{1}{2}\sqrt{3gl}\times \frac{\pi }{2\sqrt{\frac{3g}{l}}}$

x-ccordinate of COM $x=\frac{\pi l}{4}$