End B of a thin rod of length ‘l’ is hinged so that it can swing in a vertical plane. ‘S’ is a stopper at a distance ‘x’ below the hinge B. Initially the rod is held in horizontal position by an external agent as shown in figure. After the rod is released it swings in a vertical plane and when it becomes vertical it strikes the stopper S and stops. If during the course of impact the horizontal component of reaction at the hinge B is zero, find x.

 

During the course of impact moment of all external forces about point S is zero and hence angular momentum of the rod about S is conserved.

Just after the impact angular momentum of the rod about S, L_f = 0.

Linear momentum of the element of length dr is given by, 

$\mathrm{dP}=\mathrm{dm}\times \mathrm{V}=\left(\frac{\mathrm{m}}{\mathrm{l}}\mathrm{dr}\right)\left(\mathrm{\omega r}\right)=\frac{\mathrm{m\omega }}{\mathrm{l}}\mathrm{rdr}$

Therefore angular momentum of the rod about S before impact is given by

$L_i={\int }_0^{\mathrm{x}}\left(\mathrm{x}-\mathrm{r}\right)\mathrm{dP}-{\int }_{\mathrm{x}}^{\mathrm{l}}\left(\mathrm{r}-\mathrm{x}\right)\mathrm{dP}$

$=\mathrm{x}{\int }_0^{\mathrm{x}}\left(\frac{\mathrm{m\omega }}{\mathrm{l}}\right)\mathrm{rdr}-{\int }_0^{\mathrm{x}}\left(\frac{\mathrm{m\omega }}{\mathrm{l}}\right){\mathrm{r}}^2\mathrm{dr}-{\int }_{\mathrm{x}}^{\mathrm{l}}\left(\frac{\mathrm{m\omega }}{\mathrm{l}}\right){\mathrm{r}}^2\mathrm{dr}+\mathrm{x}{\int }_{\mathrm{x}}^{\mathrm{l}}\left(\frac{\mathrm{m\omega }}{\mathrm{l}}\right)\mathrm{rdr}$

$=\left(\frac{\mathrm{m\omega }}{\mathrm{l}}\right)\left[\frac{{\mathrm{x}}^3}{2}-\frac{{\mathrm{x}}^3}{3}-\frac{{\mathrm{l}}^3}{3}+\frac{{\mathrm{x}}^3}{3}+\frac{{\mathrm{xl}}^2}{2}-\frac{{\mathrm{x}}^3}{2}\right]$

$=\frac{\mathrm{m\omega }}{\mathrm{l}}\left[\frac{{\mathrm{xl}}^2}{2}-\frac{{\mathrm{l}}^3}{3}\right]$

Now   $L_i={\mathrm{L}}_{\mathrm{f}}\Rightarrow \frac{\mathrm{m\omega }}{\mathrm{l}}\left(\frac{{\mathrm{xl}}^2}{2}-\frac{{\mathrm{l}}^3}{3}\right)=0\Rightarrow \mathrm{x}=\frac{2\mathrm{l}}{3}$