A thing long rod of length ‘l’ is hinged at end A and the other end is tied to a string hanging from ceiling as shown in figure. Now the string is burnt. The possibility of breaking the rod is maximum at some section S find the distance of S from end B.

Just after the string is burnt, angular acceleration of the rod

$\mathrm{\alpha }=\frac{\mathrm{mg}.\mathrm{l}/2}{\frac{1}{3}{\mathrm{ml}}^2}=\frac{3\mathrm{g}}{2\mathrm{l}}$

Consider an element of length dx at a distance x from the hinge point.

Tangential pseudo force acting on the element

$\mathrm{dF}=\mathrm{\alpha }.\mathrm{x}.\mathrm{dm}=\mathrm{\alpha }.\mathrm{x}.\frac{\mathrm{m}\mathrm{dx}}{\mathrm{l}}$

$\mathrm{Weight}\mathrm{of}\mathrm{the}\mathrm{element}=\mathrm{dm}\times \mathrm{g}=\frac{\mathrm{m}}{\mathrm{l}}\mathrm{dx}.\mathrm{g}$

Bending moment produced by $\mathrm{dF}\mathrm{and}\frac{\mathrm{mg}}{\mathrm{l}}\mathrm{dx}$

about section S, $\mathrm{dM}=\mathrm{dF}\times \left(\mathrm{x}-\mathrm{y}\right)-\frac{\mathrm{mg}}{\mathrm{l}}\mathrm{dx}\times \left(\mathrm{x}-\mathrm{y}\right)$

$\therefore$   Total bending moment at section S,

$\mathrm{M}=\frac{\mathrm{m\alpha }}{\mathrm{l}}{\int }_{\mathrm{y}}^{\mathrm{l}}\left(\mathrm{x}-\mathrm{y}\right)\mathrm{xdx}-\frac{\mathrm{mg}}{\mathrm{l}}{\int }_{\mathrm{y}}^{\mathrm{l}}\left(\mathrm{x}-\mathrm{y}\right)\mathrm{dx}$

$\Rightarrow \mathrm{M}=\frac{\mathrm{m\alpha }}{\mathrm{l}}\left[{\int }_{\mathrm{y}}^{\mathrm{l}}{\mathrm{x}}^2\mathrm{dx}-\mathrm{y}{\int }_{\mathrm{y}}^{\mathrm{l}}\mathrm{x}\mathrm{dx}\right]-\frac{\mathrm{mg}}{\mathrm{l}}\left[{\int }_{\mathrm{y}}^{\mathrm{l}}\mathrm{xdx}-\mathrm{y}{\int }_{\mathrm{y}}^{\mathrm{l}}\mathrm{dx}\right]$

$=\frac{\mathrm{m}}{\mathrm{l}}.\frac{3\mathrm{g}}{2\mathrm{l}}.\left[\frac{\mathrm{l}}{3}\left({\mathrm{l}}^3-{\mathrm{y}}^3\right)-\frac{\mathrm{y}}{2}\left({\mathrm{l}}^2-{\mathrm{y}}^2\right)\right]$

$-\frac{\mathrm{m}}{\mathrm{l}}\left[\frac{1}{2}\left({\mathrm{l}}^2-{\mathrm{y}}^2\right)-\mathrm{y}\left(\mathrm{l}-\mathrm{y}\right)\right]$

The rod may break at section S if M is maximum or if $\frac{\mathrm{dM}}{\mathrm{dy}}=0$

$\Rightarrow \frac{3\mathrm{mg}}{2{\mathrm{l}}^2}\left[-{\mathrm{y}}^2-\frac{{\mathrm{l}}^2}{2}+\frac{3}{2}{\mathrm{y}}^2\right]-\frac{m}{l}\left[-\mathrm{y}-\mathrm{l}+2\mathrm{y}\right]=0$

$\Rightarrow \frac{3}{2{\mathrm{l}}^2}\left[\frac{{\mathrm{y}}^2}{2}-\frac{{\mathrm{l}}^2}{2}\right]-\frac{1}{\mathrm{l}}\left(\mathrm{y}-\mathrm{l}\right)=0$

$\Rightarrow \frac{3}{4{\mathrm{l}}^2}\left(\mathrm{y}-\mathrm{l}\right)\left(\mathrm{y}+\mathrm{l}\right)-\frac{1}{\mathrm{l}}\left(\mathrm{y}-\mathrm{l}\right)=0$

$\Rightarrow \frac{3}{4{\mathrm{l}}^2}\left(\mathrm{y}+\mathrm{l}\right)-\frac{1}{\mathrm{l}}=0$

$\Rightarrow \mathrm{y}=\frac{4\mathrm{l}}{3}-\mathrm{l}=\frac{\mathrm{l}}{3}.$