Question

A uniform rod of mass $M$ and length $L$ is held vertically upright on a horizontal surface as shown in Fig. Assuming zero potential energy at the base of the rod, determine the potential energy of the rod.

Answer 1

Mass per unit length of the rod $=\frac{M}{L}$

Potential energy of a small mass dm at a height $x=gxdm$

Now,$dm=\frac{M}{L}dx$

Potential energy of the rod,

${\int }_0^L\frac{Mg}{L}xdx=\frac{Mg}{2L}{x}^2\stackrel{L}{\underset{0}{|}}=\frac{MgL}{2}$