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

Explanation

Since the different parts of the rod are at different levels with respect to the horizontal surface, therefore, we have to use the integration method to find its potential energy. Consider a small element of length dy at a height y from the horizontal .
K = Mass per unit length = M/L
Therefore mass of the element is `dm = M/L dy`

`dU = M/L g.ydy `
`U = M/L g int _(0)^(L) y.dy`
` = (Mg)/L ((L^(2))/2 - 0) = (MgL)/2`