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

Choose a small element of length `dy`, its mass
`dm-m/ldy`
The potential energy of the element
`dU=(dm)gy`
The potential energy of the entire rod,
`U=underset0oversetlint(dm)gy=m/lgunderset0oversetlintydy=m/lg|y^2/2|_0^l=(mgl)/(2)`