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A rod of mass m and length l is held ver...

A rod of mass `m` and length `l` is held vertical. Find its gravitational potential energy with respect to zero potential energy at its lowest end.

Text Solution

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Since parts of the rod are at different levels from the horizontal surface, therefore, we have to use the integration to find its potential energy.
Consider a small element of length `dy` at a height `y` from the surface.
Mass of the element is `dm=M/Ldy`
Potential energy of the element is
`dU=(dm)gy=M/Lgydy`
On integrating, we get
`U=(Mg)/(L)underset0oversetLintydy=(Mg)/(L)[y^2/2]_0^L=1/2MgL`
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