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

We can not write the potential energy of the rod directly. All the points of the rod are situated at different height from the ground. Therefore we divide it into many point masses with potential energy of a small mass being `dU`,
`dU=(dm)gy`
Total P.E. would be the summation of P.E. of all elements frem `y=0` to `y=l`.
`rArr" "int_(0)^(l)"dm g y"`
Mass of small element, of length dy is dm:
`m` is mass of length `l`,
`therefore` mass of unit length `=(m)/(l)`
mass of length dy `=(m)/(l)dy`
`dm=(m)/(l)dy`
`U=gint_(0)^(l)yd.y`
`=(mg)/(l)int_(0)^(l)ydy" "=(mg)/(l)xx((l^(2))/(2))`
`=mg((1)/(2))`
In this case, rather than solving this problem, we could have said that the mass of the rod is concentrated at some height `Y` from the ground. And we shall replace the rod with a point mass at that height.
To determine that height `y`, compare `U` with Mg Y
`U=MgY=mg(1)/(2)`
we get `Y=(1)/(2)`
If we had known this position earlier, we could have solved the problem in no time without integration. This position `Y` is the center of mass of the rod. Such problem would have been really cumbersome for more complicated bodies like ring, disc, sphere, cone etc. but if their centre of mass is known, we could have solved them easily without calculations. Potential energy was an example, there are many more physical quantities of a system of particles which can be calculated by this concept of centre of centre of mass. Now we shall explore the concept of centre of mass in detail calculation of its position and application on physical quantities.