A midpoint of a thin uniform rod AB of mass `m` and length `l` is rigidly fixed to a rotation axle `OO^'` as shown in figure. The rod is set into rotation with a constant angular velocity `omega`. Find the resultant moment of the centrifugal forces of inertia relative to the point C in the reference frame fixed to the axle `OO^'` and to the end.

Explanation

Consider a small element of length `dx` at a distance x from the point C, which is rotating in a circle of radius `r=xsintheta`
Now, mass of the element `=(m/l)dx`
So, centrifugal force acting on this element `=(m/l)dxomega^2sintheta` and moment of this force about C,
`|dN|=(m/l)dxomega^2xsintheta*xcostheta`
`=(momega^2)/(2l)sin2thetax^2dx`
and hence, total moment
`N=2underset(0)overset(l//2)int(momega^2)/(2l)sin2thetax^2dx=(1)/(24)momega^2lsin2theta`,