Question

A rod of mass $M$ and length $l$ attached with a small bob of mass $m$ at its end is freely rotating about a vertical axis passing through its other end with a constant angular speed $\omega$. Find the force exerted by the system (rod and bob) on the pivot. Assume that gravity is absent.

Answer 1

When select a rotating frame of reference with respect to the rod. The centrifugal force experienced by the elementary segment of length $dr$ & mass $\delta m$ at a distance $r$ from the axis is given by the expression $dF=\left(\delta m\right){r\omega }^2\Rightarrow$ The total centrifugal force experienced by the rod$F={\omega }^2\int r\left(\delta m\right)={\omega }^2{\int }_0^{l}r\left(M/l\right).dr\Rightarrow F=\frac{M{\omega }^2}{l}{\left[\left(\frac{r^2}{2}\right)\right]}_0^l=\frac{M{\omega }^{2}l}{2}$

The centrifugal force exerted by the bob$F^{\prime }=ml{\omega }^2$$\therefore$

The total centrifugal force$=F_t=F+F=\frac{M{\omega }^{2}l}{2}+m{\omega }^{2}l$$F_t=(\frac{M}{2}+m){\omega }^{2}l$;

(centrifugal force is a pseudo force being experienced in a non-inertial plane.)

Hence, the required force$=(\frac{M}{2}+m){\omega }^{2}l$