Question

One end of a uniform rod of mass m and length l is clamped. The rod lies on a smooth horizontal surface and rotates on it about the clamped end at a uniform angular velocity $\omega$. The force exerted by the clamp on the rod has a horizontal component
(a) $m{\omega }^{2}l$
(b) zero
(c) mg
(d) $\frac{1}{2}m{\omega }^{2}t$.

Answer 1

(d) $\frac{1}{2}m{\omega }^{2}l$.

Consider a small portion of rod at a distance x from the clamped end (as shown in fig.) with width dx and mass dm.

Centripetal force on this portion =${\omega }^{2}xdm$
$dm=\left(\frac{m}{l}\right)ldx$
Force on the whole rod = F =${\int }_0^l{\omega }^{2}x\frac{m}{l}dx$

$\therefore$ F = $\frac{1}{2}m{\omega }^{2}l$