Q. To mop-clean a floor, a cleaning machine presses a circular mop of radius R vertically down with a total force F and rotates it with a constant angular speed about its axis. If the force F is distributed uniformly over the mop and if coefficient of friction between the mop and the floor is $\mu$, the torque, applied by the machine on the mop is :

Solution:

Consider a strip of radius x & thickness dx, Torque due to friction on this strip.
$\int d\tau =\int^{R}_{0} \frac{x\mu F.2\pi x dx }{\pi R^{2}} $
$\tau = \frac{ 2\mu F}{R^{2}} . \frac{R^{3}}{3} $
$ \tau = \frac{2 \mu FR}{ 3} $
$\therefore$ correct answer is (1)

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