$A$ car is moving along a circular path having a coefficient of friction $0.5$ and a radius of curvature $16.2 \,m$. The maximum velocity of the car that can travel without skidding outwards is (Acceleration due to gravity $= 10 \,ms^{-2}$)

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 the coefficient of friction between the mop and the floor is $\mu$,the torque applied by the machine on the mop is

$A$ cyclist on a level road takes a sharp circular turn of radius $3 \; m$ $(g = 10 \; m \cdot s^{-2})$. If the coefficient of static friction between the cycle tyres and the road is $0.2$,at which of the following speeds will the cyclist not skid while taking the turn?

$A$ block of mass $10\; kg$ is in contact with the inner wall of a hollow cylindrical drum of radius $1\; m$. The coefficient of friction between the block and the inner wall of the cylinder is $0.1$. The minimum angular velocity needed for the cylinder to keep the block stationary when the cylinder is vertical and rotating about its axis will be: ......$rad/s$ $(g = 10\; m/s^2)$

The maximum speed that can be achieved without skidding by a car on a circular unbanked road of radius $R$ and coefficient of static friction $\mu$ is

$A$ large drum having radius $R$ is spinning around its axis with angular velocity $\omega$, as shown in the figure. The minimum value of $\omega$ so that a body of mass $M$ remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass $M$ as $\mu$, is: