$A$ circular path of radius $75 \ m$ is banked at an angle of $\tan^{-1}(0.2)$. If the coefficient of static friction between the tyres of the car and the circular path is $0.1$,then the maximum permissible speed of the car to avoid slipping is (in $m/s$)

$A$ circular racetrack of radius $300 \; m$ is banked at an angle of $15^{\circ}$. If the coefficient of friction between the wheels of a race-car and the road is $0.2$,what is the
$(a)$ optimum speed of the racecar to avoid wear and tear on its tyres,and
$(b)$ maximum permissible speed to avoid slipping?

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

Obtain the formula for the maximum safe speed $(v_{max})$ of a vehicle on a level curved road.

$A$ car sometimes overturns while taking a turn. When it overturns,which wheel leaves the ground first?

If the coefficient of friction between the tires and the road is $\mu$,the maximum safe speed is $10\;m/s$. If the coefficient of friction becomes $\mu' = \frac{\mu}{2}$,what will be the new maximum safe speed?