The radius of a horizontal curved road is $20 \ m$ and the coefficient of friction between the road and the vehicle's tires is $0.25$. What is the safe speed of the vehicle on this road (in $m/s$)? $(g = 9.8 \ m/s^2)$

For a vehicle moving on a banked curved road,using a free body diagram $(FBD)$,obtain the formula for the maximum safe speed $(v_{max})$.

$A$ gramophone record is revolving with an angular velocity $\omega$. $A$ coin is placed at a distance $r$ from the centre of the record. The static coefficient of friction is $\mu$. The coin will revolve with the record if

$A$ car has to move on a level turn of radius $450\,m.$ If the coefficient of static friction between the tyre and the road is $\mu = 0.2,$ find the maximum speed the car can take without skidding in $m/s.$

$A$ person is driving a vehicle at a uniform speed of $5 \text{ ms}^{-1}$ on a level curved track of radius $5 \text{ m}$. The coefficient of static friction between the tyres and the road is $0.1$. Will the person slip while taking the turn with the same speed? (Take $g = 10 \text{ ms}^{-2}$)

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