What is the wear and tear on the tires of a vehicle moving at the optimum speed on a banked curved road?

$A$ disc revolves with a speed of $33 \frac{1}{3} \; rev/min$,and has a radius of $15 \; cm$. Two coins are placed at $4 \; cm$ and $14 \; cm$ away from the centre of the record. If the coefficient of friction between the coins and the record is $0.15$,which of the coins will revolve with the record?

$A$ car is moving on a horizontal circular road of radius $0.1 \, km$ with constant speed. If the coefficient of friction between the tyres of the car and the road is $0.4$,then the speed of the car may be ......... $m/s$ $(g = 10 \, m/s^2)$.

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

$A$ vehicle is moving with speed $v$ on a curved road of radius $r$. The coefficient of friction between the vehicle and the road is $\mu$. The angle $\theta$ of banking needed is given by

$A$ plank is resting on a horizontal ground in the northern hemisphere of the earth at a $45^{\circ}$ latitude. Let the angular speed of the earth be $\omega$ and its radius $r_e$. The magnitude of the frictional force on the plank will be