In a motorcycle stunt called the "well of death",the track is a vertical cylindrical surface of $18\, m$ diameter. What should be the minimum speed of the motorcyclist in $m/s$ to prevent him from sliding down? The coefficient of friction is $0.8$ and take $g = 10\, m/s^2$.

$A$ long horizontal rod has a bead which can slide along its length,and is initially placed at a distance $L$ from one end $A$ of the rod. The rod is set in angular motion about $A$ with constant angular acceleration $\alpha$. If the coefficient of friction between the rod and the bead is $\mu$,and gravity is neglected,then the time after which the bead starts slipping is

$A$ particle on the inner rough surface of a cone rotating about its vertical axis is at rest relative to the cone at a height of $1 \ m$ above its vertex. If the coefficient of friction is $\mu = 0.5$ and the semi-vertical angle of the cone is $45^\circ$,what is the maximum angular velocity $\omega$ of the cone?

The maximum safe speed of a car on a horizontal road with a radius of $500\,m$ and a coefficient of friction of $0.5$ is ...... $m/s$.

The coefficient of static friction between the road and tyres of a car is $0.4$. The maximum permissible speed of the car is $10 \,ms^{-1}$ on a curved unbanked road. Then the maximum radius of curvature of the road is (acceleration due to gravity $= 10 \,ms^{-2}$)

For a banked curved road,if the velocity of the vehicle $v < v_0$ (where $v_0$ is the optimum speed),what is the direction of the frictional force?