For a body moving in a circular path,what is the condition for no skidding if $\mu$ is the coefficient of friction?

$A$ circular road of radius $1000 \, m$ has a banking angle of $45^\circ$. The maximum safe speed of a car having mass $2000 \, kg$ will be,if the coefficient of friction between the tyre and road is $0.5$,equal to ....... $m/s$.

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 $= 10 \,ms^{-2}$)

$A$ 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?

$A$ motorcyclist has to rotate in horizontal circles inside the cylindrical wall of inner radius '$R$' metre. If the coefficient of friction between the wall and the tyres is '$\mu_{s}$',then the minimum speed required is ($g=$ acceleration due to gravity).

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?