$A$ car is moving along a circular path having a coefficient of friction $0.5$ and a radius of curvature $16.2 \,m$. The maximum velocity of the car that can travel without skidding outwards is (Acceleration due to gravity $= 10 \,ms^{-2}$)

What is the maximum safe speed of a car on a flat road of radius $40 \, m$ in $m \, s^{-1}$? The coefficient of friction between the road and the tires is $0.25$. (Take $g = 10 \, m \, s^{-2}$)

$A$ car is moving on a horizontal circular track of radius $0.2 \, km$ with a constant speed. If the coefficient of friction between the tyres of the car and the road is $0.45$,then the maximum safe speed of the car is ........ $m/s$. [Take $g = 10 \, m/s^2$]

Is the maximum safe speed of a vehicle on a banked curved road higher or lower than on a flat curved road?

$A$ small object placed on a rotating horizontal turntable just slips when it is placed at a distance $4\, cm$ from the axis of rotation. If the angular velocity of the turntable is doubled,the object slips when its distance from the axis of rotation is (in $, cm$)

$A$ 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?