$A$ car is moving on a circular track banked at an angle of $45^{\circ}$. If the maximum permissible speed of the car to avoid slipping is twice the optimum speed of the car to avoid the wear and tear of the tyres, then the coefficient of static friction between the wheels of the car and the road is

$A$ small mass $m$ rests at the edge of a horizontal disc of radius $R$. The coefficient of static friction between the mass and the disc is $\mu$. The disc is rotated about its axis at an angular velocity such that the mass slides off the disc and lands on the floor $h$ meters below. What was its horizontal distance of travel from the point it left the disc?

$A$ child is on a merry-go-round, standing at a distance $2 \ m$ from the centre. The coefficient of static friction between the child and the surface of the merry-go-round is $0.8$. At what maximum angular velocity can the merry-go-round be rotated before the child slips (in $rad/s$)? (Take $g = 10 \ m/s^2$)

$A$ 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 $, cm$)

$A$ large drum having radius $R$ is spinning around its axis with angular velocity $\omega$, as shown in the figure. The minimum value of $\omega$ so that a body of mass $M$ remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass $M$ as $\mu$, is:

$A$ car is crossing a turn at a speed of $10\,m/s$. If the coefficient of friction is $0.5$,then the minimum radius of the turn will be ........ $m$.