$A$ 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 $, cm$)

$A$ boy is sitting on the horizontal platform of a joy wheel at a distance of $5 \, m$ from the center. The wheel begins to rotate and when the angular speed exceeds $1 \, rad/s$,the boy just slips. Find the coefficient of friction between the boy and the wheel. (Take $g = 10 \, m/s^2$)

$A$ car of mass $m$ moves on a banked road having radius $r$ and banking angle $\theta$. To avoid slipping from the banked road,the maximum permissible speed of the car is $v_0$. The coefficient of friction $\mu$ between the wheels of the car and the banked road is:

$A$ cyclist on a level road takes a sharp circular turn of radius $3 \; m$ $(g = 10 \; m \cdot s^{-2})$. If the coefficient of static friction between the cycle tyres and the road is $0.2$,at which of the following speeds will the cyclist not skid while taking the turn?

The radius of a horizontal curved road is $20 \ m$ and the coefficient of friction between the road and the vehicle's tires is $0.25$. What is the safe speed of the vehicle on this road (in $m/s$)? $(g = 9.8 \ m/s^2)$

$A$ particle describes a horizontal circle on the smooth inner surface of a cone as shown in the figure. If the height of the circle above the vertex is $10 \ cm$,find the speed of the particle. (Given: acceleration due to gravity $g = 10 \ m/s^2$ and assume the semi-vertical angle $\theta = 45^\circ$ based on the geometry of the cone). (in $m/s$)