$A$ 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 $, cm$)

Find the maximum velocity for skidding for a car moved on a circular track of radius $100 \, m$. The coefficient of friction between the road and tyre is $0.2$.

$A$ 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 $g = 10 \, m/s^2$)

$A$ stone of mass $1\,kg$ is tied to the end of a massless string of length $1\,m$. If the breaking tension of the string is $400\,N$,then the maximum linear velocity the stone can have without breaking the string,while rotating in a horizontal plane,is $.......\,ms^{-1}$.

$A$ circular path of radius $75 \ m$ is banked at an angle of $\tan^{-1}(0.2)$. If the coefficient of static friction between the tyres of the car and the circular path is $0.1$,then the maximum permissible speed of the car to avoid slipping is (in $m/s$)

$A$ vehicle is moving with speed $v$ on a curved road of radius $r$. The coefficient of friction between the vehicle and the road is $\mu$. The angle $\theta$ of banking needed is given by