$A$ motorcycle is travelling on a curved track of radius $500\,m$. If the coefficient of friction between the road and tyres is $0.5$,the maximum speed to avoid skidding will be ....... $m/s$. (Take $g = 10\,m/s^2$)

The maximum tension which an inextensible ring of mass $0.1\, kg/m$ can bear is $10\,N$. The maximum velocity in $m/s$ with which it can be rotated is ........ $m/s.$

$A$ hollow vertical cylinder of radius $R$ is rotated with angular velocity $\omega$ about an axis through its center. What is the minimum coefficient of static friction necessary to keep the mass $M$ suspended on the inside of the cylinder as it rotates?

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?

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

The maximum speed of a car on a road turn of radius $30\,m$; if the coefficient of friction between the tyres and the road is $0.4$; will be ........ $m/s$. (in $.84$)