$A$ car moves on a horizontal circular road of radius $16 \ m$ with an increasing speed at a constant rate of $3 \ m \ s^{-2}$. If the coefficient of friction between the road and the tyres is $0.5$, then the speed at which the car will skid is (assume $g = 10 \ m \ s^{-2}$): (in $m \ s^{-1}$)

The coefficient of friction between the tyres and the road is $0.25$. The maximum speed with which a car can be driven round a curve of radius $40 \,m$ without skidding is ........ $ms^{-1}$ (assume $g = 10 \,ms^{-2}$)

$A$ block of mass $m$ is kept on a horizontal turntable at a distance $x$ from the center. If the coefficient of friction between the block and the surface of the turntable is $\mu$,what is the maximum angular speed of the table so that the block does not slip?

$A$ 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

$A$ body of mass $200 \text{ g}$ is tied to a spring of spring constant $12.5 \text{ N/m}$,while the other end of the spring is fixed at point '$O$'. If the body moves about '$O$' in a circular path on a smooth horizontal surface with a constant angular speed of $5 \text{ rad/s}$,then the ratio of the extension in the spring to its natural length will be:

$A$ disc of radius $5\, m$ is rotating with an angular frequency of $10\, rad/s$. $A$ block of mass $2\, kg$ is to be placed on the disc. If the coefficient of friction between the disc and the block is $\mu_s = 0.4$,find the maximum distance from the axis where the block can be placed without sliding (in $cm$).