The maximum safe speed of a car on a horizontal road with a radius of $500\,m$ and a coefficient of friction of $0.5$ is ...... $m/s$.

$A$ body of mass $M \text{ kg}$ is at the top point of a smooth hemisphere of radius $5 \text{ m}$. It is released to slide down the surface of the hemisphere. It leaves the surface when its velocity is $5 \text{ m/s}$. At this instant, the angle made by the radius vector of the body with the vertical is (Acceleration due to gravity $g = 10 \text{ m/s}^2$) (in $^{\circ}$)

$A$ motorcyclist wants to drive in horizontal circles on the vertical inner surface of a large cylindrical wooden well of radius $8.0 \ m$,with a minimum speed of $5 \sqrt{5} \ m \ s^{-1}$. The minimum value of the coefficient of friction between the tyres and the wall of the well must be (take $g = 10 \ m \ s^{-2}$):

$A$ gramophone record is rotating with an angular velocity $\omega$. $A$ coin is placed at a distance $r$ from the center of the record. The coefficient of static friction is $\mu$. The coin will revolve with the record if .......

$A$ car of mass $1500 \ kg$ is moving with a speed of $12.5 \ m/s$ on a circular path of radius $20 \ m$ on a level road. What should be the coefficient of friction between the car and the road,so that the car does not slip?

$A$ circular racetrack of radius $300 \; m$ is banked at an angle of $15^{\circ}$. If the coefficient of friction between the wheels of a race-car and the road is $0.2$,what is the
$(a)$ optimum speed of the racecar to avoid wear and tear on its tyres,and
$(b)$ maximum permissible speed to avoid slipping?