$A$ bead of mass $m$ stays at point $P(a, b)$ on a wire bent in the shape of a parabola $y = 4Cx^2$ and rotating with angular speed $\omega$ (see figure). The value of $\omega$ is (neglect friction).

$A$ $0.5 \ kg$ mass is in contact against the inner wall of a cylindrical drum of radius $4 \ m$ rotating about its vertical axis. The minimum rotational speed of the drum to enable the mass to remain stuck to the wall (without falling) is $5 \ rad/s$. The coefficient of friction between the drum's inner wall surface and mass is . . . . . . . (Take $g = 10 \ m/s^2$)

$A$ 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 $^{\circ}$)

$A$ car turns a corner on a slippery road at a constant speed of $10\,m/s$. If the coefficient of friction is $0.5$,the minimum radius of the arc in meters in which the car turns is:

$A$ $100 \, kg$ car is moving with a maximum velocity of $9 \, m/s$ across a circular track of radius $30 \, m$. The maximum force of friction between the road and the car is ........ $N$.

$A$ motor car has a width of $1.1 \ m$ between its wheels. Its center of gravity is $0.62 \ m$ above the ground and the coefficient of friction between the wheels and the road is $0.8$. What is the maximum possible speed in $m/s$ if the center of gravity moves in a circle of radius $15 \ m$ on a horizontal road surface?