$A$ car is driven on a banked road of radius of curvature $20 \ m$ with maximum safe speed. In order to increase its safe speed by $20 \%$,without changing the angle of banking,the increase in the radius of curvature will be (Assume friction is same on the road). (in $m$)

For a vehicle moving on a banked curved road,using a free body diagram $(FBD)$,obtain the formula for the maximum safe speed $(v_{max})$.

$A$ particle describes a horizontal circle on the smooth inner surface of a cone as shown in the figure. If the height of the circle above the vertex is $10 \ cm$,find the speed of the particle. (Given: acceleration due to gravity $g = 10 \ m/s^2$ and assume the semi-vertical angle $\theta = 45^\circ$ based on the geometry of the cone). (in $m/s$)

$A$ particle on the inner rough surface of a cone rotating about its vertical axis is at rest relative to the cone at a height of $1 \ m$ above its vertex. If the coefficient of friction is $\mu = 0.5$ and the semi-vertical angle of the cone is $45^\circ$,what is the maximum angular velocity $\omega$ of the cone?

$A$ man is standing on a rough $(\mu = 0.5)$ horizontal disc rotating with a constant angular velocity of $5 \ rad/s$. At what distance from the center should he stand so that he does not slip on the disc?

$A$ vehicle is moving with a velocity $v$ on a curved road of width $b$ and radius of curvature $R$. For counteracting the centrifugal force on the vehicle,the difference in elevation required between the outer and inner edges of the road is