A car of mass m is moving on a level circular track of radius R. If $$\mu $$_s represents the static friction between the road and tyres of the car, the maximum speed of the car in circular motion is given by

A circular platform is mounted on a frictionless vertical axle. Its radius R = 2 m and its moment of inertia about the axle is 200 kg m². It is initially at rest. A 50 kg man stands on the edge of the platform and begins to walk along the edge at the speed of 1 ms^$$-$$1 relative to the ground. Time taken by the man to complete one revolution is

ABC is an equilateral triangle with O as its centre. $${\overrightarrow F _1},{\overrightarrow F _2}$$ and $${\overrightarrow F _3}$$ represent three forces acting along the sides AB, BC and AC respectively. If the total torque about O is zero then magnitude of $${\overrightarrow F _3}$$ is

A car of mass 1000 kg negotiates a banked curve of radius 90 m on a frictionless road. If the banking angle is 45^o, the speed of the car is