Question

the angle made by a cyclist with the vertical while while taking a circular turn of radius R also state the force which are providing centripetal acceleration and maintain vertical equilibrium

Answer 1

To provide the needed centripetal force, the cyclist will make an angle ∅ with the ground in reaction R. The reaction R will be a force acting on the cyclist at an angle ∅ to the vertical.

So, the vertical component R cos ∅ that has been made by the normal reaction R will help in balancing the weight mg of the cyclist. So, the horizontal force indicated by R sin ∅ will provide the cyclist with all the needed centripetal force.

The picture has been attached to show the detailed angles.

Answer 2

While remaining upright may be the primary goal of beginning riders, a bike must lean in order to maintain balance in a turn: the higher the speed or smaller the turn radius, the leaner is required. This balances the roll torque of the wheel contact patches generated by centrifugal force due to the turn when a body moves in a circular motion with constant speed, it has acceleration because of the direction of the velocity changes.

The frictional component is directed forward when the car is moving with constant velocity or when it is accelerating forward.