Mass in Vertical Circle

Let us consider a body of mass m tied to one end of the string which is fixed at O and it is moving in a vertical circle of radius r about the point O. The motion is circular but is not uniform, since the body speeds up while coming down and slows down while going up. Suppose the body is at P at any instant of time t, the tension T in the string always acts towards 0. The weight mg of the body at P is resolved along the string as mg cos θ which acts outwards and mg sin θ, perpendicular to the string. When the body is at P, the following forces acts on it along the string.

• mg cos θ acts along OP (outwards)
• tension T acts along PO (inwards)

If the velocity of the body at the highest point B is below this critical velocity, the string becomes slack and the body falls downwards instead of moving along the circular path. In order to ensure that the velocity vB at the top is not lesser than the critical velocity rg , the minimum velocity vA at the lowest point should be in such a way that vB should be rg .

While rotating in a vertical circle, the stone must have a velocity greater than 5gr or tension greater than 6mg at the lowest point, so that its velocity at the top is greater than gr or tension > 0. An aeroplane while looping a vertical circle must have a velocity greater than 5gr at the lowest point, so that its velocity at the top is greater than gr. In that case, pilot sitting in the aeroplane will not fall.

When a vehicle goes round a level curved path, it should be acted upon by a centripetal force. While negotiating the curved path, the wheels of the car have a tendency to leave the curved path and regain the straight-line path. Frictional force between the tyres and the road opposes this tendency of the wheels. This frictional force, therefore, acts towards the centre of the circular path and provides the necessary centripetal force.