Vertical circular motion

Vertical Circular Motion:

Two types of vertical circular motions are commonly observed in practice:

(a) A controlled vertical circular motion such as a giant wheel or similar games. In this case the speed is either kept constant or not totally controlled by gravity.

(b) Vertical circular motion controlled only by gravity. In this case, we initially supply the necessary energy (mostly) at the lowest point. Then onwards, the entire kinetics is governed by the gravitational force. During the motion, there is interconversion of kinetic energy and gravitational potential energy. 

Point Mass Undergoing Vertical Circular Motion Under Gravity:

Case I: Mass tied to a string: The figure shows a bob tied to a massless, inextensible and flexible string. It is whirled along a vertical circle so that the bob performs a VCM and the string rotates in a vertical plane. At any position of the bob, there are only two forces acting on the bob:

(a) its weight mg, vertically downwards, which is constant and

(b) the force due to the tension along the string, directed along the string and towards the centre. Its magnitude changes periodically with time and location.

As the motion is non uniform, the resultant of these two forces is not directed towards the center except at the uppermost and the lowermost positions of the bob. At all the other positions, part of the resultant is tangential and is used to change the speed. 

Arbitrary positions: Force due to the tension and weight are neither along the same line, nor perpendicular. Tangential component of weight is used to change the speed. It decreases the speed while going up and increases it while coming down. 

Case II: Mass tied to a rod: Consider a bob tied to a rigid rod and whirled along a vertical circle. The basic difference between the rod and the string is that the string needs some tension at all the points, including the uppermost point. Thus, a certain minimum speed is necessary at the uppermost point in the case of a string. In the case of a rod, as the rod is rigid, such a condition is not necessary. Thus (practically) zero speed is possible at the uppermost point. Therefore, 

Watch this video to get a clearer understanding of VCM:

Sphere of Death:

During this popular circus show, two-wheeler rider (or riders) undergo rounds inside a hollow sphere. Starting with small horizontal circles, they eventually perform revolutions along vertical circles. The dynamics of this vertical circular motion is the same as that of the point mass tied to the string, except that the force due to tension T is replaced by the normal reaction force N. The linear speed is more for larger circles but angular speed (frequency) is more for smaller circles (while starting or stopping). This is as per the theory of conical pendulum. 

Vehicle at the Top of a Convex OverBridge:

Figure shows a vehicle at the top of a convex over bridge, during its motion (part of vertical circular motion). Forces acting on the vehicle are:

(a) Weight mg and

(b) Normal reaction force N, both along the vertical line (topmost position).

The resultant of these two must provide the necessary centripetal force if the vehicle is at the uppermost position. Thus, if v is the speed at the uppermost point, 

In order to gain a better understanding of the sphere of death, watch this video: