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Circular Motion

Uniform circular motion

Let us consider an object of mass m tied to an inextensible string and whirled in a vertical circle of radius r. Then, due to the earth's gravitational field, its velocity varies from its maximum value v2 at the lowest point B to its lowest value v1 at the highest point A.



Let us make the following assumptions:

A = Highest position
B = Lowest position
C = Midway position
v1 = Velocity at highest point A
v2 = Velocity at lowest point B
v3 = Velocity at midway point C
T1 = Tension when object is at A
T2 = Tension when object is at B

The various forces acting on the body when it is at the highest position A are:
(1)   Weight mg acting vertically downward
(2)   Tension T1 acting vertically downward

Net downward force acting on the body = mg + T1
This force will provide the necessary centripetal force for the body to move in the circle.
  T1 + mg = mv12r   T1 = mv12r - mg

The body will perform circular motion only when T1 is greater than or equal to zero. If T1 is less than zero the string will slacken and the body will fall down. The velocity of the body at the highest point can be calculated as:
When (T1 =0)
  mv12r = mg  v1 =  rg   .....1

Total energy of the particle at A = K.E at A + P.E at A
E=12mv12 + 2mgr
Using equation (1) we get:
 E = 12mgr + 2mgr E = 52mgr   .....2
Similarly, various forces acting on the body at B will be
(1)  Tension T2 acting vertically upward
(2)  Weight mg acting vertically downward

Net upward force acting on the body = T2 - mg
This will provide the required centripetal force to the body.
  T2 - mg = mv22r      T2  = mv22r +  mg
This is the tension ac…

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