Mon February 20, 2012 By: Shantanu Bharadwaj

I want to understand about motion in a verticle circle..??

Expert Reply
Mon February 20, 2012

The motion of a mass on a string in a vertical circle includes a number of mechanical concepts. It must satisfy the constraints of centripetal force to remain in a circle, and must satisfy the demands of conservation of energy as gravitational potential energy is converted to kinetic energy when the mass moves downward. The velocity must increase as the mass moves downward from the top of the circle.

A particle of mass m is attached to a light and inextensible string. The other end of the string is fixed at O and the particle moves in a vertical circle of radius r equal to the length of the string as shown in the figure.

Consider the particle when it is at the point P and the string makes an angle ? with vertical. Forces acting on the particle are:

        T = tension in the string along its length, and

        mg = weight of the particle vertically downward.


Hence, net radial force on the particle is FR = T - mg cos ?

=>     T - mg cos ? = mv2/R

=>     T = mv2/R + mg cos ?

Since speed of the particle decreases with height, hence tension is maximum at the bottom, where cos ? = 1 (as ? = 0).

=>     Tmax = mv2/R + mg; Tmin = mv'2/R - mg (at the top)

Here,  v' = speed of the particle at the top.


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