Motion in a Vertical Circle

Motion in a Vertical 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 θ 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 F_R = T - mg cos θ

=>     T - mg cos θ = mv²/R

=>     T = mv²/R + mg cos θ

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

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

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

Critical Velocity

It is the minimum velocity given to the particle at the lowest point to complete the circle.

The tendency of the string to become slack is maximum when the particle is at the topmost point of the circle.

At the top, tension is given by T = mv_T²/R - mg, where v_T = speed of the particle at the top.

=>   mv_T²/R  = T + mg

For v_T to be minimum, T ≈ 0 => v_T = √gR

If v_B be the critical velocity of the particle at the bottom, then from conservation of energy

        Mg(2R) +  1/2 mv_T²  = 0 + 1/2 mv_B²

As     v_T = √gR => 2mgR + 1/2 mgR = 1/2 mv_B²

=>     v_B = √5gR

Note: In case the particle is attached with a light rod of length l, at the highest point its minimum velocity may be zero. Then, the critical velocity is 2√gl.