Question

A conical pendulum consists of a mass m suspended by a massless string of length l as shown. The mass rotates in a horizontal circle at fixed angular velocity ω so that the string makes a constant angle β with the vertical. Show that the angular velocity of rotation is given by ω = √g/l cos β.

Answer 1

Let m be the mass of the bob of the pendulum, l be the length of the string, T be the tension in the string and r be the radius of the circle. The angle between the string and vertical is . The forces acting on a conical pendulum is shown below.

The necessary centripetal force required for the circular motion is provided by the component of tension T in the x-direction. Therefore,

Also from the figure,

Dividing both the equations, we get

From the figure,

Therefore,

The above expression gives the angular velocity of rotation of the conical pendulum.