Question

The ball has a mass \(m\) and is attached to the cord of length \(I\). The cord is tied at the top to a swivel and the ball is given a velocity \(\mathbf{v}_{0}\). Show that the angle \(\theta\) which the cord makes with the vertical as the ball travels around the circular path must satisfy the equation tan \(\theta \sin \theta=v_{0}^{2} / g l\). Neglect air resistance and the size of the ball.

Answer 1

Given mass of ball m

Length of cord is l

Velocity of ball

Let T be the tension along cord

Let the radius of circular path be r

Free-body diagram of ball:

Equation of motion:

The particle normal acceleration always acts in the positive n direction and there is no acceleration in the bi-normal direction.

Consider normal motion

Consider bi-normal motion

From the geometry,

From equations (1) and (2), we get,

Therefore it is shown that