A ball of mass m is attached to the end of a very light rod of length L. The other end of the rod is pivoted so that the ball can move in a vertical circle. The rod is pulled aside to the horizontal and given a downward push a shown. So that the rod swings down and just reaches the vertically upward position. Then, the initial speed imparted to the ball is- (1) underroot l/g (2) underroot gl (3) underroot 2gl (4) gL?

Question

Answer

Problem:

A ball of mass m is attached to the end of a very light rod of length L. The other end of the rod is pivoted so that the ball can move in a vertical circle. The rod is pulled aside to the horizontal and given a downward push a shown. So that the rod swings down and just reaches the vertically upward position. Then, the initial speed imparted to the ball is-

underroot l/g

underroot gl

underroot 2gl

gL

Solution:

The diagram of the situation is shown below:

Conservation of Energy:

The total energy of the ball is conserved throughout its motion. At the highest point, the kinetic energy of the ball is zero and the potential energy is maximum. At the lowest point, the potential energy of the ball is zero and the kinetic energy is maximum.

Let the initial speed given to the ball be v. At the highest point, the speed of the ball is zero. Hence, the total energy of the ball at the highest point is equal to the potential energy:

$E_{max} = mgL$

At the lowest point, the potential energy is zero and the total energy is equal to the kinetic energy:

$E_{min} = \frac{1}{2}mv^2$

Thus, we have:

$mgL = \frac{1}{2}mv^2$

Simplifying the above equation, we get:

$v = \sqrt{2gL}$

Hence, the initial speed imparted to the ball is underroot 2gL, which is option (3).

Answer: Option (3) underroot 2gL

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