Question

A particle of mass m is attached to a light string of length l, the other end of which is fixed. Initially the string is kept horizontal and the particle is given an upward velocity v. The particle is just able to complete a circle.

• A. The string becomes slack when the particle reaches its highest point
• B. The velocity of the particle becomes zero at the highest point.
• C. The kinetic energy of the ball in initial position was
• D. The particle again passes through the initial position.

Answer 1

Answer: (A), (D)

The correct options are
A

The string becomes slack when the particle reaches its highest point

D

The particle again passes through the initial position.

Let us assume that the particle has PE = 0 at initial position.

At the highest point net force acting on the particle = mg +T

As the particle goes in a circle and let the velocity of the particle at its highest point be $v_H$.

Then by newton's law,

T + mg = $\frac{m{v}_{H^2}}{l}$

But for this to be minimum, T = 0

Mg will provide the required centripetal acceleration

$mg=\frac{m{v}_{H^2}}{l}=v_H=\sqrt{gl}$

So keeping the initial condition in mind i.e., the particle is just able to complete the circle.

At highest point T = 0 so option (a) follows.

From above equation,

(b) Is wrong as particle has to have some velocity at maximum height.

(c) Again doesn't follow as at highest point, $\frac{1}{2}m{v}_H^2+mgl=\frac{1}{2}m{v}^2$

(d) Follows from conservation of energy.