Question

A mass m is attached to the end of a rod of length l. The mass goes around a vertical circular path. What should be the minimum velocity of mass at the bottom of the circle, so that the mass completes the circle?

• A. $\sqrt{4gl}$
• B. $\sqrt{3gl}$
• C. $\sqrt{5gl}$
• D. $\sqrt{gl}$

Answer 1

The correct option is C $\sqrt{5gl}$

When a particle is moved in a circle under the action of a torque, it acquires an angular acceleration, angular velocity of the particle and hence its angular momentum will change. Also, the linear velocity, linear momentum and kinetic energy of particle will change. Such a motion is therefore called non-uniform circular motion.
applying principle of conservation of energy, total mechanical energy at L = Total mechanical energy at H
$\therefore \frac{1}{2}m{v}_L^2=\frac{1}{2}m{v}_H^2+mg\left(2l\right)$
but, $v_H^2=gl$
$\therefore \frac{1}{2}m{v}_l^2=\frac{1}{2}m\left(gl\right)+2mgl$
$\Rightarrow v_L^2=5gl$
$\Rightarrow v_L=\sqrt{\left(5gl\right)}$
Hence, for looping the vertical loops, the minimum velocity at the lowest point L is $\sqrt{5gl}$.