Q. A mass m is attached to the end of a rod of length l. The mass goes around a vertical circular path with the other end hinged at the centre. What should be the minimum velocity of mass at the bottom of the circle, so that the mass complete the circle?
A) $\sqrt{4gl}$
B) $\sqrt{3gl}$
C) $\sqrt{5gl}$
D) $\sqrt{gl}$
Explanation
(c) When a particle is moved in a circle under the action of a torque then such motion is non-uniform circular motion.
Applying principle of conservation of energy, total mechanical energy at L
=total mechanical energy at H
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$\therefore\ \frac{1}{2}mv_{L}^{2} = \frac{1}{2}mv_{H}^{2} + MG(2l)$
But $v_{H}^{2} = gl$
$\therefore\frac{1}{2}mv_{L}^{2} = \frac{1}{2}m(gl) + 2mgl$
Or $v_{L}^{2} = 5gl\ $
Or $v_{L} = \sqrt{5gl}$
Hence for looping the vertical loop, the minimum velocity at the lowest point L IS $\sqrt{5gl.}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.