First slide

Conservation of mechanical energy

Question

A body is tied at an end of a string of length ’l’ and rotated in a vertical circle. The string is just taut when the body is at the highest point.Velocity of the body when the string is in the horizontal position is

Moderate

A

$3 \sqrt{g r}$
B

$\sqrt{g r}$
C

$\sqrt{5 g r}$
D

$\sqrt{3 g r}$

Solution

$v e l o c i t y o f b o d y e x e c u t i n g v e r t i c a l c i r c u l a r m o t i o n i s V_{θ} = \sqrt{{v_{1}}^{2} - 2 g r (1 - \cos θ)} h e r e θ i s a n g l e m a d e b y t h e s t r i n g w i t h r e s t p o s i t i o n o f b o d y; θ = 90^{0} v_{1} = v e l o c i t y a t l o w e s t p o i n t; r = r a d i u s o f t h e c i r c l e; g = a c c e l e r a t i o n d u e t o g r a v i t y f o r h o r i z o n t a l p o s i t i o n, w i t h b o d y h a v i n g c r i t i c a l v e l o c i t y = \sqrt{5 g r} a t l o w e s t p o i n t V_{θ = 90} = \sqrt{5 g r - 2 g r} V_{θ = 90} = \sqrt{3 g r}$

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