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Question

# What minimum horizontal speed should be given to the bob of a simple pendulum of length l so that it describes a complete circle?

A

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B

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C

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D

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Solution

## The correct option is B Suppose the bob is given a horizontal speed v0 at the bottom and it describes a complete vertical circle. Let its speed at the highest point be v. Taking the gravitational potential energy to be zero at the bottom, the conservation of energy gives. 12mv20=12mv2+2mgl or, mv2=mv20−4mgl --------------(i) The forces acting on the bob at the highest point are mg due to the gravity and T due to the tension in the string. The resultant force towards the centre is, therefore, mg + T. As the bob is moving in a circle, its acceleration towards the centre is v2l. Applying Newton's second law and using (i), mg + T = mv2l=1l(mv20−4mgl) Or mv20=5mgl+Tl. Now, for v0 to be minimum, T should be minimum. As the minimum value of T can be zero, for minimum speed, mv20=5mgl or, v0=√5gl.

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