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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=mv204mgl --------------(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(mv204mgl)

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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