Question

A simple pendulum is constructed by attaching
a bob of mass m to a string of length L fixed at its upper end.
The bob oscillates in a vertical circle. It is found that the speed
of the bob is v when the string makes an angle α(alpha) with the vertical. Find the tension in the string and the magnitude of net force on the bob at that instant.

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Answer 1

We know that,

There are two forces acting on the Bob. They are tension T and Weight mg.

The bob always rotates in the circle.

Here,

The bob is rotating with radius L from the centre O.

The centrifugal force is applied towards O. This force will be provided by the resultant of Tension T and mg cos α.

$\implies \sf T \ - \ mg \ cos \alpha \ = \ \dfrac {mv^2}{L}$

$\implies \sf T \ = \ m \bigg( g \ cos \alpha \ + \ \dfrac {v^2}{L} \bigg)$

$\implies \sf \bigg| \overrightarrow{F_{net}} \bigg| = \sqrt{(mg \: sin \: \alpha )^2 + \bigg( \dfrac{mv^2}{L} \bigg)}$

$\implies \sf m \sqrt{g^2 sin^2 \alpha \ + \ \dfrac {L^4}{L^2}}$

Answer 2

I have attached the full solution

hope it helped you