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 $\alpha$ with the vertical. Find the tension in the string and the magnitude of net force on the bob at that instant.

Answer 1

(i)The forces acting on the bob are
(a)the tension T
(b)the weight mg
AS the bob moves in a circle of radius L with centre at O. A centripetal force of magnitude $\frac{m{v}^2}{L}$ is required towards O. This force will be provided by the resultant of T and mg $\cos \alpha .$ Thus,
or $T-mg\cos \alpha =\frac{m{v}^2}{L}$
$T=m(g\cos \alpha +\frac{v^2}{L})$
(ii)$\left|{\overrightarrow{F}}_{net}\right|=\sqrt{{\left(mg\sin \alpha \right)}^2+\left(\frac{m{v}^2}{L}\right)}=m\sqrt{g^2{\sin }^2\alpha +\frac{v^4}{L^2}}$