Question

A simple pendulum of length l is suspended from the ceiling of a car moving with a speed v on a circular horizontal road of radius r. (a) Find the tension in the string when it is at rest with respect to the car. (b) Find the time period of small oscillation.

Answer 1

(a) From the free body diagram

$T=\sqrt{(mg{)}^2+{\left(\frac{m{v}^2}{r}\right)}^2}$

$=m\sqrt{\frac{g^2+v^4}{r^2}}$

$=m\sqrt{\frac{g^2+v^4}{r^2}}=ma,$

where a = acceleration

$={\left(\frac{g^2+v^4}{r^2}\right)}^{1/2}$

The time period of small acceleration is given by,

$T=2\pi \sqrt{\frac{l}{a}}$

$=2\pi \sqrt{\frac{1}{{\left(\frac{g^2+v^4}{r^4}\right)}^{1/2}}}$