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

It is given that a car is moving with speed v on a circular horizontal road of radius r.
(a) Let T be the tension in the string.

According to the free body diagram, the value of T is given as,
$T=\sqrt{{\left(mg\right)}^2+{\left(\frac{m{v}^2}{r}\right)}^2}$
$=m\sqrt{g^2+\frac{v^4}{r^2}}=ma,$
where acceleration, a $=\sqrt{g^2+\frac{v^4}{r^2}}$
The time period $\left(T\right)$ is given by,
$$\begin{gathered} T=2\mathrm{\pi }\sqrt{\frac{l}{g}} \\ \mathrm{On}\mathrm{substituting}\mathrm{the}\mathrm{respective}\mathrm{values},\mathrm{we}\mathrm{have}: \\ T=2\mathrm{\pi }\sqrt{\frac{l}{{\left({\displaystyle g^2+\frac{v^4}{r^2}}\right)}^{1/2}}} \end{gathered}$$