Question

A simple pendulum is suspended from the ceiling of a car accelerating uniformly on a horizontal road. If the acceleration is $a_0$ and the length of the pendulum is $l$, find the time period of small oscillations about the mean position

• A. $\pi \frac{\sqrt{l}}{{(a_0^2+g^2)}^{1/2}}$
• B. $2\pi \frac{\sqrt{l}}{{(a_0^2+2g^2)}^{1/2}}$
• C. $2\pi \frac{\sqrt{l}}{{(a_0^2+g^2)}^{1/4}}$
• D. $\pi \frac{\sqrt{l}}{{(a_0^2+g^2)}^{1/4}}$

Answer 1

The correct option is C $2\pi \frac{\sqrt{l}}{{(a_0^2+g^2)}^{1/4}}$

Time period of simple pendulum $T=2\pi \sqrt{\frac{l}{g_{eff}}}$

In the car frame, the pendulum experiences a pseudo force in the backward direction.

Thus effective acceleration due to gravity $g_{eff}=\sqrt{a_o^2+g^2}$

$⟹$ $T=2\pi \sqrt{\frac{l}{\sqrt{a_o^2+g^2}}}=2\pi \frac{\sqrt{l}}{(a_o^2+g^2{)}^{1/4}}$