Question

A simple pendulum of length of mass m and length I = 0.5 m is hanging from ceiling of car. The car is kept on a horizontal plane. The car starts accelerating on the horizontal road with acceleration of $5m/s^2$. Find the time period of oscillations of the pendulum for small amplitudes about the means position.

• A. $\frac{12}{5^{1/4}}$ sec
• B. $\frac{2}{5^{3/4}}$ sec
• C. $\frac{2}{5}$ sec
• D. $\frac{2}{5^{1/4}}$ sec

Answer 1

The correct option is D $\frac{2}{5^{1/4}}$ sec

Time period of a simple pendulum in a accelerating vehicle is given by
$T=2\pi \sqrt{\frac{l}{g_{eff}}}\Rightarrow T=2\pi \sqrt{\frac{l}{\sqrt{(g^2+a^2)}}}\Rightarrow T=2\sqrt{\frac{{\pi }^{2}l}{\sqrt{(g^2+a^2)}}}$
Here we need to have following approximations in calculations to reach at exact solution,
${\pi }^2\cong 10$ and $g\cong 10$
$\Rightarrow T=2\sqrt{\frac{{\pi }^{2}l}{\sqrt{(g^2+a^2)}}}=2\sqrt{\frac{10\times 0.5}{\sqrt{{10}^2+5^2}}}\Rightarrow T=2\sqrt{\frac{5}{\sqrt{{10}^2+5^2}}}=2\sqrt{\frac{1}{\sqrt{2^2+1}}}=2\sqrt{\frac{1}{\sqrt{5}}}=\frac{2}{5^{1/4}}sec$