Question

In an experiment, a boy plots a graph between $\left(v_{max}\right)^{2}$ and $\left(a_{max}\right)^{2}$ for a simple pendulum for different values of (small) amplitudes, where $v_{max}$ and $a_{max}$ is the maximum velocity and the maximum acceleration respectively. He found the graph to be a straight line with a negative slope, making an angle of $30^\circ $ when the experiment was conducted on the earth surface. When the same experiment was conducted at a height $h$ above the surface, the line was at an angle of $60^\circ $ . The value of $h$ is [radius of the earth = $R$ ]

• A. $0.5R$
• B. $0.24R$
• C. $0.73R$
• D. $R$

Answer 1

Answer: (C)

For a simple pendulum,
$\omega =\sqrt{\frac{g}{l}}$
$v_{max}=A\omega $
$a_{max}=-A\omega ^{2}$
$\left(\frac{v_{\max }}{a_{\max }}\right)^2=\frac{1}{\omega^2}=\tan \theta$
$\Rightarrow \omega =\sqrt{\frac{g}{l} =}\frac{1}{\sqrt{\text{tan } \theta }}$
$\therefore g=\frac{l}{\text{tan } \theta }$
$\therefore \frac{g_{surface}}{g_{height}}=\frac{\text{tan60°}}{\text{tan30°}}=\frac{\left(R + h\right)^{2}}{R^{2}}$
$\Rightarrow h=0.73R$