Question

A particle executing SHM while moving from one extremity is found to be at distances $x_1,x_{2}and{x}_3$ from the mean position at the end of three successive seconds. The time period of oscillation is ($\theta$ used in the following choices is given by $cos\theta =\frac{x_1+x_3}{2x_2}$

• A. $\frac{2\pi }{\theta }$
• B. $\frac{\pi }{\theta }$
• C. $\theta$
• D. $\frac{\pi }{2\theta }$

Answer 1

The correct option is A

$\frac{2\pi }{\theta }$

$x_1=Acos\left(\omega \right)$

$x_2=Acos\left(2\omega \right)$

$x_3=Acos\left(3\omega \right)$

$\therefore x_1+x_3=2Acos\left(2\omega \right)cos\omega =2A\frac{x_2}{A}\frac{x_1}{A}$

$\Rightarrow \frac{x_1}{A}=\frac{x_1+x_2}{2x_2}\frac{x_1}{A}$

Hence $T=\frac{2\pi }{\omega }=\frac{2\pi }{\theta }$