Question

A particle moving with SHM from an extreme of the path towards centre is observed to be at distances $x_1,x_2,x_3$ from the center at the ends of three successive seconds. The time period of the SHM is

• A. $\frac{2\pi }{{\cos }^{-1}\left[\frac{x_1+x_3}{2x_2}\right]}$
• B. $\frac{2\pi }{{\cos }^{-1}\left[\frac{x_1+x_3}{2x_3}\right]}$
• C. $\frac{2\pi }{{\cos }^{-1}\left[\frac{x_1+x_3}{x_2}\right]}$
• D. $\frac{2\pi }{{\cos }^{-1}\left[\frac{x_1+x_3}{x_3}\right]}$

Answer 1

Answer: (A)

$\frac{2\pi }{{\cos }^{-1}\left[\frac{x_1+x_3}{2x_2}\right]}$

$x=A\cos wt$
$\text{Let}{x}_1\text{occur at time}{t}_1$
$x_1=A\cos \omega t_1,$

$\text{Let}{x}_2\text{occur at time}{t}_1+1$
$x_2=A\cos \omega (t_1+1),$

$\text{Let}{x}_3\text{occur at time}{t}_1+2$
$x_3=A\cos \omega (t_1+2),$
$x_1+x_3=2A\cos (\omega t_1+\omega )\cos \omega =2x_2\cos (\omega t_1+\omega )$

$[\text{Apply}\cos A+\cos B=2\cos \left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2}\right)\text{formula}]$

$\text{Therefore,}2x_2=2A\cos \omega$

$\Rightarrow \cos \omega =\frac{2x_2}{2A}$
$\text{Thus}\omega \text{can be written as}{\cos }^{-1}\frac{x_1+x_3}{2x_2}$
$T=\frac{2\pi }{w}=\frac{2\pi }{{\cos }^{-1}\left(\frac{x_1+x_3}{2x_2}\right)}$