Question

A particle performs SHM given by the equation $x=3\cos \omega t+4\sin \omega t$. Find the phase constant and amplitude.

• A. ${50}^{\circ },5\text{units}$
• B. ${37}^{\circ },3.5\text{units}$
• C. ${53}^{\circ },3.5\text{units}$
• D. ${37}^{\circ },5\text{units}$

Answer 1

The correct option is D ${37}^{\circ },5\text{units}$

General expression for SHM of a particle is given by
$x=x_0\sin (\omega t+\varphi )=x_0\sin \omega t\cos \varphi +x_0\cos \omega t\sin \varphi x=3\cos \omega t+4\sin \omega t$(given)

Comparing we get, $x_0\cos \varphi =4\dots \dots \left(1\right)$
and $x_0\sin \varphi =3\dots \dots \left(2\right)$
Dividing (2) by (1), $\tan \varphi =\frac{3}{4}$ or $\varphi ={37}^{\circ }$

Squaring and adding (1) and (2), we get $x_0=5\text{units}$.