Question

The displacement of a particle varies with time according to the relation $y=asin\omega t+bcos\omega t.$

• A. The motion is oscillatory but not SHM.
• B. The motion is SHM with amplitude a + b.
• C. The motion is SHM with amplitude $a^2+b^2$
• D. The motion is SHM with amplitude $\sqrt{a^2+b^2}$

Answer 1

The correct option is D The motion is SHM with amplitude $\sqrt{a^2+b^2}$

Given $x=asin\omega t+bcos\omega t$ .......(i)

Let $a=Acos\varphi$ ........(ii)

and $bAsin\varphi$ ..........(iii)

Squaring adding (ii) and (iii), we get

$a^2+b^2=A^{2}co{s}^2\varphi +A^{2}si{n}^2\varphi =A^2$

Eq.(i) can be written as

$x=Acos\varphi sin\omega t+Asin\varphi cos\omega t=Asin(\omega t+\varphi )$

It is equation of SHM with amplitude $A=\sqrt{a^2+b^2}$