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

Answer: (D)

Given :$x=asin\omega t+bcos\omega t...\left(i\right)$
Let $a=Acos\varphi ...\left(ii\right)$
and $b=Asin\varphi ...\left(iii\right)$
Squaring and 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
$A=\sqrt{a^2+b^2}$.