Q. The motion of a particle varies with time according to the relation $y = a\sin{\omega t + b\cos{\omega t}}$
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 $\sqrt{a^{2} + b^{2}}$
Explanation
(d) $y = a\sin{\omega t +}b\cos{\omega t}$
Let $a = r\cos\theta\ \ \ and\ \ \ \ b = r\sin\theta$
$y = r\cos\theta\sin{\omega t} + r\sin\theta\cos{\omega t}$
$= r\sin{(\omega t + \theta)}$
$\therefore\ \ \ a^{2} + b^{2} = r^{2}\left( \cos^{2}{\theta + \sin^{2}\theta} \right) = r^{2}$
or $r = \sqrt{a^{2} + b^{2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.