Question

The displacement of a particle varies with time according to the relation y = a sin wt + 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 a2 + b2
• D. The motion is SHM with amplitude $\sqrt{a^{2} + b^{2}}$

Answer 1

Answer: (D)

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

Answer 2

Given : $x = a\sin\omega t + b\cos\omega t$ … (i)

Let $a = A\cos\varphi$ … (ii)

And $b = A\sin\varphi$ ….. (iii)

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

$a^{2} + b^{2} = A^{2}\cos^{2}\varphi + A^{2}\sin^{2}\varphi = A^{2}$

Eq. (i) can be written as

$x = A\cos\varphi\sin\omega t + A\sin\varphi\cos\omega t = A\sin(\omega t + \varphi)$ it is equation of SHM with amplitude $A = \sqrt{a^{2} + b^{2}}$