First slide

Simple hormonic motion

Question

When two displacements represented by $y_{1} = a \sin (ωt) and y_{2} = b \cos (ωt)$ are superimposed the motion is :

Moderate

A

simple harmonic with amplitude $\frac{a}{b}$
B

simple harmonic with amplitude $\sqrt{a^{2} + b^{2}}$
C

simple harmonic with amplitude $\frac{(a + b)}{2}$
D

not a simple harmonic

Solution

$y_{1} = a \sin ωt and y_{2} = b \cos ωt = - b \sin (ωt + \frac{π}{2})$
Since the frequencies for both SHMs are same, resultant motion will be SHM. Now
Amplitude $A = \sqrt{A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} \cos θ}$
Here $A_{1} = a, A_{2} = b and ϕ = \frac{π}{2}$
So $A = \sqrt{a^{2} + b^{2}}$

$A \sqrt{a^{2} + b^{2}}$

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