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

Not a simple harmonic
B

Simple harmonic with amplitude $\frac{a}{b}$
C

Simple harmonic with amplitude $\sqrt{a^{2} + b^{2}}$
D

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

Solution

The two displacement equations are $y_{1} = a \sin (ωt) and y_{2} = b \cos (ωt) = b \sin (ωt + \frac{π}{2})$
$y_{eq} = y_{1} + y_{2}$
= $a \sin ωt + b \cos ωt$
= $a \sin ωt + b \sin (ωt + \frac{π}{2})$
Since the frequencies for both SHMs are same, resultant motion will be SHM.
Now $A_{eq} = \sqrt{a^{2} + b^{2} + 2 abcos \frac{π}{2}}$
$\Rightarrow A_{eq} = \sqrt{a^{2} + b^{2}}$

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