First slide

Simple hormonic motion

Question

A particle is subjected to two SHMs, $X_{1} = A_{1} \sin ωt and X_{2} = A_{2} \sin (ωt + \frac{π}{4})$ . The resultant SHM will have an amplitude of:

Moderate

A

$\frac{(A_{1} + A_{2})}{2}$
B

$\sqrt{A_{1}^{2} + A_{2}^{2}}$
C

$\sqrt{A_{1}^{2} + A_{2}^{2} + \sqrt{2} A_{1} A_{2}}$
D

$\sqrt{\frac{A_{1}^{2} + A_{2}^{2}}{2}}$

Solution

Resultant amplitude,
$A \sqrt{A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} \cos ϕ}$
Here $ϕ = \frac{π}{4}$
$\Rightarrow A = \sqrt{A_{1}^{2} + A_{2}^{2} + \sqrt{2} A_{1} A_{2}}$

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