First slide

Simple hormonic motion

Question

The resulting amplitude A' and the phase of the vibrations
S = $A \cos (ωt) + \frac{A}{2} \cos (ωt + \frac{π}{2}) + \frac{A}{4} \cos (ωt + π) + \frac{A}{8} \cos (ωt + \frac{3 π}{2}) = A^{'} \cos (ωt + δ)$
then A' and $δ$ are respectively.

Moderate

A

$\frac{\sqrt{5}}{8} A, \tan^{- 1} (\frac{1}{2})$
B

$\frac{3 \sqrt{5}}{8} A, \tan^{- 1} (\frac{1}{2})$
C

$\frac{3 \sqrt{5}}{8} A, \tan^{- 1} (\frac{1}{4})$
D

$\frac{3}{8} A, \tan^{- 1} (\frac{1}{4})$

Solution

$\tan θ = \frac{1}{2}$
$A_{net} = \sqrt{{(\frac{3 A}{4})}^{2} + {(\frac{3 A}{8})}^{2}} = \frac{3 A}{4} \sqrt{1 + {(\frac{1}{2})}^{2}} = \frac{3 \sqrt{5}}{8} A$

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