Question

22. Two simple harmonic motions are represented by the equations \( y_{1}=10 \sin \left(3 \pi t+\frac{\pi}{4}\right) \) and \( y_{2}=5(3 \sin 3 \pi t+\sqrt{3} \cos 3 \pi t) \). Their amplitudes are in the ratio of a. \( \sqrt{3} \) b. \( 1 / \sqrt{3} \) C. 2 d. \( 1 / 6 \)

Answer 1

Given,

y₁ = 10 sin (3πt + \(\frac{\pi}{4}\))

y₂ = 5 (3 sin3πt + \(\sqrt 3\) cos3πt)

y₂ = 15 sin3πt + \(5\sqrt 3\) cos3πt

Amplitude of y₁ is A₁ = 10

Amplitude of y₂ is A₂ = \(\sqrt{(15)^2+(5\sqrt3)^2}\)

A₂ = \(\sqrt{225+(8.6)^2}\)

A₂ = \(\sqrt{225+73.96}\)

A₂ = \(\sqrt{298.96}\)

A₂ = 17.29

Then ratio of amplitude is :

\(\frac{A_1}{A_2}\) = \(\frac{10}{17.29}\)

\(\frac{A_1}{A_2}\) = \(\frac{1}{\sqrt 3}\) 

Hence, 

Option (b) is correct.

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