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Two simple harmonic motions are represented by ${y_1} = 10\sin \left( {4\pi t + \dfrac{\pi }{4}} \right)$ and ${y_2} = 5\left( {\sin 4\pi + \sqrt 3 \cos 4\pi t} \right)$​. The ratio of the amplitudes of the two SHM's is:
A) 1:1
B) 1:2
C) 2:1
D) $1:\sqrt 3 $

Last updated date: 17th Apr 2024
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Hint: Simple Harmonic Motion is a type of periodic motion in which the restoring force on the moving object will be directly proportional to the displacement magnitude and will act towards the equilibrium position of the object.

Complete step by step solution:
Given data:
${y_1} = 10\sin \left( {4\pi t + \dfrac{\pi }{4}} \right)................\left( 1 \right)$
${y_2} = 5\left( {\sin 4\pi + \sqrt 3 \cos 4\pi t} \right)$
The ratio of the amplitudes =?
Thus ${y_2}$ can also be written as.
$\Rightarrow {y_2} = 10\left( {\left( {\dfrac{1}{2}} \right)\sin 4\pi + \left( {\dfrac{{\sqrt 3 }}{2}} \right)\cos 4\pi t} \right)$
$ \Rightarrow {y_2} = 10\left( {\cos \left( {\dfrac{\pi }{3}} \right)\sin 4\pi + \sin \left( {\dfrac{\pi }{{\sqrt 3 }}} \right)\cos 4\pi t} \right)$
\[ \Rightarrow {y_2} = 10\sin \left( {\dfrac{\pi }{4} + 4\pi t} \right)\]
\[ \Rightarrow {y_2} = 10\sin \left( {4\pi t + \dfrac{\pi }{4}} \right)\]
\[\therefore {y_2} = 10\sin \left( {4\pi t + \dfrac{\pi }{4}} \right)..............\left( 2 \right)\]
Thus from equations 1 and 2, it is clear that the amplitudes of both the simple harmonic motions are equal and hence they will be in the ratio of 1:1.

Hence the correct option is A.

Note: 1. The applications of simple harmonic motion are guitar, violin, bungee jumping, clock, diving boards, etc and it is always oscillatory. All periodic motions are not oscillatory but every oscillatory motion is periodic.
2. There are two main characteristics of simple harmonic motion.
i) The acceleration will be directly proportional to the displacement.
ii) The direction of the acceleration will always be towards the mean position.
3. Amplitude is defined as the distance moved by the point on a wave which is measured from its equilibrium position. It is measured in meters. It is inversely proportional to frequency and also inversely proportional to the distance. Amplitude depends on two factors, an elastic factor, and an inertial factor.
4. The amplitude of the sound is created by the number of molecules that are displaced by a vibration. The amplitude of the sound wave does not depend on the wavelength, velocity, frequency. It is an important parameter for the waves.