Question

A particle performs SHM of amplitude A along a straight line. When it is at a distance $\dfrac{{\sqrt 3 }}{2}A$ from the mean position, its kinetic energy gets increased by an amount $\dfrac{1}{2}m{\omega ^2}{A^2}$ due to an impulsive force. Then its new amplitude becomes:
A) \[\dfrac{{\sqrt 5 }}{2}A\]
B) \[\dfrac{{\sqrt 3 }}{2}A\]
C) \[\sqrt 2 A\]
D) \[\sqrt 5 A\]

Answer 1

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:
Amplitude = A
The distance from the mean position $ = \dfrac{{\sqrt 3 }}{2}A$
Increase in the kinetic energy $ = \dfrac{1}{2}m{\omega ^2}{A^2}$
New amplitude, \[{A^{^{_1}}}\] =?
Whenever the particle performs Simple Harmonic Motion with amplitude A along the straight line, then at any point the total energy will be,
$\Rightarrow E = K + P = \dfrac{1}{2}K{A^2} = \dfrac{1}{2}m{\omega ^2}{A^2}$
If additional kinetic energy is taken into consideration,
$\Rightarrow E = E + \dfrac{1}{2}m{\omega ^2}{A^2}$
\[ \Rightarrow E = \dfrac{1}{2}m{\omega ^2}{A^2} + \dfrac{1}{2}m{\omega ^2}{A^2}\]
\[ \Rightarrow \dfrac{1}{2}m{\omega ^2}{{A’}^{2}} = \dfrac{1}{2}m{\omega ^2}2{A^2}\]
\[ \Rightarrow \dfrac{1}{2}m{\omega ^2}{{A’}^{2}} = \dfrac{1}{2}m{\omega ^2}2{A^2}\]
\[ \Rightarrow {{A’}^2} = 2{A^2}\]
\[ \Rightarrow {A’} = \sqrt 2 A\]
Thus new amplitude, \[ A’ = \sqrt 2 A\].

Hence the correct option is C.

Additional Information:
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.

Note: 1. 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.
2. 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.