Question

A particle performs simple harmonic motion with amplitude A. Its speed is trebled at the instant that it is at a distance $\dfrac{{2A}}{3}$ from equilibrium position. The new amplitude of the motion is:
A.\dfrac{A}{3}\sqrt {41} \\\ B. 3A \\\ C. A\sqrt 3 \\\ D. \dfrac{{7A}}{3} \\\

Answer 1

In order to solve this question, we first let a particle of amplitude A and then according to the question we take the speed as thrice, then dividing both the condition to get the new amplitude. Direction is always towards the mean position.it is a type of oscillatory motion .This includes sinusoidal functions. Velocity, displacement and acceleration all are described by these functions. Here the mean position in simple harmonic motion is a stable equilibrium.

Complete step by step solution:
In the given question,
Let a particle is doing SHM with amplitude A,
Velocity of the particle in SHM is;
$v = \omega \sqrt {{A^2} - {x^2}}$
Here x is the displacement from the mean position.
Also,
x = \dfrac{{2A}}{3} \\\ v = \omega \sqrt {{A^2} - {{(\dfrac{{2A}}{3})}^2}} \\\
In the given question the speed is tripled so;
$3v = \omega \sqrt {{A_1}^2 - {{(\dfrac{{2A}}{3})}^2}}$
Now dividing equation first from equation second, we get;
$\dfrac{{3v}}{v} = \dfrac{{\omega \sqrt {{A_1}^2 - {{(\dfrac{{2A}}{3})}^2}} }}{{\omega \sqrt {{A^2} - {{(\dfrac{{2A}}{3})}^2}} }}$
\Rightarrow 3 = \dfrac{{\sqrt {{A_1}^2 - {{(\dfrac{{2A}}{3})}^2}} }}{{\sqrt {{A^2} - {{(\dfrac{{2A}}{3})}^2}} }} \\\ \Rightarrow 9 = \dfrac{{{A_1}^2 - {{(\dfrac{{2A}}{3})}^2}}}{{{A^2} - {{(\dfrac{{2A}}{3})}^2}}} \\\
Solving it further,
9 = \dfrac{{{A_1}^2 - \dfrac{{4{A^2}}}{9}}}{{{A^2} - \dfrac{{4{A^2}}}{9}}} \\\ 9 = \dfrac{{9{A_1}^2 - 4{A^2}}}{{9{A^2} - 4{A^2}}} \\\ \Rightarrow 81{A^2} - 36{A^2} = 9{A_1}^2 - 4{A^2} \\\ \Rightarrow 49{A^2} = 9{A_1}^2 \\\ {A_1}^2 = \dfrac{{49}}{9}{A^2} \\\
So the value for ${A_1}$ is equal to $\dfrac{7}{3}A$ .
Option D is the correct answer.

Note:
Simple harmonic motion is a type of motion in which the restoring force is directly proportional to the displacement of the body from its mean position.
Direction is always towards the mean position.it is a type of oscillatory motion .This includes sinusoidal functions. Velocity, displacement and acceleration all are described by these functions. Here the mean position in simple harmonic motion is a stable equilibrium.
Another important concept is the path of an object needs to be a straight line. Linear SHM and angular SHM are the two classification of simple harmonic motion.