Question

A particle moves along a straight line to follow the equation $a{x^2} + b{v^2} = k$, where $a,b$ and $k$ are constants and $x$ and $v$ are x-coordinate and velocity of the particle respectively. Find the amplitude
A. $\sqrt {\dfrac{k}{b}} $
B. $\sqrt {\dfrac{b}{k}} $
C. $\sqrt {\dfrac{a}{k}} $
D. $\sqrt {\dfrac{k}{a}} $

Answer 1

Hint: Here we have to use the concepts of simple harmonic motion.
Simple harmonic motion can be defined as an oscillatory motion in which the acceleration of the particle at any point is directly proportional to the displacement of the mean position.

Complete step by step answer:
Given equation is:
$a{x^2} + b{v^2} = k$
Where $a,b$ and $k$ are constants and $x$ and $v$ are x-coordinate and velocity of the particle respectively.
The x-coordinate and velocity equations of simple harmonic motions are given by:
$ x = A\sin \omega t \\
 \implies {x^2} = {A^2}{\sin ^2}\omega t \\
 \implies {x^2}{\omega ^2} = {A^2}{\sin ^2}\omega t \times {\omega ^2} \\
 \implies v = A\omega \cos \omega t \\
\implies {v^2} = {A^2}{\omega ^2}{\cos ^2}\omega t \\
$
Now adding the values of ${x^2}{\omega ^2}$ and ${v^2}$ we get:
${x^2}{\omega ^2} + {v^2} = {A^2}{\omega ^2}\left( {{{\sin }^2}\omega t + {{\cos }^2}\omega t} \right)$
${x^2}{\omega ^2} + {v^2} = {A^2}{\omega ^2}$ ...... (i)
From the given equation we get:
$
a{x^2} + b{v^2} = k \\
\dfrac{a}{b}{x^2} + {v^2} = \dfrac{k}{b} \\
$
Putting the values in equation (i), we get:
$
{A^2} = \dfrac{{\dfrac{k}{b}}}{{\dfrac{a}{b}}} \\
\implies {A^2} = \dfrac{k}{a} \\
\therefore A = \sqrt {\dfrac{k}{a}} \\
$

So, the correct answer is “Option d”.

Additional Information:
Not all oscillatory motions are simple harmonic in nature whereas all harmonic motions are periodic and oscillatory. Oscillatory motion is often referred to as the harmonic motion of all oscillatory movements, the most important of which is basic harmonic motion.
In oscillatory motion displacement, velocity and acceleration and force differ in a manner that can be represented by either sine or cosine functions usually referred to as sinusoids.
In a simple harmonic motion the return of force or acceleration acting on the particle should always be equal to the displacement of the particle and be oriented towards the equilibrium state.
The amplitude is actually the maximum displacement of the object from the direction of equilibrium. In other words, the same equation refers to the position of an object experiencing basic harmonic motion and to one component of the position of an object experiencing uniform circular motion.

Note:
Here we have to remember the simple harmonic motion equations for x-coordinate and velocity to get the answer.
Also we have to be careful while putting the values on the given equation.