Question

A particle is executing SHM. Then the graph of acceleration as a function of displacement is
A. Straight line
B. Circle
C. Ellipse
D. Hyperbola

Answer 1

Hint:Here, we need to recall the term SHM. A simple harmonic motion is a good example of periodic motion. In simple harmonic motion, a particle will be accelerated towards a fixed point and the acceleration of the particle is proportional to the magnitude of the displacement of the particle. We have to use the formula of the velocity of the particle, and we need to find in which form the graph will be.

Formula Used:
The formula for velocity is,
\[v = \omega \sqrt {{A^2} - {Y^2}} \]
Where,
v is velocity of a wave
Y is displacement of a wave
\[\omega \] is angular velocity of a wave
A is amplitude of a wave

Complete step by step solution:
Suppose we have a particle that is executing a simple harmonic motion. Then we need to find how the velocity and displacement related in the graph that we will discuss now.
Here the velocity is maximum at a mean position and is zero at an extreme position. Now we consider a formula for the velocity that is,
\[v = \omega \sqrt {{A^2} - {Y^2}} \]

We consider this equation in order to discuss the graph between velocity versus displacement. Now squaring both sides of the above equation, we get,
\[{v^2} = {\omega ^2}\left( {{A^2} - {Y^2}} \right) \\ \]
\[\Rightarrow \dfrac{{{v^2}}}{{{\omega ^2}}} + {Y^2} = {A^2} \\ \]
\[\therefore \dfrac{{{v^2}}}{{{\omega ^2}{A^2}}} + \dfrac{{{Y^2}}}{{{A^2}}} = 1 \\ \]
Here, Y is displacement, v is velocity. If we look at the equation it is in the standard form of an ellipse. Therefore, if we plot a graph of velocity versus displacement it will be in the form of an ellipse.

Hence, option C is the correct answer.

Note: Always remember that the displacement of the particle executing SHM will be dependent on amplitude, angular frequency, and time. Moreover, a particle's movement away from its equilibrium location in a medium while it transmits a sound wave is measured as particle displacement or displacement amplitude.