Question

The displacement time graph of a particle executing S.H.M. is as shown in the figure. Then what is the corresponding force-time graph of the particle?

A.
B.
C.
D.

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.

Formula Used:
The formula acceleration in SHM is,
\[a = - {\omega ^2}x\]
Where, x is distance of a wave and \[\omega \] is angular velocity of a wave.

Complete step by step solution:
The displacement versus time graph of a simple harmonic motion is represented as shown in the figure. We have to pick the correct graph in which it represents the force versus the time graph. Here, the x-axis is the time scale, and the y-axis of velocity, that is the graph as shown in this question is nothing but the displacement of a simple harmonic motion with respect to time. We need to find its corresponding force versus the time graph.

We know hat, force is the product of mass and acceleration that is,
\[F = ma\]………… (1) and
Acceleration is, \[a = - {\omega ^2}x\]
Substitute the value of acceleration in above equation (1), we get,
\[F = - m{\omega ^2}x\]
From the above equation, we can say that, if the displacement is in one direction and the force acting is on the other direction that is, as per the above equation since it contains the negative sign on the right-hand side, the displacement is in the negative y-axis and the force is in the positive x-axis. Therefore, here the displacement is on the negative y-axis and the force acting on a particle is in the positive x-axis.

Hence, option D is the correct answer.

Note: Here, in this problem it is important to remember the sign convention of the given quantity in order to get the correct answer in the given graph. That is displacement having the negative sign and force being positive.