Question

Which of the following quantities is non-zero at the mean position for a particle executing SHM?
$(A)$ Force
$(B)$ acceleration
$(C)$velocity
$(D)$displacement

Answer 1

When a particle with a periodic motion travels frequently through the same path, then the motion of the particle is called oscillation or vibration. If the acceleration of this type of vibrated or oscillated particle has a direct relation with the displacement from its equilibrium or mean position and always has the direction toward the mean position, then the motion of the particle is called Simple Harmonic Motion or S.H.M. Note that, the mean position means from which point the particle starts its journey. It should be known that the velocity can be found by derivating the displacement w.r.t time.

Complete step-by-step solution:
For a particle participating SHM has some conditions i.e. the acceleration of this type of vibrated or oscillated particle is directly proportional to the displacement from its mean position and always towards the equilibrium or the mean position.
The mean position means the starting point of the particle. Hence, At the mean position the displacement $x = 0$
Now, let us consider the given options to find the non-zero quantities at the mean position.
Displacement is zero in the mean position since the particle starts its journey from this particular point.
The acceleration of the particle in SHM is directly proportional to the displacement and maintaining a relationship such as $a = - {\omega ^2}x$ . so. If $x = 0$ at the mean position the acceleration is also zero at that position.
The Force depends on the acceleration with a relation $F = ma$. Hence at the mean position if the acceleration is zero force must be zero.
The velocity can be represented as, $v = \dfrac{{dx}}{{dt}}$ , so in the mean position since $x = 0$, the velocity should have maximum value. Hence, the velocity of the particle is a non-zero quantity at the mean position.
So, among the given options option $(C)$ (velocity) is the right answer.

Note: Other properties of simple harmonic motion:

The total energy of the particle participating in SHM is conserved.
The velocity of the particle in SHM when just cross the mean position its velocity becomes maximum at that moment; it becomes zero for a while at the highest point of the displacement and then starts moving towards the opposite direction.
The simple harmonic motion is a periodic motion that moves frequently through the same path.
SHM can be denoted by a single oscillating function of sine or cosine.