Question

The acceleration (a) of SHM at mean position is:
A. Zero
B. \[ \propto x\]
C. \[ \propto {x^2}\]
D. None of these

Answer 1

Hint: The restoring force is directly proportional to the displacement of the particle from its mean position and it acts in the opposite direction of the displacement. This force is also equal to the product of mass and acceleration. At mean position, the displacement of the particle is zero.

Complete step by step answer:
As we know in the simple harmonic motion (SHM), the restoring force is directly proportional to the displacement of the particle from its mean position and it acts in the opposite direction of the displacement.We can write the expression for the restoring force as,
\[F = - kx\] …… (1)
Here, k is the force constant and x is the displacement from the mean position.
According to Newton’s second law of motion, the force acting on the particle is,
\[F = ma\] …… (2)
 Here, m is the mass of the particle and a is the acceleration.
Equating equation (1) and (2), we get,
\[ - kx = ma\]
\[ \Rightarrow a = - \dfrac{{kx}}{m}\]
\[ \Rightarrow a \propto - x\]
As we know, at the mean position, the displacement of the particle is zero. Therefore, the acceleration of the particle,
\[\therefore{a_{mean}} = 0\]
Therefore, the acceleration of the particle at mean position is zero.

So, the correct answer is option A.

Note: Students can also answer this question by referring to the formula for acceleration in SHM, \[a = - {\omega ^2}x\], where, \[\omega \] is the angular velocity and x is the displacement. Since the displacement of the particle at mean position is zero, the acceleration of the particle is also zero. One should always take the displacement of the particle in SHM from the mean position where there is no restoring force acting on the particle.