What is the acceleration of a particle executing S.H.M. at its mean position.?

Options

A. infinity
B. Varies
C. maximum
D. zero

Answer

To find the acceleration of a particle executing Simple Harmonic Motion (S.H.M.) at its mean position, we can follow these steps:

### Step 1: Understand the concept of S.H.M.
In S.H.M., a particle moves back and forth around a central point (mean position) and experiences a restoring force that is proportional to its displacement from this mean position.

### Step 2: Identify the mean position
The mean position is the central point where the particle has zero displacement. At this point, the particle is neither compressed nor stretched.

### Step 3: Use the formula for restoring force
The restoring force \( F \) in S.H.M. is given by Hooke's Law:
\[ F = -kx \]
where:
- \( k \) is the spring constant,
- \( x \) is the displacement from the mean position.

### Step 4: Determine the displacement at the mean position
At the mean position, the displacement \( x \) is equal to zero:
\[ x = 0 \]

### Step 5: Calculate the restoring force at the mean position
Substituting \( x = 0 \) into the restoring force equation:
\[ F = -k(0) = 0 \]
Thus, the restoring force is zero at the mean position.

### Step 6: Relate force to acceleration
According to Newton's second law, the force acting on an object is also related to its mass \( m \) and acceleration \( a \):
\[ F = ma \]

### Step 7: Set the force equal to mass times acceleration
Since we found that the restoring force \( F \) is zero at the mean position:
\[ 0 = ma \]

### Step 8: Solve for acceleration
Since mass \( m \) cannot be zero, we can conclude:
\[ a = 0 \]

### Conclusion
The acceleration of a particle executing S.H.M. at its mean position is:
\[ \text{Acceleration} = 0 \]