Choose the correct statement regarding SHM

Options

A. Acceleration is ahead of displacement by phase single of `(pi)/(2)`
B. Acceleration is ahead of displacement by phase angle of `pi`
C. Acceleration is behind displacement by phase angle of `pi`
D. Acceleration is behind displacement by phase angle of `(pi)/(2)`

Answer

To solve the question regarding the correct statement about Simple Harmonic Motion (SHM), we can follow these steps:

### Step 1: Understand the basic equations of SHM
In SHM, the displacement \( x \) of a particle from its mean position can be expressed as:
\[ x = A \sin(\omega t) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( t \) is time.

### Step 2: Differentiate the displacement to find velocity
To find the velocity \( v \), we differentiate the displacement equation with respect to time \( t \):
\[ v = \frac{dx}{dt} = A \omega \cos(\omega t) \]

### Step 3: Differentiate the velocity to find acceleration
Next, we differentiate the velocity equation to find the acceleration \( a \):
\[ a = \frac{dv}{dt} = -A \omega^2 \sin(\omega t) \]

### Step 4: Relate acceleration to displacement
We can express acceleration in terms of displacement. Since \( x = A \sin(\omega t) \), we can substitute this into the acceleration equation:
\[ a = -\omega^2 x \]

### Step 5: Analyze the phase relationship
From the equations for displacement and acceleration:
- Displacement: \( x = A \sin(\omega t) \)
- Acceleration: \( a = -A \omega^2 \sin(\omega t) \)

We can see that acceleration is proportional to the negative of the displacement. This indicates that acceleration is 180 degrees (or \( \pi \) radians) out of phase with displacement.

### Conclusion
Thus, the correct statement regarding SHM is that acceleration is ahead of displacement by a phase angle of \( \pi \) radians.

### Final Answer
The correct statement regarding SHM is: "Acceleration is ahead of displacement by a phase angle of \( \pi \)."

---