The phase difference between the displacement and acceleration of a particle executing simple harmonic motion is

Options

A. zero
B. `pi//2`
C. `pi`
D. `2pi`

Explanation

To determine the phase difference between the displacement and acceleration of a particle executing simple harmonic motion (SHM), we can follow these steps:

### Step 1: Write the equation for displacement
The displacement \( x \) of a particle in SHM can be expressed as:
\[ x(t) = A \sin(\omega t + \phi) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.

**Hint:** Remember that the sine function is used to represent the displacement in SHM.

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

**Hint:** The derivative of \( \sin \) is \( \cos \), and remember to multiply by the derivative of the argument \( \omega t + \phi \).

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

**Hint:** The derivative of \( \cos \) is \( -\sin \), and again multiply by the derivative of the argument.

### Step 4: Analyze the phase of displacement and acceleration
From the equations:
- Displacement: \( x(t) = A \sin(\omega t + \phi) \)
- Acceleration: \( a(t) = -A \omega^2 \sin(\omega t + \phi) \)

We can see that the acceleration is proportional to the negative of the displacement. This indicates that when the displacement is positive, the acceleration is negative, and vice versa.

**Hint:** The negative sign indicates that acceleration is directed opposite to the displacement.

### Step 5: Determine the phase difference
The phase of the displacement is \( \omega t + \phi \), and the phase of the acceleration is \( \omega t + \phi + \pi \) (due to the negative sign). Therefore, the phase difference \( \Delta \phi \) between displacement and acceleration is:
\[ \Delta \phi = \pi \text{ radians} \]

**Hint:** A phase difference of \( \pi \) radians means that the two quantities are 180 degrees out of phase.

### Final Answer
The phase difference between the displacement and acceleration of a particle executing simple harmonic motion is \( \pi \) radians.

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