The frequency of a particle performing `SHM` is `12Hz`. Its amplitude is `4cm`. Its initial displacement is `2cm` towards positive exterme positions. Its equation for displacement is

Options

A. `x = 0.04 cos (24pi t+(pi)/(6))m`
B. `x = 0.04sin (24 pit)m`
C. `x = 0.04 sin(24pit+(pi)/(6))m`
D. `x = 0.04 cos (24 pit)m`

Explanation

To find the equation of motion for a particle performing Simple Harmonic Motion (SHM) given the frequency, amplitude, and initial displacement, we can follow these steps:

### Step 1: Identify the given values
- Frequency (f) = 12 Hz
- Amplitude (A) = 4 cm = 0.04 m (convert to meters for standard units)
- Initial displacement (x₀) = 2 cm = 0.02 m (also convert to meters)

### Step 2: Calculate the angular frequency (ω)
The angular frequency (ω) is related to the frequency (f) by the formula:
\[ \omega = 2\pi f \]
Substituting the given frequency:
\[ \omega = 2\pi \times 12 = 24\pi \, \text{rad/s} \]

### Step 3: Write the general equation for SHM
The general equation for the displacement in SHM can be expressed as:
\[ x(t) = A \sin(\omega t + \phi) \]
Substituting the values of A and ω:
\[ x(t) = 0.04 \sin(24\pi t + \phi) \]

### Step 4: Determine the phase constant (φ)
To find the phase constant (φ), we use the initial condition. At time \( t = 0 \), the initial displacement \( x(0) = 0.02 \, \text{m} \):
\[ x(0) = 0.04 \sin(24\pi \cdot 0 + \phi) = 0.04 \sin(\phi) \]
Setting this equal to the initial displacement:
\[ 0.02 = 0.04 \sin(\phi) \]
Dividing both sides by 0.04:
\[ \sin(\phi) = \frac{0.02}{0.04} = \frac{1}{2} \]
The angle φ that satisfies this equation is:
\[ \phi = \frac{\pi}{6} \, \text{(since sin(π/6) = 1/2)} \]

### Step 5: Write the final equation
Substituting the value of φ back into the equation:
\[ x(t) = 0.04 \sin(24\pi t + \frac{\pi}{6}) \]

### Final Answer
The equation for displacement is:
\[ x(t) = 0.04 \sin(24\pi t + \frac{\pi}{6}) \, \text{meters} \]

---