A particle moves along the x-axis according to the equation `x=4+3sin(2pit)`
Hence, x in cm and t in seconds. Select the correct alternatives

Options

A. The motion of the particle is simple harmonic with mean position at x=0
B. the motion of the particle is simple harmonic with mean position at x=4cm
C. The motion of the particle is simple harmonic with mean position at x=`-4cm`
D. Amplitude of osciallation is 3cm

Answer

To solve the problem, we need to analyze the given equation of motion for the particle:

**Given Equation:**
\[ x = 4 + 3 \sin(2 \pi t) \]

### Step 1: Identify the form of the equation
The equation can be rewritten as:
\[ x - 4 = 3 \sin(2 \pi t) \]

This is of the form:
\[ x = A \sin(\omega t) + x_0 \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( x_0 \) is the mean position.

### Step 2: Determine the mean position
From the equation \( x - 4 = 3 \sin(2 \pi t) \):
- The mean position (equilibrium position) is given by \( x_0 = 4 \) cm.

### Step 3: Determine the amplitude
From the equation, we see that the coefficient of the sine function is the amplitude:
- Amplitude \( A = 3 \) cm.

### Step 4: Determine the angular frequency
The angular frequency \( \omega \) can be identified from the sine function:
- Here, \( \omega = 2 \pi \) rad/s.

### Step 5: Conclusion
Based on the analysis:
1. The motion of the particle is Simple Harmonic Motion (SHM).
2. The mean position is \( x = 4 \) cm.
3. The amplitude of the motion is \( 3 \) cm.

### Final Answers:
- The correct alternatives are:
- Mean position: \( x = 4 \) cm (Correct)
- Amplitude: \( 3 \) cm (Correct)