A particle moves along x-axis according to relation `x= 1+2 sin omegat`. The amplitude of S.H.M. is

Options

A. 2
B. 1
C. `sqrt(5)`
D. 3

Answer

To find the amplitude of the simple harmonic motion (S.H.M.) described by the equation \( x = 1 + 2 \sin(\omega t) \), we can follow these steps:

### Step 1: Identify the general form of S.H.M.
The general form of the equation for simple harmonic motion is:
\[ x = A \sin(\omega t + \phi) + C \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant,
- \( C \) is the equilibrium position.

### Step 2: Analyze the given equation
In the given equation \( x = 1 + 2 \sin(\omega t) \):
- The term \( 2 \sin(\omega t) \) indicates that the amplitude \( A \) is 2.
- The constant term \( 1 \) shifts the motion vertically, indicating that the equilibrium position is at \( x = 1 \).

### Step 3: Determine the extreme positions
To find the extreme positions of the particle:
- The maximum value of \( \sin(\omega t) \) is 1, so:
\[ x_{\text{max}} = 1 + 2 \cdot 1 = 3 \]
- The minimum value of \( \sin(\omega t) \) is -1, so:
\[ x_{\text{min}} = 1 + 2 \cdot (-1) = 1 - 2 = -1 \]

### Step 4: Calculate the amplitude
The amplitude \( A \) is half the distance between the maximum and minimum positions:
\[ \text{Distance between extreme positions} = x_{\text{max}} - x_{\text{min}} = 3 - (-1) = 3 + 1 = 4 \]
Thus, the amplitude \( A \) is:
\[ A = \frac{\text{Distance between extreme positions}}{2} = \frac{4}{2} = 2 \]

### Final Answer
The amplitude of the S.H.M. is \( \boxed{2} \).

---