A particle executing simple harmonic motion along y -axis has its motion described by the equation `y = A sin (omega t )+ B`. The amplitude of the simple harmonic motion is

Options

A. A
B. B
C. A+B
D. `sqrtA+B`

Answer

To find the amplitude of a particle executing simple harmonic motion (SHM) described by the equation \( y = A \sin(\omega t) + B \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Equation of Motion**:
The given equation is \( y = A \sin(\omega t) + B \). Here, \( A \) is the coefficient of the sine function, and \( B \) is a constant.

2. **Identify the Components**:
- The term \( A \sin(\omega t) \) represents the oscillatory part of the motion.
- The term \( B \) represents a vertical shift in the motion, which indicates the mean position or equilibrium position of the particle.

3. **Determine the Mean Position**:
The mean position (or equilibrium position) is where the particle would be if it were not oscillating. In this case, it is given by \( y = B \).

4. **Define Amplitude**:
The amplitude of simple harmonic motion is defined as the maximum displacement from the mean position. In this case, the maximum displacement occurs when \( \sin(\omega t) = 1 \) and \( \sin(\omega t) = -1 \).

5. **Calculate Maximum and Minimum Values**:
- Maximum value of \( y \) occurs when \( \sin(\omega t) = 1 \):
\[ y_{\text{max}} = A(1) + B = A + B \]
- Minimum value of \( y \) occurs when \( \sin(\omega t) = -1 \):
\[ y_{\text{min}} = A(-1) + B = -A + B \]

6. **Find the Amplitude**:
The amplitude \( A \) is the distance from the mean position \( B \) to the maximum or minimum position:
- From the mean position \( B \) to the maximum position \( A + B \) is \( A \).
- From the mean position \( B \) to the minimum position \( -A + B \) is also \( A \).

7. **Conclusion**:
Therefore, the amplitude of the simple harmonic motion described by the equation \( y = A \sin(\omega t) + B \) is simply \( |A| \).

### Final Answer:
The amplitude of the simple harmonic motion is \( |A| \).

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