The displacement of a particle executing simple harmonic motion is given by `y=A_(0)+A sin omegat+B cos omegat`. Then the amplitude of its oscillation is given by

Options

A. `A+B`
B. `A_(0)+sqrt(A^(2)+B^(2))`
C. `sqrt(A^(2)+B^(2))`
D. `sqrt(A_(0)^(2)+(A+B)^(2))`

Explanation

To find the amplitude of a particle executing simple harmonic motion given by the displacement equation \( y = A_0 + A \sin(\omega t) + B \cos(\omega t) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components**:
The displacement equation is given as:
\[ y = A_0 + A \sin(\omega t) + B \cos(\omega t) \]
Here, \( A_0 \) is a constant (the mean position), \( A \) is the coefficient of the sine term, and \( B \) is the coefficient of the cosine term.

2. **Recognize the Form of SHM**:
The general form of simple harmonic motion can be expressed as:
\[ y = A \sin(\omega t + \phi) \]
where \( A \) is the amplitude and \( \phi \) is the phase angle.

3. **Combine the Sine and Cosine Terms**:
We can express \( A \sin(\omega t) + B \cos(\omega t) \) as a single sinusoidal function. The resultant amplitude \( R \) can be calculated using the formula:
\[ R = \sqrt{A^2 + B^2} \]

4. **Calculate the Resultant Amplitude**:
Since the sine and cosine functions can be represented as a single sinusoidal function, we can ignore the constant \( A_0 \) when calculating the amplitude of the oscillation. Thus, the amplitude of the oscillation is:
\[ \text{Amplitude} = \sqrt{A^2 + B^2} \]

5. **Final Result**:
Therefore, the amplitude of the oscillation is given by:
\[ \text{Amplitude} = \sqrt{A^2 + B^2} \]