The displacement of a particle executing simple harmonic motion is given by
`y=A_(0)+Asinomegat+Bcosomegat`
Then the amplitude of its oscillation is given by :

Options

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

Explanation

To find the amplitude of the oscillation given the displacement of a particle executing simple harmonic motion, we start with the equation:

\[ y = A_0 + A \sin(\omega t) + B \cos(\omega t) \]

### Step 1: Identify the components of the equation
The equation consists of a constant term \( A_0 \) and two oscillatory terms \( A \sin(\omega t) \) and \( B \cos(\omega t) \). The constant term \( A_0 \) represents the mean position of the oscillation and does not affect the amplitude.

**Hint:** Focus on the oscillatory terms to determine the amplitude.

### Step 2: Recognize the form of the oscillatory terms
The terms \( A \sin(\omega t) \) and \( B \cos(\omega t) \) can be combined to find the resultant amplitude of the oscillation. The general form for the resultant of two simple harmonic motions is given by:

\[ A_r = \sqrt{A^2 + B^2 + 2AB \cos(\delta \phi)} \]

where \( \delta \phi \) is the phase difference between the two components.

**Hint:** Remember that the phase difference between sine and cosine functions is \( \frac{\pi}{2} \).

### Step 3: Calculate the phase difference
The phase difference \( \delta \phi \) between \( \sin(\omega t) \) and \( \cos(\omega t) \) is \( \frac{\pi}{2} \). Therefore, we can substitute this into the equation.

Since \( \cos\left(\frac{\pi}{2}\right) = 0 \), the term \( 2AB \cos(\delta \phi) \) becomes zero.

**Hint:** Use the known trigonometric identity for cosine at \( \frac{\pi}{2} \).

### Step 4: Simplify the resultant amplitude
With the phase difference considered, the resultant amplitude simplifies to:

\[ A_r = \sqrt{A^2 + B^2} \]

This means the amplitude of the oscillation is determined solely by the coefficients \( A \) and \( B \) of the sine and cosine terms.

**Hint:** Focus on the coefficients of the oscillatory terms for the final amplitude.

### Final Answer
The amplitude of the oscillation is given by:

\[ A_r = \sqrt{A^2 + B^2} \]