The motion of a particle is given by `x=A sin omegat+Bcos omegat`. The motion of the particle is

Options

A. not simple harmonic
B. SHM with amplitude `A+B`
C. SHM with amplitude `(A+B)//2`
D. SHM with amplitude `sqrt((A^2+B^2))`

Explanation

To solve the problem regarding the motion of a particle given by the equation \( x = A \sin(\omega t) + B \cos(\omega t) \), we will analyze the equation step by step to determine the nature of the motion.

### Step-by-Step Solution:

1. **Identify the Given Equation**:
The motion of the particle is described by the equation:
\[ x = A \sin(\omega t) + B \cos(\omega t) \]

2. **Rewrite the Equation**:
We can express the equation in a different form by using the trigonometric identity for a sine function:
\[ x = R \sin(\omega t + \phi) \]
where \( R \) is the amplitude and \( \phi \) is the phase angle.

3. **Calculate the Amplitude**:
To find \( R \), we use the formula:
\[ R = \sqrt{A^2 + B^2} \]
This represents the amplitude of the motion.

4. **Determine the Phase Angle**:
The phase angle \( \phi \) can be determined using:
\[ \tan(\phi) = \frac{B}{A} \]
This gives us the angle that accounts for the contributions of both sine and cosine components.

5. **Conclusion on the Nature of Motion**:
The rewritten equation \( x = R \sin(\omega t + \phi) \) indicates that the motion is periodic and oscillatory, which is characteristic of Simple Harmonic Motion (SHM).

6. **Final Answer**:
Therefore, the motion of the particle is Simple Harmonic Motion.