A particle moves in the `x-y` plane according to the equation `vec(r)=(vec(i)+vec(j)) (A sin omega t +B cos omega t)`. Motion of particle is

Options

A. periodic
B. `SHM`
C. along a straight line
D. all of the above

Answer

To analyze the motion of the particle given by the equation \(\vec{r} = (\vec{i} + \vec{j})(A \sin \omega t + B \cos \omega t)\), we can break it down step by step:

### Step 1: Understand the Equation
The position vector \(\vec{r}\) can be expressed in terms of its components in the \(x\) and \(y\) directions. Here, \(\vec{i}\) represents the \(x\)-axis and \(\vec{j}\) represents the \(y\)-axis. Thus, we can rewrite the equation as:
\[ \vec{r} = (A \sin \omega t + B \cos \omega t) \vec{i} + (A \sin \omega t + B \cos \omega t) \vec{j} \]

### Step 2: Identify the Motion
The terms \(A \sin \omega t\) and \(B \cos \omega t\) indicate that the motion in both \(x\) and \(y\) directions is periodic. This suggests that the particle undergoes oscillatory motion.

### Step 3: Combine the Terms
We can combine the two components into a single expression. If we let:
\[ C = \sqrt{A^2 + B^2} \]
and use the angle \(\theta\) such that:
\[ \cos \theta = \frac{B}{C}, \quad \sin \theta = \frac{A}{C} \]
we can rewrite the motion as:
\[ \vec{r} = C \left( \sin(\omega t + \theta) \right) \vec{i} + C \left( \sin(\omega t + \theta) \right) \vec{j} \]

### Step 4: Analyze the Type of Motion
The motion described by the equation is a combination of sine and cosine functions, which are periodic functions. Therefore, the motion of the particle is periodic and can be classified as Simple Harmonic Motion (SHM).

### Step 5: Conclusion
Since both components of the motion are periodic and can be expressed in terms of sine and cosine, we conclude that the particle exhibits Simple Harmonic Motion in the \(x-y\) plane.

### Final Answer
The motion of the particle is Simple Harmonic Motion (SHM).

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