A particle moves in the `x-y` plane , accoding to the equation, `r = (hati + 2hatj) A cos omegat`. The motion of the particle is

Options

A. on a straight line
B. on an ellipse
C. periodic
D. simple harmonic

Answer

To analyze the motion of the particle described by the equation \( \mathbf{r} = (1 \hat{i} + 2 \hat{j}) A \cos(\omega t) \), we can break down the problem step by step.

### Step 1: Understanding the Equation
The equation given is in vector form, where:
- \( \hat{i} \) and \( \hat{j} \) are unit vectors in the x and y directions, respectively.
- \( A \) is the amplitude of the motion.
- \( \omega \) is the angular frequency.
- \( t \) is time.

This means that the position vector \( \mathbf{r} \) can be expressed in terms of its components:
\[ \mathbf{r} = A \cos(\omega t) \hat{i} + 2A \cos(\omega t) \hat{j} \]

### Step 2: Analyzing the Motion in the x-y Plane
From the equation, we can see that both the x and y components of the motion depend on \( \cos(\omega t) \). Specifically:
- The x-component is \( x(t) = A \cos(\omega t) \)
- The y-component is \( y(t) = 2A \cos(\omega t) \)

### Step 3: Identifying the Type of Motion
Since both components are proportional to \( \cos(\omega t) \), we can conclude that the motion is periodic. The particle moves back and forth along a straight line defined by the vector \( (1, 2) \).

### Step 4: Determining the Nature of the Motion
1. **Straight Line Motion**: The motion occurs along the line connecting the origin (0,0) to the point (1,2). This indicates that the particle moves in a straight line.
2. **Simple Harmonic Motion (SHM)**: The equations for both x and y components are of the form \( A \cos(\omega t) \), which is characteristic of SHM.
3. **Periodic Motion**: Since SHM is inherently periodic, the motion repeats itself after a certain interval.

### Conclusion
Based on the analysis:
- The motion of the particle is along a straight line (Option 1).
- The motion is periodic (Option 3).
- The motion is simple harmonic (Option 4).

Thus, the correct options are 1, 3, and 4, while option 2 (ellipse) is incorrect.