If a particle is moving as `vec(r) = ( vec(i) +2 vec(j))cosomega _(0) t ` then,motion of the particleis

Options

A. Elliptical
B. Along a straight line
C. Periodic
D. Simple harmonic

Answer

To analyze the motion of the particle given by the vector equation \(\vec{r} = ( \hat{i} + 2 \hat{j}) \cos(\omega_0 t)\), we can break down the problem step by step.

### Step 1: Identify the Components of Motion
The position vector can be expressed in terms of its components:
\[ \vec{r} = x \hat{i} + y \hat{j} \]
where \(x = \cos(\omega_0 t)\) and \(y = 2 \cos(\omega_0 t)\).

### Step 2: Relate the Components
From the expressions for \(x\) and \(y\), we can establish a relationship:
\[ y = 2x \]
This indicates that the motion of the particle is constrained to a linear relationship between \(x\) and \(y\).

### Step 3: Determine the Type of Motion
The equation \(y = 2x\) represents a straight line in the \(xy\)-plane. Therefore, the motion of the particle is along a straight line.

### Step 4: Analyze the Periodicity
The function \(\cos(\omega_0 t)\) is periodic with a period \(T = \frac{2\pi}{\omega_0}\). Since both \(x\) and \(y\) depend on this periodic function, the motion of the particle is periodic.

### Step 5: Check for Simple Harmonic Motion (SHM)
The general form of simple harmonic motion is given by:
\[ x = A \cos(\omega_0 t) \]
For our case:
- \(x = \cos(\omega_0 t)\) (which is SHM with amplitude 1)
- \(y = 2 \cos(\omega_0 t)\) (which is also SHM with amplitude 2)

Since both components exhibit simple harmonic motion, the overall motion can be considered as a combination of two simple harmonic motions in the \(x\) and \(y\) directions.

### Step 6: Resultant Motion
The resultant motion can be analyzed using the Pythagorean theorem. The resultant amplitude can be calculated as:
\[ R = \sqrt{(1)^2 + (2)^2} = \sqrt{5} \]
The angle \(\theta\) with respect to the \(x\)-axis can be determined using:
\[ \tan(\theta) = \frac{y}{x} = 2 \implies \theta = \tan^{-1}(2) \]
Thus, the particle performs simple harmonic motion along a straight line at an angle \(\theta\).

### Conclusion
The motion of the particle is:
1. Along a straight line.
2. Periodic.
3. Simple harmonic motion.