The position vector of a particle `vec(R )` as a funtion of time is given by:
`vec(R )= 4sin(2pit)hat(i)+4cos(2pit)hat(j)`
Where `R` is in meters, `t` is in seconds and `hat(i)` and `hat(j)` denote until vectors along x-and y- directions, respectively Which one of the following statements is wrong for the motion of particle ?

Options

A. Path of the particle is a circle of radius `4 meter`
B. Acceleration vector is along `-vec(R )`
C. Magnitude of acceleration vector is `(V_(2))/(R )` where `v`is the velocity of particle.
D. Magnitude of the Velocity of particle is `8 meter//second`

Explanation

To solve the problem, we need to analyze the given position vector of the particle and evaluate the statements provided. The position vector is given as:

\[ \vec{R} = 4 \sin(2\pi t) \hat{i} + 4 \cos(2\pi t) \hat{j} \]

### Step 1: Identify the path of the particle

To find the path of the particle, we can express the x and y coordinates in terms of the sine and cosine functions:

- \( x = 4 \sin(2\pi t) \)
- \( y = 4 \cos(2\pi t) \)

### Step 2: Determine the locus of motion

Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can relate x and y:

\[ \left(\frac{x}{4}\right)^2 + \left(\frac{y}{4}\right)^2 = 1 \]

This simplifies to:

\[ x^2 + y^2 = 16 \]

This equation represents a circle with a radius of 4 meters centered at the origin.

### Step 3: Analyze the first statement

The first statement claims that the path of the particle is a circle of radius 4 meters. Since we derived that the particle indeed moves in a circle of radius 4 meters, this statement is **correct**.

### Step 4: Calculate the velocity vector

Next, we find the velocity vector by differentiating the position vector with respect to time:

\[ \vec{V} = \frac{d\vec{R}}{dt} = \frac{d}{dt}(4 \sin(2\pi t) \hat{i} + 4 \cos(2\pi t) \hat{j}) \]

Using the chain rule:

\[ \vec{V} = 4(2\pi \cos(2\pi t)) \hat{i} - 4(2\pi \sin(2\pi t)) \hat{j} = 8\pi \cos(2\pi t) \hat{i} - 8\pi \sin(2\pi t) \hat{j} \]

### Step 5: Calculate the acceleration vector

Now, we differentiate the velocity vector to find the acceleration vector:

\[ \vec{A} = \frac{d\vec{V}}{dt} = \frac{d}{dt}(8\pi \cos(2\pi t) \hat{i} - 8\pi \sin(2\pi t) \hat{j}) \]

Applying the chain rule again:

\[ \vec{A} = -8\pi(2\pi \sin(2\pi t)) \hat{i} - 8\pi(2\pi \cos(2\pi t)) \hat{j} = -16\pi^2 \sin(2\pi t) \hat{i} - 16\pi^2 \cos(2\pi t) \hat{j} \]

### Step 6: Express acceleration in terms of position vector

We can express the acceleration vector in terms of the position vector \(\vec{R}\):

\[ \vec{A} = -4\pi^2 \vec{R} \]

This shows that the acceleration vector is indeed directed along \(-\vec{R}\), confirming that the second statement is **correct**.

### Step 7: Analyze the third statement

The third statement claims that the magnitude of the acceleration vector is given by \( \frac{v^2}{r} \).

First, we find the magnitude of the acceleration:

\[ |\vec{A}| = 16\pi^2 \]

Next, we compute the magnitude of the velocity:

\[ |\vec{V}| = \sqrt{(8\pi \cos(2\pi t))^2 + (-8\pi \sin(2\pi t))^2} = 8\pi \]

Now, we calculate \( \frac{v^2}{r} \):

\[ \frac{v^2}{r} = \frac{(8\pi)^2}{4} = 16\pi^2 \]

Since both magnitudes match, the third statement is **correct**.

### Step 8: Analyze the fourth statement

The fourth statement claims that the magnitude of the velocity of the particle is 8 m/s. However, we found that:

\[ |\vec{V}| = 8\pi \approx 25.13 \text{ m/s} \]

Thus, this statement is **incorrect**.

### Conclusion

The wrong statement regarding the motion of the particle is the fourth one.