The position vector of a particle as a function of time is given by: R = 4sin(2πt)i + 4cos(2πt)j, where R is in meters, t is in seconds and i and j denote unit vectors along x- and y- directions, respectively. Which one of the following statements is wrong for the motion of particle?

Given the position vector of the particle as a function of time:

R = 4sin(2πt)i + 4cos(2πt)j

We can find the velocity vector by differentiating the position vector with respect to time:

v = dR/dt = 8πcos(2πt)i - 8πsin(2πt)j

The magnitude of the velocity is:

|v| = √[(8πcos(2πt))² + (-8πsin(2πt))²] = √[64π²(cos²(2πt) + sin²(2πt))] = 8π m/s

Now, let's find the acceleration vector by differentiating the velocity vector with respect to time:

a = dv/dt = -16π²sin(2πt)i - 16π²cos(2πt)j

The magnitude of the acceleration is:

|a| = √[(-16π²sin(2πt))² + (-16π²cos(2πt))²] = √[256π⁴(sin²(2πt) + cos²(2πt))] = 16π² m/s²

We can also express the acceleration vector as:

a = -4π²(4sin(2πt)i + 4cos(2πt)j) = -4π²R

This shows that the acceleration vector is along -R. The magnitude of the acceleration is |a| = 16π². The magnitude of the velocity is |v| = 8π. The radius of the circular path is 4m.

Let's check the given options:

1. Magnitude of acceleration vector is v²/R. This is a correct statement for uniform circular motion. v²/R = (8π)²/4 = 16π², which is equal to |a|.

2. Magnitude of the velocity of particle is 8π m/s. This is correct, as calculated above.

3. Path of the particle is a circle of radius 4 m. This is correct. The equation represents a circle with radius 4 in the x-y plane.

4. Acceleration vector is along -R. This is correct, as shown above.

Therefore, there seems to be an error in either the original problem statement or the given options. All statements appear to be correct for the given position vector. The option "Magnitude of the velocity of particle is 8 m/s" is wrong. It should be 8π m/s.