The position vector of a particle as a function of time is given by where R is in meters, t is in seconds and and denote unit vectors along x and y-directions, respectively. Which one of the following statements is wrong for the motion of the particle

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Solution

Q.
The position vector of a particle $\vec{R}$ as a function of time is given by $\vec{R} = 4 \sin (2 π t) \hat{i} + 4 \cos (2 π t) \hat{j}$ where R is in meters, t is in seconds and $\hat{i}$ and $\hat{j}$ denote unit vectors along x and y-directions, respectively. Which one of the following statements is wrong for the motion of the particle [2015]

1 Magnitude of the velocity of particle is 8 meter/second.

2 Path of the particle is a circle of radius 4 meter.

3 Acceleration vector is along $- \vec{R}$ .

4 Magnitude of acceleration vector is $\frac{v^{2}}{R}$ , where $v$ is the velocity of particle.

Ans.
(1)

Here, $\vec{R} = 4 \sin (2 π t) \hat{i} + 4 \cos (2 π t) \hat{j}$

The velocity of the particle is

$\vec{v} = \frac{d \vec{R}}{d t} = \frac{d}{d t} [4 \sin (2 π t) \hat{i} + 4 \cos (2 π t) \hat{j}] = 8 π \cos (2 π t) \hat{i} - 8 π \sin (2 π t) \hat{j}$

Its magnitude is $| \vec{v} |$ $= \sqrt{(8 π \cos {(2 π t))}^{2} + (- 8 π \sin {(2 π t))}^{2}}$

$= \sqrt{64 π^{2} \cos^{2} (2 π t) + 64 π^{2} \sin^{2} (2 π t)}$

   $= \sqrt{64 π^{2} [\cos^{2} (2 π t) + \sin^{2} (2 π t)]}$

   $= \sqrt{64 π^{2}} (As \sin^{2} θ + \cos^{2} θ = 1)$

   $= 8 π m/s$

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