Motion in a Plane - Result Question 43

46. The position vector of a particle $\vec{R}$ as a function of time is given by:

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

Where $R$ is in meter, $t$ 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 particle?

[2015 RS]

(a) Magnitude of acceleration vector is $\frac{v^{2}}{R}$, where $v$ is the velocity of particle

(b) Magnitude of the velocity of particle is 8 meter/second

(c) path of the particle is a circle of radius 4 meter.

(d) Acceleration vector is along $\vec{R}$

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Answer:

Correct Answer: 46. (b)

Solution:

  1. (b) Here, $x=4 \sin (2 \pi t)$

$ \begin{equation*} y=4 \cos (2 \pi t) \tag{i} \end{equation*} $

Squaring and adding equation (i) and (ii) $x^{2}+y^{2}=4^{2} \Rightarrow R=4$

Motion of the particle is circular motion, acceleration vector is along $-\vec{R}$ and its magnitude $=\frac{V^{2}}{R}$

Velocity of particle, $V=\omega R=(2 \pi)(4)=8 \pi$