JEE PYQ: Mathematics In Physics Question 17
Question 17 - 2020 (08 Jan Shift 2)
A particle moves such that its position vector $\vec{r}(t) = \cos\omega t;\hat{i} + \sin\omega t;\hat{j}$ where $\omega$ is a constant and $t$ is time. Then which of the following statements is true for the velocity $\vec{v}(t)$ and acceleration $\vec{a}(t)$ of the particle:
(1) $\vec{v}$ is perpendicular to $\vec{r}$ and $\vec{a}$ is directed away from the origin
(2) $\vec{v}$ and $\vec{a}$ both are perpendicular to $\vec{r}$
(3) $\vec{v}$ and $\vec{a}$ both are parallel to $\vec{r}$
(4) $\vec{v}$ is perpendicular to $\vec{r}$ and $\vec{a}$ is directed towards the origin
Show Answer
Answer: (4)
Solution
Given, Position vector, $\vec{r} = \cos\omega t;\hat{i} + \sin\omega t;\hat{j}$ Velocity, $\vec{v} = \frac{d\vec{r}}{dt} = \omega(-\sin\omega t;\hat{i} + \cos\omega t;\hat{j})$ Acceleration, $\vec{a} = \frac{d\vec{v}}{dt} = -\omega^2(\cos\omega t;\hat{i} + \sin\omega t;\hat{j})$ $\vec{a} = -\omega^2\vec{r}$ $\therefore \vec{a}$ is antiparallel to $\vec{r}$ Also $\vec{v} \cdot \vec{r} = 0$ $\therefore \vec{v} \perp \vec{r}$ Thus, the particle is performing uniform circular motion.
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