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STANDARDPhysics-Motion in Two Dimensions

STANDARD Physics Circular Motion MCQ Question

Type: MCQ-conceptual-Medium

A particle moves so that its position vector is given by r=cos⁡ωti+sin⁡ωtj\mathbf{r} = \cos \omega t \mathbf{i} + \sin \omega t \mathbf{j}, where ω\omega is a constant. Which of the following is true?

A

Velocity is perpendicular to r\mathbf{r} and acceleration is directed towards the origin.

B

Velocity is perpendicular to r\mathbf{r} and acceleration is directed away from the origin.

C

Velocity and acceleration both are perpendicular to r\mathbf{r}.

D

Velocity and acceleration both are parallel to r\mathbf{r}.

Correct Answer

Option A

Detailed Explanation

Given, r=cos⁡ωti+sin⁡ωtj\mathbf{r} = \cos \omega t \mathbf{i} + \sin \omega t \mathbf{j}. The velocity v=drdt=−ωsin⁡ωti+ωcos⁡ωtj\mathbf{v} = \frac{d\mathbf{r}}{dt} = -\omega \sin \omega t \mathbf{i} + \omega \cos \omega t \mathbf{j} is perpendicular to r\mathbf{r}. The acceleration a=dvdt=−ω2cos⁡ωti−ω2sin⁡ωtj=−ω2r\mathbf{a} = \frac{d\mathbf{v}}{dt} = -\omega^2 \cos \omega t \mathbf{i} - \omega^2 \sin \omega t \mathbf{j} = -\omega^2 \mathbf{r} is directed towards the origin.

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