First slide

Circular motion

Question

Position vector of a particle moving in xy plane at time t is $\vec{r} = a (1 - \cos ω t) \hat{i} + a \sin ω t \hat{j}$ . The path of the particle is

Moderate

A

a circle of radius a and center at (a,0)
B

a circle of radius a and center at (0, 0)
C

an ellipse
D

neither a circle nor an ellipse

Solution

Particle is moving in a circle of radius a and center (a, 0) with constant angular velocity $ω$ . At time t=0 particle is at origin and it starts r = rotating clockwise. At time r it has
rotated an angle $θ$ given by:
$\begin{array}{l} θ = ω t \\ y = a \sin θ = a \sin ω t \\ and x = a - a \cos θ = a (1 - \cos ω t) \\ ∴ \vec{r} = a (1 - \cos ω t) \hat{i} + a \sin ω t \hat{j} \end{array}$

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