How do I get the time it takes to travel between two points in a circular motion?

Question

I have two points on a circle.

Circle with two points

Given that I have the constant angular velocity, cartesian coordinates of the two points and center, and the radius. How would I get the time it would take to travel point A to point B?

The center is not 0, 0

Answer 1

To solve, you'd need to now the angular displacement going from point $A$ to point $B$ on the circle of the radius you've been given.

Let the center of the circle be $C$.

There are a couple of ways to do this, but the easiest (IMO) is to draw a triangle $ABC$ connecting the three points $A,B,C$ and use the distance formula to find the length $|AB|$. Let this distance be $d$.

Note that $$|AC|=|BC|=R$$ since the segments |AC| and |BC| are from the center of the circle to points on the circle.

From this, apply the law of cosines to obtain your angle $$d^2=R^2+R^2-2R^2\cos\Delta\theta\to \Delta\theta=\dots$$ Then use the angular kinematic equation to solve for the time: $$\Delta \theta=\omega t +\frac{1}{2}\alpha t^2$$ I'm assuming that $\alpha=0$ (since you haven't given one in the problem), so then: $$\Delta \theta=\omega t$$