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Getting angular speed from turn speed and linear speed

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OK, so picture an object moving in a circle. The object rotates left at a specific speed, and has a set linear speed. If you give it enough time, the object will be right back where it started, its path forming a circle. 

 

 

I want to get the angular speed from these 2 variables. By angular speed, I mean the amount of rotation that's happening from the perspective of the centre of the circle, as if the object were a spoke on a wheel.

 

I've been trying to figure this out, but it's driving me bonkers. Here's a few equations I've been looking at:

 

 

length of an arc = angle * r

linear speed = angular speed * r

 

angular speed = linear speed/r

 

 

 

 

 

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The angular speed from the center of the circle is the same as the turn rate of the moving object. It makes sense intuitively if you think about it. If you have an object with a stick attached to it where the other end of the stick is attached to a fixed point you can swing the object in a circle around that fixed point. The rotation of the object is tied to the rotation of the stick.

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So angular velocity be the same. In 3D the angular velocity vector is an axis around which an object spins, and the magnitude of this vector is the the speed radians/seconds.

If a big object makes a full circle (360 degrees/2 pi radians) in one second, all the points on this object are also spinning at 360 degrees per second.

The linear velocity however, is different for for points closer to the axis than further. If this point is at a distance R from the center of the spin, the angular velocity vector A has a magnitude specifying rotational speed in radians per seconds, and V0 is the linear speed of the entire object then, the linear velocity V for point i is:

V(i) = V0 + R(i) x A

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