# Linear speed of a rotating sphere

**1. The problem statement, all variables and given/known data**

Hello everyone.

Each minute, I have 3d coordinates of points at the surface of a unit sphere (with center at (0;0;0)) rotating with an axis which can (slightely?) change over time. I want to know the linear speed (s) of this sphere. I don’t know how to find r at each time.

**2. Relevant equations**

[itex]s=r*\omega[/itex]

with [itex] \omega = [/itex] angular speed

**3. The attempt at a solution**

I found solutions which doesn’t imply directly r :

[itex]\cos{s} = \cos(\theta_1)*\cos(\theta_2) + \sin(\theta_1)*\sin(\theta_2) * \cos(\phi_2-\phi_1)[/itex]

with [itex]\theta = colatitude [/itex]

and [itex]\phi = longitude [/itex].

Is it a good way to calculate s ?

To have a good linear speed according to positions on the sphere, we also told me to find "dynamically" the plane [itex]\pi[/itex] containing severals points and to project these points on the parallel plane to [itex]\pi[/itex] passing through the center of the sphere. Then, to calculate the angular speed [itex]\omega[/itex].

But it doesn’t work since my linear speed was sometimes higher when points are close together ([itex]\omega~0[/itex]) compared at when they describe a small circle ([itex]\omega~0.08 rad/sec[/itex]).

Thank you.

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