# Computing angular velocity on a different point of body

Suppose you have a rigid body and you have two local coordinate systems on the body at $$P_1$$ and $$P_2$$ and we can write a vector in the $$P_2$$ coordinate system using $$P_1$$ via the transformation $$v_{p2} = R*v_{p1} + T$$.

We know the instantaneous angular velocity vector of the object written with respect to $$P_1$$, how do we compute the angular velocity vector respect to $$P_2$$? Additionally how do we compute the orientation of the object with respect to $$P_2$$ assuming we know its orientation with respect to $$P_1$$? In other words suppose we know euler angles $$(\alpha, \beta, \gamma )$$ to rotate the object into the reference frame $$P_1$$ from a coordinate system located at $$P_1$$ which is parallel to the global coordinate system and we want to know the euler angles to transform the coordinate system at $$P_2$$ parallel to the global coordinate system to orient the object the same as it is oriented in $$P_1$$.

• Angular and tangential accelerations are related like : $$\alpha ={\frac {a_{\perp }}{r}}.$$ So having one, you will deduce the other. Commented Apr 11, 2023 at 7:33
• I should have written angular velocity, but basically I have a gyro and I want to compute the orientation on the body using another coordinate frame attached to the body. Commented Apr 11, 2023 at 8:40
• Please state your problem better. What kind of body? magnitude and-or direction of angular velocity? what is " the orientation on the body " Commented Apr 11, 2023 at 11:59

If the two coordinate systems have different orientations, and $$R$$ is the rotation matrix going from one to the other, then the angular velocity vector is simply transformed as
$$\omega_{\rm p2} = R\; \omega_{\rm p1}$$
Angular velocity is not location dependent, as it is shared by the entire body and you are just expressing the same quantity in different basis-vectors. The location of the two coordinate systems (and the associated translation $$T$$) plays no role here.