Here, rP, Up and ap are the position, velocity and acceleration vectors of point P, as observed by O: To is the position vector of point O; and rp,(up)or and (ap)o are the position, velocity and acceleration vectors of point P, as observed by O. Relative to point O, all the points in the body describe a circular orbit(rp=constant), and hence we can easily calculate the velocity (up)or=rpe=rw or in vector form where w is the angular velocity vector. The acceleration has a circumferential and a radial component (ap)one=rpa=rpa (apor=-rpe-=-rpe Noting that w and w are perpendicular to the plane of motion (i.e. w can change magnitude but not direction), we can write an expression for the acceleration vector as (ap)o=山×r}p+u×(uxrp) Recall here that for any three vectors A, B and C, we have Ax(BxC)=(A C)B-(A B)C. Therefore (w x rp)=(w.rp)w-wr'p=-w2r'p. Finally, equations 2 and 3 become P vor+w x ao+×r}+wx(u×rp) Body Axes An alternative description can be obtained using body axes. Now, let r'y be a set of axes which are rigidly attached to the body and have the origin at point OHere, rP , vP and aP are the position, velocity and acceleration vectors of point P, as observed by O; rO′ is the position vector of point O′ ; and r ′ P , (vP )O′ and (aP )O′ are the position, velocity and acceleration vectors of point P, as observed by O′ . Relative to point O′ , all the points in the body describe a circular orbit (r ′ P = constant), and hence we can easily calculate the velocity, (vP )O′ = r ′ P ˙θ = rω , or, in vector form, (vP )O′ = ω × r ′ P , where ω is the angular velocity vector. The acceleration has a circumferential and a radial component, ((aP )O′ )θ = r ′ P ¨θ = r ′ P ω, ˙ ((aP )O′ )r = −r ′ P ˙θ 2 = −r ′ P ω 2 . Noting that ω and ω˙ are perpendicular to the plane of motion (i.e. ω can change magnitude but not direction), we can write an expression for the acceleration vector as, (aP )O′ = ω˙ × r ′ P + ω × (ω × r ′ P ) . Recall here that for any three vectors A, B and C, we have A×(B × C) = (A·C)B −(A·B)C. Therefore ω × (ω × r ′ P ) = (ω · r ′ P )ω − ω 2r ′ P = −ω 2r ′ P . Finally, equations 2 and 3 become, vP = vO′ + ω × r ′ P (4) aP = aO′ + ω˙ × r ′ P + ω × (ω × r ′ P ) . (5) Body Axes An alternative description can be obtained using body axes. Now, let x ′y ′ be a set of axes which are rigidly attached to the body and have the origin at point O′ . 2