Reference Frame
The position r˚ a , the velocity v˚ a , and the acceleration a˚ a of a particle A of mass ma relative
to a reference frame S, are given by:
r˚ a = ra + r˚ S
v˚ a = va + ω˚ S × ra + v˚ S
a˚ a = aa + 2 ω˚ S × va + ω˚ S × (ω˚ S × ra ) + α˚ S × ra + a˚ S
where ra , va , and aa are the position, the velocity, and the acceleration of particle A relative to
the reference frame S; r˚ S , v˚ S , a˚ S , ω˚ S , and α˚ S are the position, the velocity, the acceleration,
the angular velocity, and the angular acceleration of the reference frame S relative to the
˚
universal reference frame S.
The position r˚ S , the velocity v˚ S , the acceleration a˚ S , the angular velocity ω˚ S , and the
angular acceleration α˚ S of a reference frame S fixed to a particle S relative to the universal
˚ are given by:
reference frame S,
r˚ S =

RR

(F0 /ms ) dt dt

R

v˚ S = (F0 /ms ) dt
a˚ S = (F0 /ms )

1/2
ω˚ S = (F1 /ms − F0 /ms )/(r1 − r0 )
α˚ S = d(ω˚ S )/dt
where F0 is the net force acting on the reference frame S in a point 0, F1 is the net force
acting on the reference frame S in a point 1, r0 is the position of the point 0 relative to
the reference frame S (the point 0 is the center of mass of particle S and the origin of the
reference frame S) r1 is the position of the point 1 relative to the reference frame S (the point
1 does not belong to the axis of rotation) and ms is the mass of particle S (the vector ω˚ S is
along the axis of rotation)
On the other hand, the position r˚ S , the velocity v˚ S , and the acceleration a˚ S of a reference
frame S relative to the universal reference frame S˚ are related to the position rcm , the velocity
vcm , and the acceleration acm of the center of mass of the universe relative to the reference
frame S.
2