Download rotating + ω r

Rotating Coordinate Systems
• For Translating Systems: We just got Newton’s 2nd Law (in
the accelerated frame): ma = F where
F = F - ma0
ma0  A non-inertial or fictitious force
Solely from the COORDINATE TRANSFORMATION!
Rotating systems are accelerating systems!
• For Rotating Systems: Follow a similar procedure as for
translating systems & similarly get: Newton’s 2nd Law
(in the accelerated frame): ma = F where
F = F - mar
mar  A non-inertial or fictitious force:
Solely from the COORDINATE TRANSFORMATION!
For rotating systems: mar = much more complicated expressions
than ma0!
• As in our treatment of rigid bodies, consider again 2
sets of axes:
– A “Fixed” or inertial set (space axes in Goldstein
notation): (x1,x2,x3) & A “Rotating” (body axes
in Goldstein notation): (x1,x2,x3).
CAUTION! The treatment here & the associated
figures are from Marion’s book. His unprimed
coordinate system = the accelerating system & his
primed system is fixed. The rotating system
treatment in Goldstein uses the OPPOSITE
notation: The primed system is accelerating
(rotating) & the unprimed is fixed.
• 2 axis sets: “Fixed”: (x1,x2,x3), “Rotating”: (x1,x2,x3).
• Consider point P in space. r in fixed system, r in
rotating system. R = position of the origin of the
rotating system with respect to the fixed axes: See fig:
• Clearly:
r = R + r
• To relate time derivatives in the fixed & rotating
systems, use the results we had for rigid bodies
(letting space = s  fixed = f & r  rotating ):
(d/dt)fixed = (d/dt)rotating + ω 
or, for G = arbitrary vector,
(dG/dt)fixed = (dG/dt)rotating + ω  G
(1)
• First, special case: Let G = ω (angular velocity)
 Relation between time derivatives of ω (between
angular accelerations) in fixed & rotating frames is:
(dω/dt)fixed = (dω/dt)rotating + ω  ω
But ω  ω = 0  (dω/dt)fixed = (dω/dt)rotating  ω
• Angular acceleration (sometimes called α  ω) is
the same in the fixed & rotating frames!
• Consider again point P: Position in fixed frame = r.
Position in rotating frame = r. Position of origin of
rotating frame in fixed frame = R.
r = R + r
(1)
• Goal: Express velocity of point P in fixed system in
terms of ω & velocity in rotating system: Moving frame
is translating
1. Differentiate (1) in the fixed system: AND rotating!
(dr/dt)fixed = (dR/dt)fixed + (dr/dt)fixed
2. Use (dr/dt)fixed = (dr/dt)rotating + ω  r

(dr/dt)fixed = (dR/dt)fixed
+ (dr/dt)rotating + ω  r
(dr/dt)fixed = (dR/dt)fixed + (dr/dt)rotating+ ω  r (2)
• Define:
vf  rf  (dr/dt)fixed = Velocity relative to the fixed axes.
V  R  (dR/dt)fixed = Velocity (linear) of the moving origin
(fixed frame).
vr  rr  (dr/dt)rotating = Velocity (linear) relative to the
rotating axes.
ω  Angular velocity
ω  r  Velocity due to the rotation of the moving axes.
 (2) becomes:
vf = V + v r + ω  r
“Fictitious” Centrifugal & Coriolis Forces
• Procedure (the same as for translational
acceleration): Use Newton’s 2nd Law & transform
from the fixed axis system to rotating the axis system,
using the operator:
(d/dt)fixed = (d/dt)rotating + ω 
(1)
on the velocity equation we just got!
vf = V + v r + ω  r
(2)
• A particle of mass m at point P under the influence of
a net force F: Newton’s 2nd Law is valid ONLY in the
fixed, inertial frame (primed coordinates!):
F = maf  m(dvf/dt)fixed
(3)
vf = V + vr + ω  r
• Differentiate (2) in fixed frame:
(dvf/dt)fixed = [d(V + vr + ω  r)/dt]fixed
Or: (dvf/dt)fixed = (dV/dt)fixed + (dvr/dt)fixed
+ [(dω/dt)fixed  r] + ω  (dr/dt)fixed
(2)
• Define: Af  (dV/dt)fixed
(5)
• Recall that (dω/dt)fixed  ω (same in both frames).
• By discussion we just had:
(dvr/dt)fixed  (dvr/dt)rotating + ω  vr
Or:
(dvr/dt)fixed = ar + ω  vr
(4)
(6)
ar = Acceleration in rotating frame.
• We know:
(dr/dt)fixed  (dr/dt)rotating + ω  r
Or:
(dr/dt)fixed = vr + ω  r
vr = Velocity in rotating frame.
(7)
• We had,
F = maf  m(dvf/dt)fixed
• Combine (4)-(7) on previous page:
(dvf/dt)fixed = Af + ar + ω  vr + ω  r + ω  [vr + ω  r]
• Put into Newton’s 2nd Law (above) & rearrange:
 F = maf = mAf + mar + m(ω  r)
+ m[ω  (ω  r)] + 2m(ω  vr)
(I)
• Observer in rotating frame. Measures mar. Insists on
writing this in Newtonian form, even rotating though frame is
not inertial! So:

mar  Feff
(II)
(I) & (II) together  We must have: mar  Feff
 F - mAf - m(ω  r) - m[ω  (ω  r)] - 2m(ω vr)
• Applying Newton’s 2nd Law to the rotating frame yields:
Feff  mar  F - mAf - m(ω  r)
- m[ω  (ω  r)] - 2m(ω  vr)
• Physical Interpretations:
- mAf : From translational acceleration of the rotating frame.
- m(ω  r): From the angular acceleration of rotating frame.
- m[ω  (ω  r)]:  Centrifugal “Force”. See figure!
- 2m(ω  vr):  Coriolis “Force”. Comes from motion
of particle in rotating system (= 0 if vr = 0)
More discussion of last two follows
- m[ω  (ω  r)]:  Centrifugal “Force”
If ω  r: This has magnitude mω2r. Outwardly
directed from
center of
rotation!
“Fictitious” Forces
• Physical discussion of “Centrifugal Force” &
“Coriolis Force”.
• These terms have entered the right side of the product
mar (mass x acceleration in rotating frame). They came
about because we wanted to write a Newton’s 2nd
Law-like eqtn in the rotating frame: Feff  mar ,
when, in fact Newton’s 2nd Law, F = maf, is valid
only in the fixed (inertial) frame.

The transformation from the fixed to the
rotating frame gave:
Feff  F - (non-inertial terms)
• Feff  F - (non-inertial terms)
• Example: A body rotating about a fixed (attractive) force
center: The only real force (defined by Newton!) is force of
attraction to the center: Causes the centripetal acceleration (in
the inertial frame!).
• However, an observer moving WITH the body (in the rotating
frame) notices that body doesn’t fall towards the force center.
To that observer, the body is stationary (in equilibrium). Total
“force” = 0 in the rotating frame:
 The observer postulates an additional “force”, the
Centrifugal “Force”. It comes solely from the attempt to extend
Newton’s 2nd Law to the non-inertial system! Only possible with a
correction “force”.
• Similar for the Coriolis “Force”: This correction “force” arises
when one attempts to describe the motion of the body relative to rotating
system using Newton’s 2nd Law.
• Bottom Line: In the sense just discussed, the
Centrifugal Force & the Coriolis Force are
“artificial” or “fictitious” forces.
• However, as long as we understand what
they really are (partially a philosophical view)
they are very useful concepts.
• They can be used with the Newtonian & also
the Lagrangian & Hamiltonian methods to
treat complicated problems involving rotating
bodies, relative motion where one body is
translating & the other is rotating, etc.
Example from Marion
• A person does measurements with a hockey puck on a large merrygo-round with frictionless horizontal surface. The merry-go-round
has constant angular velocity ω & rotates counterclockwise as seen
from above. Find the effective force on hockey puck after it is given
a push. Plot the path for various initial directions & velocities of the
puck, as observed by a person on the merry-go-round who pushes
the puck (how does he stay on??). See figure: