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): (x1,x2,x3) & 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”: (x1,x2,x3), “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:

1. Differentiate (1) in the fixed system:

(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

Moving frameis translatingAND rotating!

(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 + vr + 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 + vr + 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 (2)

• 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 (4)

• 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 (6)

ar = Acceleration in rotating frame.

• We know: (dr/dt)fixed (dr/dt)rotating + ω r

Or: (dr/dt)fixed = vr + ω r (7)

vr = Velocity in rotating frame.

• 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 merry-

go-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: