Rotating Coordinate Systems For Translating Systems We just

Rotating Coordinate Systems • For Translating Systems: We just got Newton’s 2 nd Law (in the accelerated frame): ma = F where F = F - ma 0 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 2 nd Law (in the accelerated frame): ma = F where F = F - mar A non-inertial or fictitious force: Solely from the COORDINATE TRANSFORMATION! For rotating systems: mar = much more complicated expressions than ma 0!

• 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): (x 1, x 2, x 3). 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”: (x 1, x 2, x 3). • 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, (d. G/dt)fixed = (d. G/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 = (d. R/dt)fixed + (dr/dt)fixed 2. Use (dr/dt)fixed = (dr/dt)rotating + ω r (dr /dt)fixed = (d. R/dt)fixed + (dr/dt)rotating + ω r

(dr /dt)fixed = (d. R/dt)fixed + (dr/dt)rotating+ ω r (2) • Define: vf rf (dr /dt)fixed = Velocity relative to the fixed axes. V R (d. R/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 2 nd 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 2 nd 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 = (d. V/dt)fixed + (dvr/dt)fixed + [(dω/dt)fixed r] + ω (dr/dt)fixed (2) • Define: Af (d. V/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 [vr + ω r] • Put into Newton’s 2 nd Law (above) & rearrange: F = maf = m. Af + mar + m(ω r) + m[ω (ω r)] + 2 m(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 - m. Af - m(ω r) - m[ω (ω r)] - 2 m(ω vr)

• Applying Newton’s 2 nd Law to the rotating frame yields: Feff mar F - m. Af - m(ω r) - m[ω (ω r)] - 2 m(ω vr) • Physical Interpretations: - m. Af : 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! - 2 m(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 mω2 r. 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 2 nd Law-like eqtn in the rotating frame: Feff mar , when, in fact Newton’s 2 nd 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 2 nd 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 2 nd 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: