11. Dynamics in Rotating Frames of Reference

Transcription

1 Unversty of Rhode Island Classcal Dynamcs Physcs Course Materals Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons Lcense Ths work s lcensed under a Creatve Commons Attrbuton-Noncommercal-Share Alke 4.0 Lcense. Follow ths and addtonal works at: Abstract Part eleven of course materals for Classcal Dynamcs (Physcs 520), taught by Gerhard Müller at the Unversty of Rhode Island. Documents wll be updated perodcally as more entres become presentable. Recommended Ctaton Müller, Gerhard, "11. Dynamcs n Rotatng Frames of Reference" (2015). Classcal Dynamcs. Paper Ths Course Materal s brought to you for free and open access by the Physcs Course Materals at DgtalCommons@URI. It has been accepted for ncluson n Classcal Dynamcs by an authorzed admnstrator of DgtalCommons@URI. For more nformaton, please contact dgtalcommons@etal.ur.edu.

2 Contents of ths Document [mtc11] 11. Dynamcs n Rotatng Frames of Reference Moton n rotatng frame of reference [mln22] Effect of Corols force on fallng object [mex61] Effects of Corols force on an object projected vertcally up [mex62] Foucault pendulum [mex64] Effects of Corols force and centrfugal force on fallng object [mex63] Lateral deflecton of projectle due to Corols force [mex65] Effect of Corols force on range of projectle [mex66] What s vertcal? [mex170] Lagrange equatons n rotatng frame [mex171] Holonomc constrants n rotatng frame [mln23] Parabolc slde on rotatng Earth [mex172]

3 Moton n rotatng frame of reference [mln22] Consder two frames of reference wth dentcal orgns. Frame R s rotatng wth angular velocty ω relatve to the nertal frame I. Coordnate axes: e (I) = const, ė (R) = ω e (R), = 1, 2, 3. Knematcs of a partcle movng n frame R. Poston: r R = r I. = r, where ri = Velocty: dr dt = 3 =1 ẋ (I) e (I) = 3 =1 ẋ (I) e (I) = 3 [ =1 ẋ (R) e (R) Acceleraton: d2 r dt = dv I 2 dt = ( ) dvr wth dt I 3 [ =1 ẋ (R) 3 =1 e (R) + ω x (R) = a R + ω v R, ω x (I) e (I), r R = + x (R) e (R) ] ė (R) ( ) dvr + dt ω r + ω I ]. 3 =1 x (R) e (R). v I = v R + ω r. ( ) dr. dt I ( ) dr = ω v R + ω ( ω r). dt I a I = a R + ω r + 2 ω v R + ω ( ω r). Dynamcs of a partcle of mass m. Inertal frame: ma I = F I Rotatng frame: ma R = F I m ω r 2m ω v R m ω ( ω r). Real and fcttous forces: F I (appled force). m ω r (due to angular acceleraton of frame R). 2m ω v R (Corols force). m ω ( ω r) (centrfugal force). If the orgn of frame R undergoes a lateral moton n addton to the rotaton, then a term m(d 2 R/dt 2 ) must be added to the fcttous forces. Here R s the vector pontng from the orgn of frame I to the orgn of frame R.

4 [mex61] Effect of Corols force on fallng object Consder a locaton at northern lattude λ on the Earth s surface. A partcle of mass m starts fallng from rest at poston r 0 = (0, 0, h) n the local coordnate system wth axes as shown n the fgure. (a) Determne the poston r(t) = (x(t), y(t), z(t)) durng the fall. Perform the calculaton to leadng order n ω, the Earth s angular velocty of rotaton. (b) If h = 100m, g = 9.8m/s 2 and λ = 45, what s the magntude and drecton of horzontal deflecton from the vertcal lne of the pont where the partcle hts the ground. N λ ez ey ex S

5 [mex62] Effects of Corols force on an object projected vertcally up A partcle of mass m s projected vertcally up from a pont on the Earth s surface at northern lattude λ. (a) Fnd the deflecton (x 1, y 1 ) of the path from the vertcal at z 1 = h, where the partcle reaches ts maxmum heght. Use the local frame of reference wth the orgn at the launch ste and the axes as shown. Express the result as a functon of λ (angle of lattude), ω (angular frequency of Earth s rotaton), g (acceleraton due to gravty), and h (maxmum heght reached by partcle). Keep only terms up to lnear order n ω. (b) Fnd the deflecton (x 2, y 2 ) of the path at z 2 = 0, when the partcle strkes the ground. N λ ez ey ex S

6 [mex64] Foucault pendulum Consder a locaton at northern lattude λ on the Earth s surface. A pendulum (mass m, length L) s free to swng n any drecton. At tme t = 0, the pendulum s set n moton from a small dsplacement x 0 > 0, y 0 = 0 wth no ntal velocty. (a) Show that the lnearzed equatons of moton ncludng the effect of the Corols force can be expressed n the form q + 2ω z q + Ω 2 q = 0; q x + y, Ω = g/l, ω z = ω sn λ, where ω s the angular frequency of the Earth s rotaton. Ths equaton of moton descrbes a harmonc oscllator wth magnary dampng. (b) Show that for the ntal condtons stated above and for ω z Ω ts soluton s of the form q(t) = x 0 cos Ωte ωzt. (c) Show that the last factor n ths soluton descrbes a precesson wth angular frequency ω z of the plane n whch the pendulum swngs.

7 [mex63] Effects of Corols and centrfugal forces on fallng object Consder a locaton at northern lattude λ on the Earth s surface. A partcle of mass m starts fallng from rest at poston r 0 = (0, 0, h) n the local coordnate system wth axes as shown n the fgure. (a) Determne the poston r(t) = (x(t), y(t), z(t)) durng the fall. Perform the calculaton to second order order n ω usng the frst-order results of [mex61]. (b) Fnd the horzontal deflectons d x, d y from the vertcal lne of the pont where the partcle strkes the ground. Express the result as a functon of λ (angle of lattude), ω (angular frequency of Earth s rotaton), g (acceleraton due to gravty), and h (heght). (c) What are the values of d x, d y f h = 100m, g = 9.8m/s 2 and λ = 45? N λ ez ey ex S

8 [mex65] Lateral deflecton of projectle due to Corols force Consder a locaton at northern lattude λ on the Earth s surface. A partcle s projected due east wth ntal speed v 0 and angle of nclnaton α above the horzontal. Fnd the lateral deflecton d x due to the Corols force of the pont where the partcle strkes the ground. Perform the calculaton to leadng order n ω, the angular frequency of the Earth s rotaton. Express d x as a functon of v 0, g, α, λ, ω. Evaluate the range R (to zeroth order n ω) and the lateral deflecton d x (to frst order n ω) for the case where a projectle s launched wth speed v 0 = 100m/s at angle α = 45. N λ ez ey ex S

9 [mex66] Effect of Corols force on range of projectle Consder a locaton at northern lattude λ on the Earth s surface. A partcle s projected due east wth ntal speed v 0 and angle of nclnaton α above the horzontal. Show that the change n the range R = (2v 2 0/g) sn α cos α of the projectle due to the Corols force s R = [ 2R 3 g ω cos λ cot 1/2 α 1 ] 3 tan3/2 α. Perform the calculaton to leadng order n ω, the angular frequency of the Earth s rotaton. N λ ez ey ex S

10 [mex170] What s vertcal? (a) Calculate the the angular devaton ɛ of a plumb lne from the drecton to the Earth s center at a pont of lattude λ. (b) At what lattude s does ɛ have ts maxmum value? (c) State ɛ max n arc seconds. Use the value g = 9.81m/s 2 for the acceleraton due to gravty.

11 [mex171] Lagrange equatons n rotatng frame From [mex79] we know that the Lagrange equatons are nvarant under a pont transformaton. Here we use ths property to transform the equaton of moton of a partcle n a potental V (r) from an nertal frame to a frame rotatng wth constant angular velocty ω. (a) Express the Lagrangan L(r, ṽ) = L(r, v) = 1 2 mv2 V (r) n terms of the rotatng-frame coordnates. (b) Derve the Lagrange equatons (d/dt)( L/ x ) ( L/ x ) = 0, = 1, 2, 3. (c) Brng the resultng Lagrange equatons nto the form mã = V r 2m ω ṽ m ω ( ω r).

12 Holonomc constrants n rotatng frame [mln23] Recpe for solvng a Lagrangan mechancs problem wth holonomc constrants between coordnates n the rotatng frame. Formulate Lagrangan n nertal frame (I) wthout mposng constrants. Transform coordnates to the rotatng frame (R). Impose holonomc constrants va ndependent generalzed coordnates. Derve Lagrange equatons n frame R. Example: partcle of mass m movng on surface of rotatng Earth n vertcal plane parallel to merdan and subject to scalar potental V. Notes: Lagrangan: L I = 1 2 m( x I 2 + y 2 I + z 2 I ) V (x I, y I, z I ). Earth s angular velocty: ω = ( ω cos λ, 0, ω sn λ). ẋ ωy sn λ Transformaton: v I = v + ω r = ẏ + ωx sn λ + ωz cos λ. ż ωy cos λ ẋ Constrant: y = 0 v I = (ẋ I, ẏ I, ż I ) = ωx sn λ + ωz cos λ ż. Substtute v I nto Lagrangan: L I = L(x, z, ẋ, ż). Lagrange equatons: d L dt ẋ L x = 0, d L dt ż L z = 0. The accelerated translatonal moton can be taken nto account by a modfed acceleraton due to gravty: g = g 0 + ω 2 r [mex170]. In the local coordnate system, e x s pontng south, e y s pontng east, and e z s pontng vertcally up. It s common practce to drop subscrpts R n the rotatng frame to keep the notaton smple.

13 [mex172] Parabolc slde on rotatng Earth A bead of mass m sldes wthout frcton along a wre of parabolc shape, z = Ay 2, n a unform gravtatonal feld g pontng n the negatve z-drecton. In generalzaton to [mex131], the effect of the Earth s rotaton rotaton must be taken nto account under the assumpton that the slde s placed at lattude λ wth ts (vertcal) plane orented perpendcular to the merdan. (a) Construct the Lagrangan L(y, ẏ). (b) Derve the Lagrange equaton.