Reading: Matin, Section. ROTATING REFERENCE FRAMES ESCI 34 Atmospheic Dnamics I Lesson 3 Fundamental Foces II A efeence fame in which an object with zeo net foce on it does not acceleate is known as an inetial efeence fame. A efeence fame attached to the Eath is a noninetial efeence fame, since a net foce is equied to keep an object in one spot with espect to the efeence fame. Use of a noninetial efeence fame equies the intoduction of appaent foces in ode to use Newton s second law. This can be demonstated as follows: Imagine that Peson A is on a otating tuntable while Peson B walks in a staight line at a constant speed towad them as pictued in the left illustation below. Since Peson B is moving in a staight line at constant speed, thee is no acceleation and theefoe no net foces acting on Peson B. Fom the standpoint of Peson A, who is in a otating efeence fame, Peson B is spialing towad him as in the illustation on the ight. Theefoe, Peson A assumes that thee must be a foce on Peson B, since Peson B is acceleating. This foce is an appaent foce, since it onl appeas in the non-inetial (otating) efeence fame.
If Peson B is not moving in the oiginal efeence fame thee is still an appaent foce in the noninetial fame since Peson B will be moving in a cicle aound Peson A (fom Peson A s efeence fame) as shown below. Appaent foces ae not eal, though the appea to be eal to someone in the noninetial efeence fame. Appaent foces cannot do wok on an object, no can the change the speed of the object; the can onl change the diection of motion of the object. DERIVATIVES OF VECTORS IN ROTATING AND NONROTATING FRAMES The appaent foces can be developed mathematicall as follows. Imagine two efeence fames, one inetial and one otating, shaing a common oigin and a common z-axis. The otating fame as an angula velocit of (ad/s) about the z- axis. A vecto A can be epesented in component fom in eithe fame as A a iˆ a ˆj aiˆ a ˆj. (1) (Pimes indicate the otating efeence fame.) x x The deivative of (1) with espect to time in the nonotating fame is da da da x ˆ ˆ daa i j. () The subscipt a is just a wa to emind us that this is the deivative fom the pespective of an obseve in the absolute (nonotating) efeence fame.
In the otating fame the deivative is da da da ˆ ˆ x ˆ ˆ di d j i j ax a. (3) dt An obseve in the otating efeence fame would peceive the deivative of A as simpl da da da x ˆ i ˆj. (4) dt (the would be unawae of the additional tems involving the deivatives of the unit vectos). The subscipt indicates this is the deivative as peceived in the otating fame. Using (4) in (3) we get da da ˆ ˆ di d j ax a. (5) The esults fom (3) and (5) must be identical, since da dt is a geometic invaiant expession. We can theefoe equate them to get da da di ˆ d ˆ j a ax a. (6) We need to evaluate the deivatives of the unit vectos in the otating fame. The tem diˆ dt is evaluated as follows: diˆ iˆ ˆ ˆ t t i t t j lim lim ˆj dt t0 t t0 t (efe to the diagam below). Similal we can show that d ˆj dt iˆ. Theefoe we have d a A d A a ˆ x j a iˆ (7) which is the same as d A d A a A. (8) Equation (8) shows how the deivative of a vecto can be tansfomed between an inetial and a otating efeence fame. Even though the deivation was done fo a vecto in onl two dimensions, it woks egadless of the numbe of dimensions. 3
Equation (8) is also valid even if the z-axes of the oigins of the two coodinate sstems do not coincide and/o thei z-axes aen t paallel. ACCELERATIONS IN ROTATING VERSUS NONROTATING FRAMES Appling (8) to the position vecto of a point in space ields d d a which is also V V (9) V a a whee is the velocit in the absolute fame and is the velocit in the otating fame. Appling (8) to the absolute velocit vecto gives d ava d V a Va. (10) Substituting fom (9) into the ight-hand side onl of (10) gives d V dv d a a V dt which simplifies to d V d V a a V. (11) Equation (11) shows the elationship between the acceleation obseved in the absolute fame and the acceleation obseved in the otating fame. If the two efeence fames did not shae a common axis then equation (11) would be d V d V a a V R (1) whee the vecto R is a vecto nomal to the axis of otation and pointing to the position of the object. V a NEWTON S SECOND LAW IN A ROTATING REFERENCE FRAME If we wee witing Newton s second law of motion fo an ai pacel in the absolute efeence fame we would onl need to include the pessue gadient foce, the gavitational foce, and the viscous foce, and would have dv a a 1 p g * Va. (13) dt Howeve, fom (11) this can be witten as dv 1 V p g * Va dt and noting that V V (see execises), we get a dv 1 p g * V V R (14) dt V 4
Equation (14) is Newton s second law fo the otating coodinate sstem. The two additional tems that appea in the otating coodinate sstem vesion, but not in the absolute coodinate sstem vesion, equation (13), ae the acceleations due to the appaent foces. The fist new tem is the Coiolis acceleation, and the second is the centifugal acceleation. THE CORIOLIS FORCE We ve seen alead that the acceleation due to the Coiolis foce is given b aco V. (15) Notice that the Coiolis foce depends on the speed of the object elative to the otating fame. If the object is at est elative to the otating fame then the Coiolis foce is zeo. Notice also that the Coiolis acts at ight angles to the motion. Relative to the otating efeence fame at the suface of the Eath, can be witten in component fom as cos ˆj sin kˆ, (16) whee is latitude. The velocit vecto is V ˆ ˆ ˆ ui v j wk. The Coiolis acceleation theefoe has components of a co v sin wcos iˆ u sin ˆj u cos kˆ. (17) An object moving towad the East (u > 0) will be a deflected upwad and southwad due to the Coiolis foce. An object moving towad the West (u < 0) will be a deflected downwad and nothwad due to the Coiolis foce. An object moving towad the Noth (v > 0) will be deflected eastwad due to the Coiolis foce. An object moving towad the South (v < 0) will be deflected westwad due to the Coiolis foce. An object moving upwad will be deflected westwad due to the Coiolis foce. An object moving downwad will be deflected eastwad due to the Coiolis foce. CENTRIFUGAL FORCE AND GRAVITY If an object is in cicula motion in an absolute efeence fame, it equies a centipetal acceleation of R, which is diected inwad towad the axis of otation. In a efeence fame otating with the object, the object is not acceleating, so thee is an appaent foce, the centifugal foce, that is invoked to balance the centipetal foce and keep the object fom acceleating. The tem R in equation (4) epesents the centifugal acceleation due to the Eath s otation. The centifugal foce is alwas diected awa fom the axis of otation. 5
On the Eath the centifugal foce appeas to pull is awa fom the suface (except at the Poles) and to make us feel lighte. This effect is most ponounced at the Equato. The centifugal foce is combined with the gavitational foce to define a new foce called gavit. The acceleation due to gavit foce is defined as g g * R. (18) Equation (14) theefoe becomes dv 1 p V g V. (19) dt When ou see gavit in an equation such as(19), keep in mind that it is a combination of the gavitational acceleation plus centifugal acceleation. Note that gavit has a plus sign in (19), but keep in mind that if witten in component fom it lies solel in the negative diection,. (0) Gavit is not diected exactl though the cente of the Eath except at the Poles and at the Equato. The centifugal foce causes the Eath to not be a pefect sphee. Instead it is an oblate spheoid, with a lage adius at the Equato than at the Poles. THE MOMENTUM EQUATON ˆk g gkˆ Equation (19) is known as the momentum equation. Fom now on we leave off the subscipt on the velocit, and just emembe that it is the velocit elative to the Eath in ou otating coodinate sstem. So we will wite it fom now on as dv 1 p V g V. (1) dt The tems in the momentum equation ae: 1 acceleation due to the pessue gadient foce: p Coiolis acceleation: V Gavit (includes centifugal acceleation): g Fiction (viscous acceleation): COORDINATE REPRESENTATION OF THE MOMENTUM EQUATION V Equation (1) is a vecto equation. It is valid as is, without modification, in an coodinate sstem. Howeve, fo calculations and othe puposes it is often necessa to wite it out in component fom. This is done b fist choosing a coodinate sstem, and then expanding each tem into its component fom. Then, all the tems fo a paticula component ae gouped togethe, esulting in thee scala equation, one fo each coodinate. 6
This is illustated hee using a local Catesian coodinate 1 sstem: Expanding each tem of (1) into this coodinate sstem we get: Fo the acceleation tem we have dv d ˆ ˆ ˆ ˆ ˆ ˆ du ˆ dv ˆ dw ˆ ui vj wk i j k u di v dj w dk, () dt dt but in a Catesian fame the unit vectos ae constants in both time and space, so thei deivative ae zeo. Theefoe, in Catesian coodinates dv du iˆ dv ˆj dw kˆ, (3) The pessue-gadient tem expands as 1 1 p ˆ 1 p ˆ 1 p p i j kˆ. (4) x z The Coiolis tem expands as V v sin wcos iˆ u sin ˆj ucos k (5) ˆ The gavit tem expands as g gkˆ (6) The viscous tem expands as ˆ V uiˆ vj ˆ wk u iˆ v ˆj w kˆ, (7) but in Catesian coodinates the tems in paentheses ae zeo, because deivatives of the unit vectos ae zeo. Theefoe, in a Catesian fame we have V uiˆ vj ˆ wkˆ. (8) Collecting the like components ( î,, and ) togethe we end up with the thee Catesian-component momentum equations, du 1 p vsin wcos u dt x (9) dv 1 p usin v dt (30) dw 1 p u cos g w dt z (31) In paameteized spheical coodinates the deivatives of the unit vectos, the tems in paentheses in () and (7), ae not zeo, and these lead to additional tems, called cuvatue tems, which involve the adius of the Eath (denoted b a). In paameteized spheical coodinates the component equations ae ĵ ˆk 1 Local Catesian coodinates ae coodinates based on a flat plane tangent to the Eath s suface at a paticula latitude,. 7
du uv tan uw 1 p vsin wcos dt a a x v 1 w u u tan tan a x a x a dv u tan vw 1 p u sin dt a a u v w w v tan 1 tan tan a x a a a dw u v 1 p ucos g dt a z u v w u 1 w v w a x a a a a x a tan (3) (33) (34) The tems in squae backets in (3), (33), and (34) ae the cuvatue tems, and ae the onl diffeence between the component epesentation in paameteized-spheical coodinates and Catesian coodinates. 8
EXERCISES 1. Show that if k ˆ then a ˆ j a iˆ x A. Show that if and. ae nomal to each othe then 3. a. Show that the magnitude of the gavit foce at latitude is given b g acos g * acos sin whee a is the adius of the Eath. (Hint: Wite g* and the centifugal acceleation in component fom and add them togethe. Then find the magnitude of the esultant vecto. You ma efe to the diagam below.) b. Using the esults fom pat a., find the magnitude of the gavit foce at the Noth Pole, 45N, and at the Equato (g* = 9.81 m/s, the adius of the Eath is 6378 km, and = 7.910 5 ad/s). c. Using ou esults fo the gavit foce fom pat a., find the geopotential height at an altitude of 5000 metes at the Noth Pole, 45N, and at the Equato (assume that the gavit foce is constant with height). 9
4. Assume that the gavit foce at the suface is g0 = 9.80665 m/s. Calculate the geopotential height at an altitude of 5,000 metes fo the following two cases: a. Gavit is constant with height. b. Gavit deceases with height accoding to the following fomula, g 0 g 1 z a whee a is the adius of the Eath (6378 km). c. Fom these esults, do ou think its impotant to include the vaiation of gavit with height? V v sin w cos iˆ u sin ˆj u cos ˆ. 5. Show that k 6. An ant is walking on a tuntable that is otating clockwise at 5 evolutions pe minute (pm). A coodinate sstem (x, ) is otating with the tuntable, the oigin of which is the cente of the tuntable, with the x and -axes pointing adiall outwad. At time t = 0, this coodinate sstem is pefectl aligned with a coodinate sstem fixed to the non-otating oom (x, ). The ant is initiall at coodinates x = x = 0, = = 1 cm, and with espect to the tuntable is taveling along the -axis at a constant speed of 0.5 cm/s. a. What is the angula velocit of the tuntable in ad/s? b. What ae the components (in the otating efeence fame) of the ant s Coiolis acceleation at time t = 0? c. What ae the components (in the otating efeence fame) of the ant s centifugal acceleation at time t = 0? d. What ae the components of the ant s velocit in the efeence fame fixed to the oom? e. What ae the components of the ant s acceleation in the efeence fame fixed to the oom? 7. A coodinate sstem is otating counte-clockwise aound it s ˆk axis at an angula speed. An object is located at position, and has a velocit such that the Coiolis acceleation is exactl balanced b the centifugal acceleation. 10
a. What ae the u and v components of the velocit of the object in the otating coodinate sstem? b. What ae the u and v components of the velocit of the object in a non-otating coodinate sstem? c. What will the path of the object look like in the otating coodinate sstem? d. What will the path of the object look like in a non-otating coodinate sstem? 1. Show that Va V. Hint: To do this, ou will need to show that 0. Use the identit A A A, and ecognize that is the tangential velocit of solid-bod otation. Also, the velocit divegence in solid-bod otation is zeo, and the voticit in solid-bod otation is constant. 11