The study of the motion of a body along a general curve. û N the unit vector normal to the curve. Clearly, these unit vectors change with time, uˆ

Size: px

Start display at page:

Download "The study of the motion of a body along a general curve. û N the unit vector normal to the curve. Clearly, these unit vectors change with time, uˆ"

Audrey Mosley
5 years ago
Views:

1 Secion. Curilinear Moion he sudy of he moion of a body along a general cure. We define û he uni ecor a he body, angenial o he cure û he uni ecor normal o he cure Clearly, hese uni ecors change wih ime, ( ), ( ) Bu, heir lenghs are always ( ) = 1, ( ) = 1 And, we can always wrie a ecor u as u = u + u he elociy is always angenial o he cure, = ( ) ( ) he acceleraion a is no always angenial o he cure: d d a = = ( ( ) ( ) ) d d = + d Bu wha is? d For sraigh-line moion, û is consan, i.e. = 0. Oherwise, û changes direcion (no magniude, always 1). Le he pah of he body include a small arc lengh ds from he poin A o he nearby poin A, urning hrough a small angle dϕ. ha is, u ˆ a A makes he angle dϕ wih û a A. he change in is d = = dϕ So, Bu d dϕ = dϕ dϕ ds dϕ = = ds ds

2 he normals o he cure a A and A mee a a poin C. he disance o C defines he radius of curaure of he arc. hen dϕ 1 dϕ ds = dϕ, =, = ds d and = Hence a = d + ˆ u ha is, he acceleraion has radial and angenial componens: a = d a = he radial componen changes he direcion of he elociy. For uniform moion along he cure, a = 0, so body moes a consan speed. Velociy aries, wih a 0 For recilinear moion, a = 0, radius is. hese resuls are needed for planeary moion a paricularly imporan applicaion. CICULA MOIO Specialise o case where pah is circular. Since is always angenial, radial direcion ds dθ = = = ω ω is angular elociy, radians per second. In ecor noaion: Le r be he posiion ecor of he body from an arbirary poin on he axis, so ha angle beween r and he axis ω is γ. hen r = r, ω = ω (oe - ω is = r sin γ and so = ω r defined o hae = = ω = ωr sin γ lengh ω)

3 Acceleraion: angenial: a d d dω = = ω =, because consan. So a = α where α dω adial: a =, or a = Hence cenripeal or cenrifugal force, from F=ma, is m F = = mω he acceleraion a is due o he cenripeal force exered on he body o keep i moing in a circle. he cenripeal force acs cenrally, i.e. is always direced o he cenre, and i is responsible for changing he direcion of he moion. I does no change he magniude. As Acion equals eacion (ewon s Law) i is perfecly correc and ofen conenien o consider he cenrifugal force, which is he force he body exers. Uniform circular moion: d a = = 0 dω α = = 0 angenial acceleraion Angular acceleraion

4 In ecor form: d d a = = ( ω r ) dω dr = r + ω = 0 (uniform circular moion) = Cenripeal acceleraion in ecor form So a ω = ω ( ω r ) oe: A (B C) (A B) C MASS O SIG Mass m aached o cenre by sring lengh r, roaing in circle herefore of radius r, a angular elociy ω. eglec graiy. We may immediaely wrie down he ension in he sring: = ma = mω his is he (inward) cenripeal force exered by he sring on he mass, responsible for he (inward) acceleraion of he mass. We may also idenify he (ouward) force (reacion) exered by he mass on he sring, he cenrifugal force, responsible for he ension in he sring. Oher aspecs of his problem will be inesigaed laer. OAIO OF HE EAH he Earh roaes on is axis, wih consan ω for he uniform circular π 1 moion of all poins. oe ω = = 7.7 μrad sec. Consider a poin A on surface a laiude λ. he angenial elociy a A is 1 = rω = ω cosλ = 463cosλ m s = 1036 cosλ mph where is he radius of he Earh (6370 km) and r is he disance of A from he axis of roaion. he cenripeal acceleraion is a = ω = rω cosλ = cosλ m s A he equaor (λ = 0), his is 0.3% of g.

5 BAKED AILWAY ACK On cures (radius r), railway rack is banked ( supereleaed ) o supply cenripeal force for rains running a speed. Wha is he required angle α of bank? equired force is m F = r his mus be he horizonal componen of he normal reacion of rack on rain, i.e. m F = F sin α = r Bu he weigh of he rain mus equal he erical componen of he normal reacion of rack on rain, i.e. mg = F cosα So mg m F = = cosα r sin α an α = gr For ypical alues, = 100 mph, r = 1 mile, 1 ( 44.7 m s ) α = arcan = = 7 gr 9.81 m s 1609 m For he sandard gauge of 4 f 8½ in, his means he ouer rail is lifed ( supereleaed ) seen inches aboe he inner rail. Exercise: Wha do he passengers feel in a rain which is saionary on his cure? In a rain which goes round he cure a 150 mph?

6 UIFOMLY OAIG FAMES OF EFEECE Consider a saionary frame of reference S, coordinaes ( x y, z, ), and origin O, and a frame S, roaing abou he z-axis a a consan angular elociy ω and wih origin O = O, which herefore has a coordinae sysem x, y, z = z, =. ( ) We wan o derie relaionships beween he quaniies such as posiion, elociy and acceleraion measured in S and measured in S. Consider a body a a poin A a res in S. Clearly in S he body is in circular moion and has a elociy = ω r If, howeer, he body, he poin A, moes a wih respec o frame S, hen ecor addiion of elociies gies is elociy in S as = + ω r And is acceleraion? [Viewed from S] As always, we need only differeniae he elociy wih respec o ime o ge he acceleraion. his can be done wih ecors. Howeer, i is a ricky example of ecor calculus and will be presened in M (Semeser B). he body has an acceleraion in S which we call a. In S we see addiionally he cenripeal acceleraion ω r. We see also he Coriolis acceleraion if here is a radial componen in. COIOLOIS FOCE Consider an air curren flowing from he orh Pole o he Equaor. I sars off wih no Eas-Wes elociy. As i flows souh, he Earh urns under i o he Eas. (he sun rises in he Eas.) If i underwen no easward acceleraion, by he ime i reached he Equaor i would consiue a 1000mph Eas wind. From he Earh as a frame of reference, i would appear ha large wesward forces had been exered on i. his is he Coriolis force. Like cenrifugal force, i is ermed ficiious. he Coriolis acceleraion can be deried wihou ecor calculus: Consider he x-y planes of he saionary and roaing frames of reference;

7 y ω y P x ωr P x A =0: x is aligned wih x, y is aligned wih y A =0 le us projec a paricle radially ouwards from he cenre o P in S a = 0 o a poin P a a radius r a ime. A ime, he paricle reaches P and is radial elociy is r =. From he poin of iew of S i sared moing owards a poin P coinciden wih P a = 0 hen follows he rajecory in red in he aboe diagram. When i reaches P, ha poin is now a disance ωr away from P angenially. he body sared wih no angenial elociy (a he cenre. So from he poin of iew of S i has acceleraed angenially, and using s = ½a and puing in he alues, we hae a s = ωr = = = = ωr ωr r ω he Coriolis force is angenial, and independen of radius, so i acs een a he cenre. Ficicious forces (someimes called pseudo-forces) o from physical ineracion bu resul from he acceleraion of a non-inerial frame of reference. We hae encounered wo: Cenrifugal acceleraion acs radially, = ω r = ω ( ω r ) Coriolis acceleraion acs angenially, a a = ω = ω

8 MOIO ELAIVE O EAH Falling Bodies Cenrifugal erm: Le g 0 be acceleraion due o graiy if Earh didn roae (i.e. graiy as iewed from S ). hen he effecie graiy, graiy as iewed from he Earh s roaing frame S, is g e = g 0 ω ( ω r ') hese are no parallel, wih g 0 poining owards he cenre of he Earh, and he cenrifugal erm poining ouwards from he Earh s axis. So graiy is reduced and iled owards he Equaor. Bodies falling owards he ground in he orhern hemisphere are displaced o he Souh, while bodies falling in he Souhern hemisphere are displaced o he orh. he displacemen anishes a he Poles and a he Equaor. Coriolis erm: ha only applies o bodies ha hae no elociy in he Earh s frame. Le he body be falling erically a elociy. hen Coriolis erm is ω ', which poins Eas in boh hemispheres. he displacemen anishes a he Poles and is maximum a he Equaor. Bodies wih angenial Velociy: orhward in orhern hemisphere, ω poins o he Eas, and he moion is defleced o he Eas. orhwards in Souhern hemisphere, ω poins o he Wes and he moion is defleced o he Wes. he effec is maximum a he Poles and anishes a he Equaor. Consequences of he Coriolis Froce for he Weaher: 1. Cyclones. A region a low pressure ends o fill as air flows in radially, a righ angles o he conours of consan pressure (isobars). he Coriolis force deflecs he radial moion, o he righ in orhern hemisphere (and o he lef in Souhern hemisphere). his ses up an ani-clockwise roaion (clockwise in Souhern). A cyclone becomes sable when he air flow is parallel o he isobars. Look for his on weaher maps.. rade Winds. he larges scale paern in he amosphere is he conecion of hea from he Equaor o he Poles, wih cold air reurning Souh a sea-leel. his curren of air is defleced o he Wes (in boh hemispheres), so ha he mos sable wind paerns are he rade Winds, a orh-eas wind in he orhern hemisphere and a Souh-Eas wind in he Souhern hemisphere. Look for hese on weaher maps.

Physics Notes - Ch. 2 Motion in One Dimension

Physics Notes - Ch. 2 Motion in One Dimension Physics Noes - Ch. Moion in One Dimension I. The naure o physical quaniies: scalars and ecors A. Scalar quaniy ha describes only magniude (how much), NOT including direcion; e. mass, emperaure, ime, olume,