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

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1 Section. Cuilinea Motion he study of the motion of a body along a geneal cue. We define u ˆ û the unit ecto at the body, tangential to the cue the unit ecto nomal to the cue Clealy, these unit ectos change with time, u ˆ ( t), uˆ ( t) But, thei lengths ae always uˆ ( t) = 1, uˆ ( t) = 1 And, we can always wite a ecto u as he elocity is always tangential to the cue, = ( t) uˆ ( t) u = u uˆ + u uˆ he acceleation a is not always tangential to the cue: d d a = = ( ( t) uˆ ( t) ) d duˆ = uˆ + duˆ But what is? Fo staight-line motion, u ˆ is constant, i.e. d uˆ = 0. Othewise, u ˆ changes diection (not magnitude, always 1). Let the path of the body include a small ac length ds fom the point A to the neaby point A, tuning though a small angle dϕ. hat is, u ˆ at A makes the angle dϕ with û at A. he change in uˆ is duˆ = uˆ uˆ = dϕ uˆ So, But duˆ dϕ = uˆ dϕ dϕ ds dϕ = = ds ds

2 he nomals to the cue at A and A meet at a point C. he distance to C defines the adius of cuatue of the ac. hen dϕ 1 dϕ ds = dϕ, =, = ds duˆ and = uˆ Hence a = d uˆ + ˆ u hat is, the acceleation has adial and tangential components: a = d a = he adial component changes the diection of the elocity. Fo unifom motion along the cue, a = 0, so body moes at constant speed. Velocity aies, with a 0 Fo ectilinea motion, a = 0, adius is. hese esults ae needed fo planetay motion a paticulaly impotant application. CICULA MOIO Specialise to case whee path is cicula. Since is always tangential, adial diection ds dθ = = = ω ω is angula elocity, adians pe second. In ecto notation: Let be the position ecto of the body fom an abitay point on the axis, so that angle between and the axis ω is γ. hen =, ω = ω (ote - ω is = sin γ and so = ω defined to hae = = ω = ω sin γ length ω)

3 Acceleation: angential: a d d dω = = ω =, because constant. So a = α whee α dω adial: a =, o a = uˆ Hence centipetal o centifugal foce, fom F=ma, is m F = = mω he acceleation a is due to the centipetal foce exeted on the body to keep it moing in a cicle. he centipetal foce acts centally, i.e. is always diected to the cente, and it is esponsible fo changing the diection of the motion. It does not change the magnitude. As Action equals eaction (ewton s Law) it is pefectly coect and often conenient to conside the centifugal foce, which is the foce the body exets. Unifom cicula motion: d a = = 0 dω α = = 0 angential acceleation Angula acceleation

4 In ecto fom: d d a = = ( ω ) dω d = + ω = 0 (unifom cicula motion) = Centipetal acceleation in ecto fom So a ω = ω ( ω ) ote: A (B C) (A B) C MASS O SIG Mass m attached to cente by sting length, otating in cicle theefoe of adius, at angula elocity ω. eglect gaity. We may immediately wite down the tension in the sting: = ma = mω his is the (inwad) centipetal foce exeted by the sting on the mass, esponsible fo the (inwad) acceleation of the mass. We may also identify the (outwad) foce (eaction) exeted by the mass on the sting, the centifugal foce, esponsible fo the tension in the sting. Othe aspects of this poblem will be inestigated late. OAIO OF HE EAH he Eath otates on its axis, with constant ω fo the unifom cicula π 1 motion of all points. ote ω = = 7.7 µad sec. Conside a point A on suface at latitude λ. he tangential elocity at A is = ω = ω cosλ = 463cosλ m s = 1036 cosλ mph whee is the adius of the Eath (6370 km) and is the distance of A fom the axis of otation. he centipetal acceleation is a 1 = ω = ω cosλ = cosλ m s At the equato (λ = 0), this is 0.3% of g.

5 BAKED AILWAY ACK On cues (adius ), ailway tack is banked ( supeeleated ) to supply centipetal foce fo tains unning at speed. What is the equied angle α of bank? equied foce is m F = his must be the hoizontal component of the nomal eaction of tack on tain, i.e. m F = F sin α = But the weight of the tain must equal the etical component of the nomal eaction of tack on tain, i.e. mg = F cosα So F mg m = = cosα sin α tan α = g Fo typical alues, = 100 mph, = 1 mile, 1 ( 44.7 m s ) o α = actan = = 7 g 9.81 m s 1609 m Fo the standad gauge of 4 ft 8½ in, this means the oute ail is lifted ( supeeleated ) seen inches aboe the inne ail. Execise: What do the passenges feel in a tain which is stationay on this cue? In a tain which goes ound the cue at 150 mph?

6 UIFOMLY OAIG FAMES OF EFEECE Conside a stationay fame of efeence S, coodinates ( x, y, z, t) and oigin O, and a fame S, otating about the z-axis at a constant angula elocity ω and with oigin O = O, which theefoe has a coodinate system ( x, y, z = z, t = t). We want to deie elationships between the quantities such as position, elocity and acceleation measued in S and measued in S. Conside a body at a point A at est in S. Clealy in S the body is in cicula motion and has a elocity = ω If, howee, the body, the point A, moes at with espect to fame S, then ecto addition of elocities gies its elocity in S as = + ω And its acceleation? [Viewed fom S] As always, we need only diffeentiate the elocity with espect to time to get the acceleation. his can be done with ectos. Howee, it is a ticky example of ecto calculus and will be pesented in M (Semeste B). he body has an acceleation in S which we call a. In S we see additionally the centipetal acceleation ω. We see also the Coiolis acceleation if thee is a adial component in. COIOLOIS FOCE Conside an ai cuent flowing fom the oth Pole to the Equato. It stats off with no East-West elocity. As it flows south, the Eath tuns unde it to the East. (he sun ises in the East.) If it undewent no eastwad acceleation, by the time it eached the Equato it would constitute a 1000mph East wind. Fom the Eath as a fame of efeence, it would appea that lage westwad foces had been exeted on it. his is the Coiolis foce. Like centifugal foce, it is temed fictitious. he Coiolis acceleation can be deied without ecto calculus: Let the body moe outwads adially in S fom the cente at t = 0 to a point P at a adius at time t. hen its adial elocity is = t Fom the point of iew of S it stated moing towads a point P coincident with P at t = 0. When it eaches P, that point is now a distance ωt away

7 fom P tangentially. he body stated with no tangential elocity (at the cente. So fom the point of iew of S it has acceleated tangentially, and using s = ½ and putting in the alues, we hae a a t s ωt ω ω = = = = = ω t t t he Coiolis foce is tangential,and independent of adius, so it acts een at the cente. MOIO ELAIVE O EAH Falling Bodies Centifugal em: Let g 0 be acceleation due to gaity if Eath didn t otate (i.e. gaity as iewed fom S ). hen the effectie gaity, gaity as iewed fom the Eath s otating fame S, is g e = g 0 ω ( ω ) hese ae not paallel, with g 0 pointing towads the cente of the Eath, and the centifugal tem pointing outwads fom the Eath; axis. So gaity is educed and tilted towads the Equato. Bodies falling towads the gound in the othen hemisphee ae displaced to the South, while bodies falling in the Southen hemisphee ae displaced to the oth. he displacement anishes at the Poles and at the Equato. Coiolis em: hat only applies to bodies that hae no elocity in the Eath s fame. Let the body be falling etically at elocity. hen Coiolis tem is ω, which points East in both hemisphees. he displacement anishes at the Poles and is maximum at the Equato. Bodies with angential Velocity: othwad in othen hemisphee, ω points to the East, and the motion is deflected to the East. othwads in Southen hemisphee, ω points to the West and the motion is deflected to the West. he effect is maximum at the Poles and anishes at the Equato.

8 Consequences of the Coiolis Foce fo the Weathe: 1. Cyclones. A egion at low pessue tends to fill as ai flows in adially, at ight angles to the contous of constant pessue (isobas). he Coiolis foce deflects the adial motion, to the ight in othen hemisphee (and to the left in Southen hemisphee). his sets up an anti-clockwise otation (clockwise in Southen). A cyclone becomes stable when the ai flow is paallel to the isobas. Look fo this on weathe maps.. ade Winds. he lagest scale patten in the atmosphee is the conection of heat fom the Equato to the Poles, with cold ai etuning South at sea-leel. his cuent of ai is deflected to the West (in both hemisphees), so that the most stable wind pattens ae the ade Winds, a oth-east wind in the othen hemisphee and a South-East wind in the Southen hemisphee. Look fo these on weathe maps.

DYNAMICS OF UNIFORM CIRCULAR MOTION

DYNAMICS OF UNIFORM CIRCULAR MOTION Chapte 5 Dynamics of Unifom Cicula Motion Chapte 5 DYNAMICS OF UNIFOM CICULA MOTION PEVIEW An object which is moing in a cicula path with a constant speed is said to be in unifom cicula motion. Fo an object