13.5. Torsion of a curve Tangential and Normal Components of Acceleration
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1 13.5 osion of cuve ngentil nd oml Components of Acceletion
2 Recll: Length of cuve '( t) Ac length function s( t) b t u du '( t) Ac length pmetiztion ( s) with '( s) 1 '( t) Unit tngent vecto '( t) Cuvtue: s '(s) t t t t t 3
3 Ac length function s( t) t u du '( t) mesues speed of motion s t s d d d d if s is c length pmete, then = = '( t) hence ( s) 1 '( t) ' '( t) If s is c length pmete, then '( s) 1 Assume tht t is pmete with '( t) 1 : If you bsepoint is t 0, then s( t) u du 1du t t 0 0 So s t, which mens t is ledy the clength pmete. If I ssume you bsepoint is t, then s mesues distnce tveled stting t t t s( t) u du 1du t t is still n c length pmete, it just mesues distnce stting t in eithe cse, distnce tveled fom s to s is simply Exmples: ) c length pmetiztion of stight line: (s) sv with v = 1 "c length pmetiztion" 0 t t "you tvel with speed 1" b) c length pmetiztion of cicle x +y = : (s) cos( ), sin( ) s s 0s
4 Q P cuvtue t P > cuvtue t Q Unit tngent vecto '( t) '( t) '( s) s s cuvtue mesue how quickly we tun if we tvel t speed 1
5 Fenet Fme: '( t) '( t) d Pincipl unit noml : ( is only defined when 0!) is lso clled the cuvtue vecto since 1, we hve ' 0 o 0 thid vecto is the binoml B B is othogonl to nd nd of unit length: since d d B sin( ) 1 Altogethe, we hve Fenet fme (o B fme),, B nd 0 is scl is othogonl to hey e ll of unit length nd othogonl to ech othe (like i, j,k) they fom moving fme:
6 osion: d 1 d db Clim : is pllel to : db B B 1 B 0 o d d d 0 B B 0 = B Since d B 0 nd d B 0 we see d B B is multiple of his multiple (up to sign) is clled tosion: B db B = db db o db
7 B is the noml vecto to the plne spnned by nd db mesue the "tilt'' of this plne since db we lso hve db (up to sign) mesues the mgnitude of the tilt
8 Exmple: cicle of dius : ( t) cos( t), sin( t),0 c length pmetiztion: ( s) cos( ), sin( ),0 s s s s '( s) sin( ),cos( ),0 s s cos( ), sin( ),0 1 s 1 s cos( ), sin( ),0 1 i j k s s s s cos( ) sin( ) 0 B = sin( ) cos( ) 0 s s sin ( ) cos ( ) k = k db 0 fo evey plne cuve B k nd tosion 0!
9 Exmple: Compute,,B of the cicul helix: ( t) cos( t), sin(t), bt sin( t), cos( t), b '( t) sin( t), cos( t), b hence b cos( t), sin( t),0 b 1 b t t cos ( ) sin ( ) b t t pinciple unit noml binoml B 1 b b 1 b cuvtue cos( t), sin( t),0 i j k sin( t) cos( t) b cos( t) sin( t) 0 1 b b bsin( t) i bcos(t) jk
10 Wht is the tosion of the cicul helix? cicul helix: ( t) cos( t), sin(t), bt sin( t), cos( t), b b cos( t), sin( t),0 1 bsin( t), bcos( t), b b B d B but t is not c length pmete s! we need fomul fo the tosion in genel pmete t whee ( t) x( t), y(t), z( t) nd v ', '' t t t t t computtion shows tht fo the helix we hve: b b
11 Decompose the cceletion vecto use v ' nd '' v d s v Recll: t ''( t) '( t) '( t) d s d s d s hence d
12 d s d tngentil cceletion: = ( ) noml cceletion: if c tvels long cuve, it feels n intenl cceletion of nd foce of mgnitude m m (centifugl foce) lge cuvtue (tight cuve) nd lge speed = poblems! if you tvel t unit speed, then othe fomuls: v v ' '' ' 0, nd foce v ' '' v ' m (ty to show this...) lso useful: Exmple: A c tvels long tck of dius with velocity d ( ) 0 1
13 13.6 Acceletion in Pol Coodintes
14 ewton s lw of gvittion (1687): F GmM Invese sque lw GM F = m '' = d ' M m G is the vecto fom the cente of the sun to the plnet is the mss of the sun is the mss of the plnet is the gvittionl constnt ' ' '' 11 G = m kg (fom 1798) '' 0 since '' is pllel to by ewton's lw hence ' is constnt vecto C in pticul C 0 the plnet moves in plne othogonl to C!
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