ME 210 Applied Mathematics for Mechanical Engineers

Transcription

1 Tangent and Ac Length of a Cuve The tangent to a cuve C at a point A on it is defined as the limiting position of the staight line L though A and B, as B appoaches A along the cuve as illustated in the figue. O (t) (t + t) A (t) T (t) B C Tangent s: Ac Length fom A to B L (t + t) (t) Pof. D. Bülent E. Platin Sping 14 Sections & 3 1/9

2 Hee, it is assumed that the cuve C is epesented b a continuousl diffeentiable (i.e., deivative eists and is continuous) vecto function t is some paamete, the points A and B coespond to t and t + t. Then, the staight line L has the diection of the vecto (t) (t + t) (t), whee O (t) (t + t) A (t) T (t) B C Tangent s: Ac Length fom A to B L Pof. D. Bülent E. Platin Sping 14 Sections & 3 /9

3 O (t) (t + t) A (t) T (t) B C Tangent s: Ac Length fom A to B L Hence, the deivative of d(t) (t) lim t t (t) defined as (t + t) (t) lim t t has the diection of the tangent to C at A. Theefoe, it is also called as the tangent vecto of C at A. The coesponding unit vecto T(t) is called a s the unit tangent vecto to C at A. Note that both tangent vecto and unit tangent vecto point the diection of inceasing values of t, hence the depend on the oientation of the cuve. Pof. D. Bülent E. Platin Sping 14 Sections & 3 3/9

4 O (t) (t + t) A (t) T (t) B C Tangent s: Ac Length fom A to B L If the segment of the cuve C between points A and B is consideed, its length s If the incement t in t is consideed to be infinitesimall small, then this epession can be ewitten as d(t) ds d(t). o equivalentl ds (t) (t) can be epessed as (t) s (t) t t Hence, the epession of what is so-called as the ac length (function) becomes s(t) Pof. D. Bülent E. Platin Sping 14 Sections & 3 4/9

5 O (t) s(t) (t + t) * A (t) T (t) whee t* is the dumm vaiable of integation. B Tangent If one wants to measue this ac length along a cuve C fom a fied point on it coesponding to t t o, then the indefinite integal ma be ewitten as the following definite integal. * * ) (t ) Note that s(t) is a scala function of the paamete t, and s itself ma seve as a paamete in the paametic epesentation of cuves. C t t t * * (t ) * * t t t s: Ac Length fom A to B * L t t As a fist and impotant case, the use of the ac length in the fomula fo the unit tangent vecto gives d(s) T(s) (s) ds Because, (s) 1 is obtained when t is eplaced b s in the equation (t * ds Pof. D. Bülent E. Platin Sping 14 Sections & 3 5/9

6 Eample: (Cicle) Conside the cicle epesented b ( asint) i + (acost) j (t) (acost) i + (asint) j a t * * * s(t) (t ) (t ) t at The paametic epesentation of this cicle as s being the paamete becomes (s) acos s a i + asin The unit tangent vecto is given as d(s) s s T(s) (s) sin i cos j ds a + a d(t) d(s) Note that a 1 But T(s) (s) 1 ds a s j a * Pof. D. Bülent E. Platin Sping 14 Sections & 3 6/9

7 Eample (Heli) Find the ac length of the cicula heli fom A(a,,) to B(a,,πc). (t) a cos(t) i + a sin(t) j + c t k [ ] [ ] L B πc A a O (t) a a cos(t) a sin(t) c t + a c t This is a heli wapped aound ais as shown in the figue. Fo the fist point A: Fo the second point B: a cos(t) a a sin(t) c t a cos(t) a a sin(t) c t π c Pof. D. Bülent E. Platin Sping 14 Sections & 3 7/9

8 Ac length: d π d d s. - a sin(t) i + a cos(t) j + c k d d. a + c s π a + c π a + c Let s epesent the heli with the ac length s as the paamete t s a + c a + c t > t s a + c s s c s (s) a cos i + a sin j + k a + c a + c a + c Pof. D. Bülent E. Platin Sping 14 Sections & 3 8/9

9 Velocit and Acceleation along a Cuve If a bod (epesented b a point P) moves along a path C whose paametic epesentation is given b its position vecto (t) whee t is time, then the successive deivatives and of (t) can be attibuted to two kinematic concepts, namel the velocit and the acceleation, espectivel. The vecto defined b v(t) d(t) lim t (t) t lim t (t + t) (t) t is tangent to the cuve C and, theefoe, points in the instantaneous diection of motion of P. Pof. D. Bülent E. Platin Sping 14 Sections & 3 9/9

10 We also know that this vecto has a magnitude d(t) ds v (t) whee s is the ac length, which measues the distance of P fom a fied point (s ) on C along the cuve C. Hence, ds/ is the speed of P, and the vecto v (t) is, theefoe, called as the velocit o the velocit vecto of the motion of P along C. Since the velocit vecto v (t) is defined onl and onl fo those points belonging to the cuve C along which P moves, it constitutes a vecto field. Similal, the speed ds/ is also defined onl and onl fo those points belonging to the cuve C along which P moves; but since it is a scala quantit, in tun, it constitutes a scala field. Both speed and velocit fields have a common domain of definition: The cuve C along which P moves. Pof. D. Bülent E. Platin Sping 14 Sections & 3 1/9

11 Anothe wa of epesenting the elationship between the velocit (a vectoial quantit) and the speed (a scala quantit) though unit tangent vecto is d(t) d(s) ds ds v(t) T(s) ds The deivative of this velocit vecto is called as the acceleation o acceleation vecto and, hence, is defined as dv(t) d (t) a(t) Again, since this vecto is defined onl and onl fo those points belonging to the cuve C along which P moves, it too constitutes a vecto field, whose domain of definition is C. Pof. D. Bülent E. Platin Sping 14 Sections & 3 11/9

12 Eample: Let (t) be the position vecto of a moving paticle whee t is the time. Descibe the geometic shape of the path and find the velocit vecto, the speed, and the acceleation vecto. 3 3 (t) (1+ t ) i + t j + ( t 3 )k Let t 3 n be a new vaiable eplacing t 3 : 1 + n n - n n n Theefoe: n This is a staight line as shown in the figue. Pof. D. Bülent E. Platin Sping 14 Sections & 3 1/9

13 Position vecto: (t) (1+ t 3 ) i + t 3 j + ( t 3 )k Velocit vecto: v(t) d 3t i + 6t j 3t k Speed: d v (t) ( ) ( ) ( ) 3t + 6t + 3t 3 6t Acceleation vecto: a(t) dv d 6t i + 1tj 6tk Rewok this poblem with (t) (1+ t) i + tj + ( t)k Pof. D. Bülent E. Platin Sping 14 Sections & 3 13/9

14 Eample Conside a paticle, which moves along a path C whose position vecto is given b (t) (cost) i + (cost) j + ( sint)k Let us fist descibe the geometic shape of the path, and then find the velocit vecto, the speed and the acceleation vecto of the paticle as it moves along this path. Note that in the epesentation given, (t) cost ; (t) cost ; (t) sint Fo a closed fom epesentation of the coesponding cuve C, b a pope elimination of the paamete t, one can easil deive two equations that eithe ield the pojection of the cuve on two Catesian planes; e.g., f(,) and g(,) o two sufaces (i.e., F(,,) and G(,,) ) whose intesection gives C. Pof. D. Bülent E. Platin Sping 14 Sections & 3 14/9

15 If the second altenative is used, one obvious choice of seveal altenatives becomes: F(,,) G(,,) F(,,) epesents a spheical suface with a adius of. G(,,) epesents a vetical plane passing though the -ais and the line on -plane. Pof. D. Bülent E. Platin Sping 14 Sections & 3 15/9 line

16 The intesection of these two sufaces is a cicle C on the plane, whose cente is at the oigin and adius is. (t) (cost) i + (cost) j + ( sint)k Note that A(1,1,) coesponds to t. this cuve has an oientation as shown while t inceases fom C (t) line A(1,1,) t The Cicula Cuve and its Position Vecto Pof. D. Bülent E. Platin Sping 14 Sections & 3 16/9

17 Note that the pojection of this cicle on the -plane is which epesents an ellipse with pincipal adii of 1 and in and coodinates, espectivel. The pojection onto the -plane is, a staight line bisecting - and -aes These two equations would also suffice fo the path desciption of the moving bod. The velocit vecto is the deivative of the position vecto: d(t) v(t) ( sint) i + ( sint) j + ( cost)k Pof. D. Bülent E. Platin Sping 14 Sections & 3 17/9

18 d(t) v(t) ( sint) i + ( sint) j + ( The diection of the velocit vecto depends on the diection of inceasing t on the cuve C. The coesponding velocit vecto field is pesented in the following figue. cost)k Velocit Vecto Field Pof. D. Bülent E. Platin Sping 14 Sections & 3 18/9

19 The speed of the moving paticle is the magnitude of the velocit vecto, hence v(t) ( ) ( ) ( ) sint + sint + cost ( sint) + ( cost) It is inteesting to note that the speed of the moving paticle is constant along its path fo the motion specified. In a moe geneal motion, the speed of the paticle comes about a function of s o t, whicheve is pefeed to use. The acceleation vecto can be found b diffeentiating the velocit vecto as dv(t) a(t) ( cost) i + ( cost) j + ( sint)k (t) Pof. D. Bülent E. Platin Sping 14 Sections & 3 19/9

20 a(t) dv(t) ( cost) i + ( cost) j + ( sint)k (t) Hence, all acceleation vectos points to the oigin. The coesponding acceleation vecto field is pesented in the following figue. Acceleation Vecto Field Pof. D. Bülent E. Platin Sping 14 Sections & 3 /9

21 It is also inteesting to note that the magnitude of the acceleation vecto of the moving paticle a(t) ( ) ( ) ( ) cost + cost + sint ( cost) + ( sint) is constant along its path fo the motion specified. Again, in a moe geneal motion, the magnitude of the acceleation of a paticle comes about a function of s o t, whicheve is pefeed to use. Pof. D. Bülent E. Platin Sping 14 Sections & 3 1/9

22 Cuvatue and Tosion of a Cuve, Nomal, Binomal, TNB Fame The velocit v (t) of a point P moving along a space cuve C is alwas tangent to this cuve taced b the tip of the position vecto, (t). Theefoe, v (t) is in the diection of the unit tangent vecto defined as d(s) T(s) (s) ds P T(s) v (t) Pof. D. Bülent E. Platin Sping 14 Sections & 3 /9 Cuve C

23 As ou epesentative point P moves along a diffeentiable cuve, the unit tangent vecto, T (s), changes its diection as the cuve bends since it must emain tangent to the cuve at P. At an point P on C, the ate of change in T (s) b the paamete s is called the cuvatue, κ( s) (a positive scala quantit), of the cuve C at its point P, epessed as P 1 T(s) dt(s) κ(s) v ds ds P T(s+ds)T(s)+dT(s) Pof. D. Bülent E. Platin Sping 14 Sections & 3 3/9 Cuve C v+dv

24 The diffeence between the unit tangent vecto at P and the unit tangent vecto T dt(s) (s) at P 1 has to be pependicula to T T (s + ds) (s) and in the diection at which T (s) otates (o towads the cuve C bends). The unit vecto N in the diection of dt(s) dt(s) dt(s) (s) ds defined as N(s) ds ds is called the unit nomal vecto (o the pincipal nomal) of the cuve C at its point P. P 1 T(s) v N(s) ds P T(s+ds)T(s)+dT(s) dt(s)/ds Pof. D. Bülent E. Platin Sping 14 Sections & 3 4/9 Cuve C v+dv

25 Using this definition, one can also wite dt(s) dt(s) 1 κ(s)n(s) o N(s) ds ds ρ(s) whee ρ(s) is called as the adius of cuvatue, which can be consideed as the adius of a cicle that is tangent to the cuve C at P whose cuvatue is identical to the cuvatue of the cuve C at P. This cicle is efeed to as the cicle of cuvatue. P 1 T(s) v Cicle of cuvatue with a adius ρ tangent to C at P 1 N(s) dt(s)/ds ds Pof. D. Bülent E. Platin Sping 14 Sections & 3 5/9 P T(s+ds)T(s)+dT(s) Cuve C v+dv

26 Note that the cuvatue κ(s) [o the adius of cuvatue ρ(s)] is a function of s, hence it changes (hence the sie of the cicle of cuvatue changes) as the epesentative point moves along the cuve C. The lage the value of κ (the smalle the value of ρ) is the moe apid change in is obseved indicating a shape bend in the cuve C. The opposite condition indicates a smoothe cuve; i.e., close to a staight line. T (s) P 1 T(s) v Cicle of cuvatue with a adius ρ tangent to C at P 1 N(s) dt(s)/ds ds Pof. D. Bülent E. Platin Sping 14 Sections & 3 6/9 P T(s+ds)T(s)+dT(s) Cuve C v+dv

27 The plane fomed b two unit vectos; namel, T (s) and N (s), is called the osculating plane. (to osculate to be in contact, to touch closel, to kiss). Since both T (s) and N (s) ae functions of s, the osculating plane changes too as the epesentative point P moves along the cuve C. To account fo this change, one can conside the changes in its unit nomal vecto which can be defined as B(s) T(s) N(s) called as the binomal vecto. Note that infinitesimal changes db(s) in binomal as taveled infinitesimall along the cuve C should occus onl in the diection of N (s) since the cuve is tangent to T (s). This esult ma be deived b using the fact that the dot poduct of T (s) and B (s) is eo B (s) T(s) Pof. D. Bülent E. Platin Sping 14 Sections & 3 7/9

28 Hence the deivative of this dot poduct with espect to s should vanish, too, to give d [ ] db dt db db B T T + B T + B N T ds ds ds ds 13 κ ds indicating that db(s) is pependicula to T (s), hence db (s) is in the diection of N (s) onl. Theefoe, the following elationship between db(s) and N (s) can be witten db(s) db(s) τ(s)n(s) whee τ(s) ds ds τ(s), a scala quantit, is called the tosion of the cuve C at its point P. The tosion of a cuve can be thought of as the ate at which the osculating plane tuns as P moves along the cuve C. In othe wods, it is a measue of how much the cuve C twists out of its osculating plane at point P. Note that τ(s) can be positive o negative. Pof. D. Bülent E. Platin Sping 14 Sections & 3 8/9

29 END OF WEEK 7 Pof. D. Bülent E. Platin Sping 14 Sections & 3 9/9