Dynamics test 1 formula sheet

Dynamics test 1 formula sheet - F orlVld5 CH.1CK.2 5Ae.e:f...

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F orlVld5. 5Ae.e:f CH.1- / F:: (;~, ~~. -r;\. .. t:- 9 V-=. VI:J+- J Ct(r) cJ t: t" x- ~ - -(: > J. 'X := V(t=) =5> f d\c ~ f V(f) d-l: :> x:::- ><~ f- fV(t)d-t ~ ~ ~ ~ a.( v);::- ¥""" 1: - )( -Yo j:= "3).,).. ~Kt'L'- = 9. ~ I """/5'L j c1v' ~ fd.-b ,/ -' -" ,.. .. . - eJ; .::)~ - - . - y-~ 7>-] t- ~.,j,.) y -= .x.2 + ff f.) ~::r.x. 1. + J i- Vi> J bo ~ ~ a. J1. V'l V :::: Vy, + Vy ) v = Vx f-'\'- )" i<cV\ e ::: VX . /9--( v) = ~ - c..A;::. c:{~:t+.?-J l./ ~::' J C<.x~-t-C{ J :;J.. rJJ. - .' V _.X - VX~_VX(o) Y..'f:: VYc -;r~ - - .> _'I9#iLJ Vf.\l " [d. "'- -)(~ XD 4 V"D t -y-~ y. + VIol - -J:3 e -- - ~ - v tJ..{V)(o V ~.:::V J..- '). ( -' II ) () - - --- - -- --- --. . -y- - .y~_. - 5. 1. 1 > OI.(x) -= W- - -- h::::-c. .;: 1/ > ) a{xJJ.Y -= f-vtJ.v 'fu VI> CK.2; > -- -- > .. X v- cJ.x ~:: cJ.a:x ;:; dv :: vdv J - ~kt J J t;;t # ?fX > U;y..~t- v~,. £\::'0 v=vb ><- ~ ~() +- Vo 1:.. .4 II ;::: aCt:) o..t -t- JVJ-v ::- f~Ci) c.lt v~ "-f:-I) > > ) Ol -::.ex..!? ) v-;;.v()!tJ.--C . )( = Xo+ V 0 t i' t D. t 'J.. vJ..-=.v/ + ~c... (Y-Xo) \i = rJ--F ~- i 'C: tJ-v -ct (). f 75.f - -ri: ~
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/ t n/ C t A\// \ ~-~:.:::- \ In ?f"""'" "- : I \ \ --J-~:::"- \n h B t ~ Figure 2/9 / / fe' Path / t \ / t \ I \ ,I '" v .,( \ jet I I "C \ :1: 1 I -."'--1. p v '- -- /'- A' d f3 '- '- -- n '- / " ' -ds=pdf3 en '- /A ./ ,/ ./ I I I I I I I ,I I I I I I I I I L______---------- (b) (a) a\ " (c) Figure 2/10 2/5 NORMAL AND TANGENTIAL COORDINATES (n-f) As we mentioned in Art. 2/1, one of the common descriptions of curvilinear motion uses path variables, which are measurements made along the tangent t and normal n to the path of the particle. These coordinates provide a very natural description for curvilinear motion and are frequently the most direct and convenient coordinates to use. The n- and t-coordinates are considered to move along the path with the particle, as seen in Fig. 2/9 where the particle advances from A to B to C. The positive direction for n at any position is always taken toward the center of curvature of the path. As seen from Fig. 2/9, the positive n-direction will shift from one side of the curve to the other side if the curvature changes direction. Velocity and Acceleration We now use the coordinates nand t to describe the velocity v and acceleration a which were introduced in Art. 2/3 for the curvilinear motion of a particle. For this purpose, we introduce unit vectors en in the n-direction and et in the t-direction, as shown in Fig. 2/lOa for the position of the particle at point A on its path. During a differential in- crement of time dt, the particle moves a differential distance ds along the curve from A to A'. With the radius of curvature of the path at this position designated by p, we see that ds = p df3, where f3 is in radians. It is unnecessary to consider the differential change in p between A and A' because a higher-order term would be introduced which disappears in the limit. Thus, the magnitude of the velocity can be written v = ds/dt = P df3/dt, and we can write the velocity as the vector ( v = vet = p~et) (2/7) The acceleration a of the particle was defined in Art. 2/3 as a
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  • Fall '08
  • LGKraige
  • Vectors, Velocity, Dee, unit vectors, yv, vector eXI velocity

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