Motions and Coordinates

Transcription

1 Chapte Kinematics of Paticles Motions and Coodinates Motion Constained motion Unconstained motion Coodinates Used to descibe the motion of paticles 1

2 ectilinea motion (1-D) Motion Plane cuvilinea motion (-D) Space cuvilinea motion (3-D) Coodinates ectangula (Catesian) coodinates ( x, y),( x, y, z) Nomal and tangential coodinates ( n t) Pola coodinates (, ) Cylindical coodinates(,, z) Spheical coodinates (,, )

3 Chapte -. ectilinea Motion Instantaneous velocity: Instantaneous acceleation: ds v s dt dv a v dt d v s dt vdv sds ads sds Gaphical Intepetations 3

4 1 ads vdv ( v v1 ) (the aea) a v dv ds High School Physics Given aconstant (and s( t ) s, v( t ) v, when t ) dv (1). a dt v v at (). vdv ads v v a( s s ) ds (3). v v at dt 1 s s v t at 4

5 when a constant dv a f( t) (1). a v v adt dt o 1 s (). vdv ads ( v v ) ads ds (3). v dt dv a f( v) (1). a f( v) dt (). vdv ads s s vdt v v t t s dv dt t f ( v) vdv ds s s f ( v) a f () s (). vdv ads v v f( s) ds ds ds v g() s (3). v dt t dt gs () v v s s s s s t s t o Sample.1 3 st () t 4t 6 (1). vt ( ) 7, t? 1 1 (). vt ( ) 3, at ( )? (3). s(4) s(3)? 5

6 Poblem /19 Small balls fall fom est though the opening at the steady ate of pe-second. Find the vetical displacement h of consecutive balls when the lowe one has dopped 3 m. Poblem /5 A bumpe povides a deceleation as shown in the figue. Suppose a tain is appoaching the bumpe at speed of 4 ft/sec. Detemine the maximum compession of the bumpe. 6

7 Chapte -3. Plane Cuvilinea Motion 1. -D motion: a special case of 3-D.. Define the position vecto measued fom a fixed point O. d dv 3. Time deivative of a position vecto: v, a v dt dt Thee coodinates systems to descibe the cuvilinea motion ectangula (Catesian) coodinates ( x, y) Nomal and tangential coodinates ( n t) Pola coodinates (, ) 7

8 Chapte.4 ectangula coodinates (x-y) Vecto epesentation xi yj v xi yj a v xi yj Pojectile motion: ax, ay -g 8

9 Sample.5 v () t 5 16, t x yt () 1 4 t. x(), y in mete and t in second. Question : when yt ( ), a? and v? Detemine such that is maximized. 9

10 Chapte.5 Nomal and Tangential Coodinates (n-t) The positive diection of n is always taken towad the cente of cuvatue of the path. ds ρdβ, ds v vet et ρβ et dt a ve ve ve e t t t ρβ n v ve t e ρ n 1

11 A special case: Cicula Motion v t v an v a Wite the vecto expession fo the acceleation a of the mass G of the simple pendulum in both n- t and x- y coodinates fo the instance when 6 o. ad/sec 4.5 ad/sec 11

12 Execise /119 A paticle moving in the x- y plane has the position vecto as: 3 3 P ( t, t ) (in) 3 Calculate the adius of the path fo the position when t sec. Chapte.6 Pola coodinates (-) elative to a fixed point e 1

13 e, v e e a (, ) e ( ) e. Sample /9 3 ( t).t. t, ( t)..4 t. v(3)? a(3)? 13

14 Execise /145 (slide) ( t).8t.5 t, ( t) 1.6. t. v(4)? a(4)? and diection (elative to x-axis) Constant speed v.6 m/s 1. m,,,,,? π when t (1 ) 3 14

15 Chapte.7 Space Cuvilinea Motion ectangula (x-y-z) Cylindical (--z) Spheical (--ψ) * n-t coodinates ectangula coodinates (x-y-z) a v v xi xi xi yj yj yj zk zk zk 15

16 16 Cylindical Coodinates (--z) zk e e v a zk e e v zk e ) ( ) ( Spheical Coodinates(--ψ) e e e v a e e v ) cos sin ( ) sin cos cos ( ) cos ( cos e e

17 Sample /11 The powe scew stats fom est and is given a otational speed which inceases unifomly accoding to kt. Suppose the lead of the scew (advancement pe evolution) is L. Detemine the expession fo the velocity and acceleation of the cente of ball A when the scew has tuned though one complete evolution fom est. Execise /169 The velocity and acceleation of a paticle ae given by v 6x 3y z a 3x 1y 5z Detemine the angle between v and a, the adius of cuvatue. v, and 17

18 Execise /181 The evolving cane has a boom of length 4 m, and is tuning about the vetical axis at a constant ate of ev/ boom is loweed at the constant ate 1. ad/ sec. Calculate the magnitudes of the velocity and acceleation of end of the boom when it is loweed to 3 o. min. At the same time, the the Chapte.8 elative Motion A B A/ B, v v v A B A/ B, a a a A B A/ B, 18

19 Sample /1 Flight A is moving east at a speed of 8 km/h Flight B is moving notheast o (45 ) at a speed v Passenges at flight A obseve that flight B moves nothwest o (6 ). Detemine v? Execise /188 Two cas A and B ae moving along staight oads. If ate of incease of the distance between the cas equals the magnitude of the elative velocity of the cas. What can be said concening the velocities of the cas? the time 19

20 Execise /194 A ship is capable of 16 knots though still wate is to maintain a tue couse due west while encounteing a 3-knots cuent unning fom noth to south. What should be the heading of the ship (measued clockwise fom the noth to the neaest degee)? How long does it take the ship to poceed 4 nautical miles due west? Chapte.9 Constained Motion x y k x y x y

21 B y A yb yd y y B y y y y A A y D D D k 1 yc k y D y c yd y c Sample /15 The tacto A is used to hoist the bale B with the pulley aangement shown. If A has a fowad velocity V, detemine an expession fo the upwad velocity V of x. B A of the bale in tems 1

22 Execise /7 If block B has a leftwad velocity of 1. m/s, detemine the velocity of cylinde A. Execise / The paticle A is mounted on a loght od pivoted at O and theefoe is constained in a cuvela ac of adius. Detemine the velocity of A in tems of the downwad velocity v B of the counteweight fo any angle.

23 Motion Chapte eview ectilinea motion (1-D) Plane cuvilinea motion (-D) Space cuvilinea motion (3-D) Coodinates ectangula (Catesian) coodinates ( x, y),( x, y, z) Nomal and tangential coodinates ( n t) Pola coodinates (, ) Cylindical coodinates Spheical coodinates (,, z) (,, ) 3