Kinematics in 2-D (II)

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1 Kinematics in 2-D (II) Unifom cicula motion Tangential and adial components of Relative velocity and acceleation a Seway and Jewett 4.4 to 4.6 Pactice Poblems: Chapte 4, Objective Questions 5, 11 Chapte 4, Poblems 39, 41, 43, 47, 53, 79 Physics 1D03 - Lectue 6

2 Unifom Cicula Motion unifom means constant speed v velocity v changes (diection changes) acceleation : find v by subtacting vectos, a v lim t 0 t = 1) Is v t as t 0? 2) What is the value of v/ t, as t 0? v v 2 cente Physics 1D03 - Lectue 6

3 Subtact velocities: v 1 θ v 2 v v θ v 2 Compae with displacements: Simila tiangles, 1 2 θ v v Note v is pependicula to = Physics 1D03 - Lectue 6

4 Fom pevious slide: as t 0, v a t v = v, v t v and so a = v 2 Diection: Since v is pependicula to, a is pependicula to v So, a is towads the cente of the cicle ( centipetal ). Physics 1D03 - Lectue 6

5 In geneal, diection and speed both change: v 1 v 2 v 1 v v = v 2 v1 2 v has components paallel and pependicula to dv v the motion ; and a = dt t Physics 1D03 - Lectue 6

6 The adial (centipetal) component is due to the change in diection. a a t a a = v 2, pependicula to path The tangential component (tangent to the path) is equal to the ate of change of speed: d v a t = dt Physics 1D03 - Lectue 6

7 Concept Quiz You ae diving at constant speed aound a cuve. Seeing a ada tap, you apply the bakes. Compae the magnitude of you acceleation just befoe and just afte you apply the bakes : a) a is zeo befoe, and non-zeo afte b) a is not zeo befoe, and inceases immediately as you apply the bakes c) a is not zeo befoe, and deceases immediately as you apply the bakes Physics 1D03 - Lectue 6

8 Relative Motion Example: A wasp is flying fom noth to south at 30 km/h. You ae iding you bicycle notheast at 20 km/h. What is the velocity of the wasp elative to you? If it acceleates, what is its acceleation elative to you? Thee ae two efeence fames (coodinate axes) fo measuing fom: the gound, and the bicycle. At left ae the velocities elative to the gound. How do these look elative to the bicycle? bicycle v b, g waspv w, g Physics 1D03 - Lectue 6

9 Ty an easie poblem in one dimension : Wind blows southwads, bicycle tavels noth. A cyclist iding at 20 km/h diectly into a 30 km/h wind should feel a 50 km/h headwind. In tems of vectos : v w, b = v w, g vb, g v w, g v b, g v w, b Use the same equation in two o thee dimensions. Re-daw the diagam and solve the oiginal poblem. Physics 1D03 - Lectue 6

10 Quiz: A wasp is flying fom noth to south at 30 km/h. You ae iding you bicycle notheast at 20 km/h. What is the velocity of the wasp elative to you? Which vecto diagam shows the thee velocities coectly? (The ed vecto is the velocity of the wasp elative to you). A B C D Physics 1D03 - Lectue 6

11 Suppose we have a stationay efeence fame (axes x,y), and anothe efeence fame x', y' moving at velocity v 0. The elative displacement of the two sets of axes is 0 (which changes with time). The positions ae elated by = + 0 Diffeentiate to get velocities: v = v + v 0 Diffeentiate again. If v 0 is constant, y y' v 0 paticle a = a o x x' The acceleation of a paticle appeas the same in both efeence fames! Physics 1D03 - Lectue 6

12 Inetial Fames A efeence fame in which Newton s Laws ae tue is called an Inetial Fame. Since Newton s mechanics is based on acceleation (and on elative positions), any fame moving at constant velocity is an inetial fame Physics looks the same in all inetial fames. ( Galileo s Pinciple of Relativity? ) Physics 1D03 - Lectue 6

13 Dopping a suga cube into a coffee cup in an aiplane taveling 600 km/h. But: Example Dopping a suga cube in a tain going aound a cuve. cube falls vetically cube moves away?? View fom inetial fame (top view) cup follows cuve (away fom cube) cube goes staight Physics 1D03 - Lectue 6

14 Concept Quiz Bicycle wheel, adius = 35 cm, speed = 10 m/s 2 and acceleation ( 10 m/s ) 2 = 286 m/s = 29 g 0.35m What about a paint spot on the im at its lowest point (almost in contact with the gound, momentaily stationay)? What is a fom the point of view of the ide, and of an ant about to be un ove? a) Both see an upwad acceleation of 29 g b) The ide sees a = 29 g [up], ant sees zeo c) Both see zeo acceleation momentaily Physics 1D03 - Lectue 6

15 Discussion : what does the motion (position and velocity) of the paint spot look like to the ant? Conside only the shot inteval of time stating and ending when the tie is a few millimetes above the ant. (you may assume the ant is in a small hole in the oad and isn t squashed --- no insects wee hamed in this example) Physics 1D03 - Lectue 6

16 Summay Acceleation has a tangential component (paallel to motion) and a adial component (pependicula to the motion) a t = ate of incease of speed v = towads the cente of the cicle o ac Inetial efeence fames move at constant velocity elative to a 2 each othe Acceleation is the same in all inetial fames, velocities obey : v a, b + vb, c = va, c Physics 1D03 - Lectue 6

Physics 111 Lecture 5 Circular Motion

Physics 111 Lecture 5 Circular Motion Physics 111 Lectue 5 Cicula Motion D. Ali ÖVGÜN EMU Physics Depatment www.aovgun.com Multiple Objects q A block of mass m1 on a ough, hoizontal suface is connected to a ball of mass m by a lightweight