Phsics 07 Lecture 5 Agend Phsics 07, Lecture 5, Sept. 0 Chpter 4 Kinemtics in or 3 dimensions Independence of, nd/or z components Circulr motion Cured pths nd projectile motion Frmes of reference dil nd tngentil ccertion Assignment: For Mond red Chpter 5 nd look t Chpter 6 WebAssign Problem Set due Tuesd net week (strt ASAP) Phsics 07: Lecture 5, Pg 1 See tet: 4-1 Chpter 4: Motion in (nd 3) dimensions 3-D D Kinemtics The position, elocit, nd ccelertion of prticle in 3-dimensions cn be epressed s: r i + j + z k i + j + z k (i, j, k unit ectors ) i + j + z k (t ) ( t ) z z( t ) d d dz z d d d z z Which cn be combined into the ector equtions: r r(t) dr / d r / Phsics 07: Lecture 5, Pg Instntneous Velocit The instntneous elocit is the limit of the erge elocit s t pproches zero The direction of the instntneous elocit is long line tht is tngent to the pth of the prticle s direction of motion. Aerge Accelertion The erge ccelertion of prticle s it moes is defined s the chnge in the instntneous elocit ector diided b the time interl during which tht chnge occurs. The mgnitude of the instntneous elocit ector is the speed. (The speed is sclr quntit) The erge ccelertion is ector quntit directed long Phsics 07: Lecture 5, Pg 3 Phsics 07: Lecture 5, Pg 4 Instntneous Accelertion The instntneous ccelertion is the limit of the erge ccelertion s / t pproches zero Motion long pth ( displcement, elocit, ccelertion ) 3-D Kinemtics : ector equtions: r r(t) dr / d r / The instntneous ccelertion is ector with components prllel (tngentil) nd/or perpendiculr (rdil) to the tngent of the pth pth Velocit : r / t dr / 1 Chnges in prticle s pth m produce n ccelertion The mgnitude of the elocit ector m chnge The direction of the elocit ector m chnge (Een if the mgnitude remins constnt) Both m chnge simultneousl (depends: pth s time) - 1 Accelertion : / t d / Phsics 07: Lecture 5, Pg 5 Phsics 07: Lecture 5, Pg 6 Pge 1
Phsics 07 Lecture 5 See tet: 4-4 Generl 3-D 3 D motion with non-zero ccelertion: Uniform Circulr Motion t pth nd time 0 + Two possible options: Chnge in the mgnitude of Chnge in the direction of Animtion Uniform Circulr Motion is one specific cse: 0 0 Wht does it men? How do we describe it? Wht cn we lern bout it? Phsics 07: Lecture 5, Pg 7 Phsics 07: Lecture 5, Pg 8 Aerge ccelertion in UCM: See tet: 4-4 Een though the speed is constnt, elocit is not constnt since the direction is chnging: must be some ccelertion! Consider erge ccelertion in time t / t See tet: 4-4 Instntneous ccelertion in UCM: Agin: Een though the speed is constnt, elocit is not constnt since the direction is chnging. As t goes to zero in / t d / 1-1 seems like (hence / t ) points towrd the origin! d / Now points in the - direction. Phsics 07: Lecture 5, Pg 9 Phsics 07: Lecture 5, Pg 10 Accelertion in UCM: This is clled Centripetl Accelertion. Clculting the mgnitude: 1 1 1 Similr tringles: So: But t for smll t t t Period nd Frequenc ecll tht 1 reolution π rdins Period (T) seconds / reolution distnce / speed π / Frequenc (f) reolutions / second 1/T () Angulr elocit (ω) rdins / second (b) B combining () nd (b) ω π f elize tht: Period (T) seconds / reolution So T 1 / f π/ω ω π / T πf ω s s Phsics 07: Lecture 5, Pg 11 Phsics 07: Lecture 5, Pg 1 Pge
Phsics 07 Lecture 5 ecp: Centripetl Accelertion UCM results in ccelertion: Mgnitude: / ω Direction: - r (towrd center of circle) ω See tet: 4-4 Lecture 5, Eercise 1 Uniform Circulr Motion A fighter pilot fling in circulr turn will pss out if the centripetl ccelertion he eperiences is more thn bout 9 times the ccelertion of grit g. If his F18 is moing with speed of 300 m/s, wht is the pproimte rdius of the tightest turn this pilot cn mke nd surie to tell bout it? (Let g 10 m/s ) UCM (recll) Mgnitude: / rˆ Direction: (towrd center of () circle) 10 m (b) 100 m (c) 1000 m (d) 10,000 m Phsics 07: Lecture 5, Pg 13 Phsics 07: Lecture 5, Pg 14 9g Lecture 5, Eercise 1 Emple: Newton & the Moon Wht is the ccelertion of the Moon due to its motion round the erth? 10000 m 1000m 9. 8 T 7.3 ds.36 10 6 s 3.84 10 8 m E 6.35 10 6 m (period ~ 1 month) (distnce to moon) (rdius of erth) Answer (c) E Phsics 07: Lecture 5, Pg 15 Phsics 07: Lecture 5, Pg 16 So ω.66 10-6 rd s -1. Moon... Clculte ngulr frequenc: 1 rot 1 d rd 6 7.3 d 86400 s π.66 10 rot rd s -1 Now clculte the ccelertion. ω 0.007 m/s.00078 g direction of is towrd center of erth (- ). dil nd Tngentil Quntities For uniform circulr motion Phsics 07: Lecture 5, Pg 17 Phsics 07: Lecture 5, Pg 18 Pge 3
Phsics 07 Lecture 5 dil nd Tngentil Quntities Lecture 5, Eercise The Pendulum Wht bout non-uniform circulr motion? θ is long the direction of motion 1m θ 30 Which sttement best describes the motion of the pendulum bob t the instnt of time drwn? i. the bob is t the top of its swing. ii. which quntities re non-zero? r is perpendiculr to the direction of motion A) r 0 r 0 θ 0 θ 0 B) V r 0 r 0 θ 0 θ 0 C) r 0 r 0 θ 0 θ 0 Phsics 07: Lecture 5, Pg 19 Phsics 07: Lecture 5, Pg 0 θ 1m θ 30 r Lecture 5, Eercise The Pendulum O NOT uniform circulr motion : is circulr motion so must be r not zero, Speed is incresing so θ not zero At the top of the swing, the bob temporril stops, so 0. eltie motion nd frmes of reference eference frme S is sttionr eference frme S is moing t o This lso mens tht S moes t o reltie to S Define time t 0 s tht time when the origins coincide C) r 0 r 0 θ 0 θ g In the net lecture we will lern bout forces nd how to clculte just wht is. Phsics 07: Lecture 5, Pg 1 Phsics 07: Lecture 5, Pg eltie Velocit Two obserers moing reltie to ech other generll do not gree on the outcome of n eperiment For emple, obserers A nd B below see different pths for the bll eltie Velocit, equtions The positions s seen from the two reference frmes re relted through the elocit r r o t The deritie of the position eqution will gie the elocit eqution o These re clled the Glilen trnsformtion equtions Phsics 07: Lecture 5, Pg 3 Phsics 07: Lecture 5, Pg 4 Pge 4
Phsics 07 Lecture 5 Centrl concept for problem soling: nd components of motion treted independentl. Agin: mn on the crt tosses bll stright up in the ir. You cn iew the trjector from two reference frmes: eference frme on the moing trin. eference frme on the ground. (t) motion goerned b 1) -g ) 0 g t 3) 0 + 0 g t / motion: t Net motion: (t) i + (t) j (ector) Accelertion in Different Frmes of eference The deritie of the elocit eqution will gie the ccelertion eqution o The ccelertion of the prticle mesured b n obserer in one frme of reference is the sme s tht mesured b n other obserer moing t constnt elocit reltie to the first frme. Phsics 07: Lecture 5, Pg 5 Phsics 07: Lecture 5, Pg 6 Lecture 5, Eercise 3 eltie Motion You re swimming cross 50 m wide rier in which the current moes t 1 m/s with respect to the shore. Your swimming speed is m/s with respect to the wter. You swim cross in such w tht our pth is stright perpendiculr line cross the rier. How mn seconds does it tke ou to get cross? ) 50 5 s b) 50 1 50 s c) 50 3 9 s d) 50 35 s m/s Lecture 5, Eercise 3 Choose is long rierbnk nd is cross rier The time tken to swim stright cross is (distnce cross) / ( ) Since ou swim stright cross, ou must be tilted in the wter so tht our component of elocit with respect to the wter ectl cncels the elocit of the wter in the direction: m/s 1 3 m/s Phsics 07: Lecture 5, Pg 7 Phsics 07: Lecture 5, Pg 8 Lecture 5, Eercise 3 The component of our elocit with respect to the wter is 3 m/s The time to get cross is 9s 3 m s 3 m/s Answer (c) ecp First mid-term em in just two weeks, Thursd Oct. 5 Chpter 4 Chpter 4 Kinemtics in or 3 dimensions Independence of, nd/or z components Circulr motion Cured pths nd projectile motion Frmes of reference dil nd tngentil ccertion Assignment: For Mond red Chpter 5 nd look t Chpter 6 WebAssign Problem Set due Tuesd net week (strt ASAP) Phsics 07: Lecture 5, Pg 9 Phsics 07: Lecture 5, Pg 30 Pge 5