Phsics 07 Lecture 7 Agend: Phsics 07, Lecture 7, Sept. 6 hpter 6: Motion in (nd 3) dimensions, Dnmics II Recll instntneous velocit nd ccelertion hpter 6 (Dnmics II) Motion in two (or three dimensions) Frmes of reference Strt hpter 7 (Dnmics III) irculr Motion Assignment: For Wednesd red hpter 7 MP Problem Set 3 due tonight MP Problem Set vilble now MidTerm Thursd, Oct., hpters 1-7, 1 90 minutes, 7:15-8:5 PM Rooms: 10 & 130 in Vn Vleck. Phsics 07: Lecture 7, Pg 1 These re vector epressions reflecting, nd z motion r = r(t) v = dr / dt = d r / dt Phsics 07: Lecture 7, Pg Kinemtics The position, velocit, nd ccelertion of prticle in 3-dimensions cn be epressed s: r = i + j + z k v = v i + v j + v z k (i, j, k unit vectors ) = i + j + z k = (t) = ( t ) z = z( t ) d v = dt v d = dt v dz z = dt d d d z = = z = dt dt dt Specil se Throwing n object with long the horizontl nd long the verticl. nd motion both coeist nd t is common to both Let g ct in the direction, v 0 = v 0 nd v 0 = 0 vs t vs t vs All this compleit is hidden w in this compct nottion r = r(t) v = dr / dt = d r / dt Phsics 07: Lecture 7, Pg 3 0 t 0 t Phsics 07: Lecture 7, Pg A different trjector n ou identif the dnmics in this picture? How mn distinct regimes re there? vs t =10 Phsics 07: Lecture 7, Pg 5 A different trjector n ou identif the dnmics in this picture? How mn distinct regimes re there? 0 < t < 3 3 < t < 7 7 < t < 10 I. v = constnt = v 0 ; v = 0 II. v = v = v 0 III. v = 0 ; v = constnt < v 0 vs t =10 Wht cn ou s bout the ccelertion? Phsics 07: Lecture 7, Pg 6 Pge 1
Phsics 07 Lecture 7 Lecture 7, Eercises 1 & Trjectories with ccelertion A rocket is moving sidews in deep spce, with its engine off, from A to. It is not ner n strs or plnets or other outside forces. Its constnt thrust engine (force is constnt) is fired t point nd left on for seconds in which time the rocket trvels from point to some point Sketch the shpe of the pth from to. At point the engine is turned off. Sketch the shpe of the pth fter point From to? A. A.. D. D E. None of these Lecture 7, Eercise 1 Trjectories with ccelertion A D Phsics 07: Lecture 7, Pg 7 Phsics 07: Lecture 7, Pg 8 Lecture 7, Eercise Trjectories with ccelertion Trjector with constnt ccelertion long the verticl After? A. A.. D. D E. None of these A D How does the trjector pper to different observers? Wht if the observer is moving with the sme velocit? vs t vs Phsics 07: Lecture 7, Pg 9 0 t Phsics 07: Lecture 7, Pg 10 Trjector with constnt ccelertion long the verticl Lecture 7, Eercise 3 Reltive Trjectories: Monke nd Hunter Wht if the observer is moving with the sme velocit? A hunter sees monke in tree, ims his gun t the monke nd fires. At the sme instnt the monke lets go. Does the bullet? This observer will onl see the motion vs t vs In ll inertil reference frmes everone sees the sme ccelertion Phsics 07: Lecture 7, Pg 11 A. Go over the monke. Hit the monke. Go under the monke Phsics 07: Lecture 7, Pg 1 Pge
Phsics 07 Lecture 7 Trjector with constnt ccelertion long the verticl Wht do the velocit nd ccelertion vectors look like? Instntneous Velocit ut how we think bout requires knowledge of the pth. The direction of the instntneous velocit is long line tht is tngent to the pth of the prticle s direction of motion. Velocit vector is lws tngent to the trjector curve! Accelertion m or m not be! vs The mgnitude of the instntneous velocit vector is the speed, s. (Knight uses v) s = (v + v + v z ) 1/ v Phsics 07: Lecture 7, Pg 13 Phsics 07: Lecture 7, Pg 1 Averge Accelertion: Review The verge ccelertion of prticle motion reflects chnges in the instntneous velocit vector (divided b the time intervl during which tht chnge occurs). Instntneous Accelertion The instntneous ccelertion is the limit of the verge ccelertion s v/ t pproches zero The verge ccelertion is vector quntit directed long v The instntneous ccelertion is vector with components prllel (tngentil) nd/or perpendiculr (rdil) to the tngent of the pth hnges in prticle s pth m produce n ccelertion The mgnitude of the velocit vector m chnge The direction of the velocit vector m chnge (Even if the mgnitude remins constnt) oth m chnge simultneousl (depends: pth vs time) Phsics 07: Lecture 7, Pg 15 Phsics 07: Lecture 7, Pg 16 Motion long pth ( displcement, velocit, ccelertion ) 3-D Kinemtics : vector equtions: r = r(t) v = dr / dt = d r / dt v pth v v 1 -v 1 Velocit : v v = r / t v = dr / dt v Accelertion : v = v / t = dv / dt Phsics 07: Lecture 7, Pg 17 Generl 3-D 3 D motion with non-zero ccelertion: t v pth nd time = 0 = + Two possible options: hnge in the mgnitude of v hnge in the direction of Animtion Uniform irculr Motion (h. 7) is one specific cse: v = 0 = 0 http://romno.phsics.wisc.edu/winokur/fll006/af_018.html Phsics 07: Lecture 7, Pg 18 Pge 3
Phsics 07 Lecture 7 Lecture 7, Eercise The Pendulum revisited Lecture 7, Eercise The Pendulum Solution O θ = 30 A) v r = 0 r = 0 v T = 0 T 0 Which set of sttements below best describes the motion of the pendulum bob t the instnt of time drwn when the bob is t the top of its swing. ) v r = 0 r 0 v T = 0 T = 0 ) v r = 0 r 0 v T = 0 T 0 Phsics 07: Lecture 7, Pg 19 mg T T O θ = 30 NOT uniform circulr motion : ut if curved motion then r not zero, if speed is incresing so T not zero However, t the top of the swing the bob temporril comes to rest, so v = 0 nd the net tngentil force is mg sin θ ) v r = 0 r = 0 v T = 0 T 0 Everwhere else the bob hs non-zero velocit nd so then (ecept t the bottom of the swing) v r = 0 r 0 v T 0 T 0 Phsics 07: Lecture 7, Pg 0 Returning to reltive velocit, equtions The positions s seen from the two reference frmes re relted through the velocit r = r v o t The derivtive of the position eqution will give the velocit eqution v = v v o These re clled the Glilen trnsformtion equtions Reltive Velocit Two observers moving reltive to ech other generll do not gree on the outcome of n eperiment (pth) For emple, observers A nd below see different pths for the bll Phsics 07: Lecture 7, Pg 1 Phsics 07: Lecture 7, Pg entrl concept for problem solving: nd components of motion treted independentl. Emple: mn on the crt tosses bll stright up in the ir. You cn view the trjector from two reference frmes: Reference frme on the moving trin. Reference frme on the ground. (t) motion governed b 1) = -g ) v = v 0 g t 3) = 0 + v 0 g t / motion: = v t Accelertion in Different Frmes of Reference The derivtive of the velocit eqution will give the ccelertion eqution v = v v o = The ccelertion of the prticle mesured b n observer in one frme of reference is the sme s tht mesured b n other observer moving t constnt velocit reltive to the first frme. Net motion: R = (t) i + (t) j (vector) Phsics 07: Lecture 7, Pg 3 Phsics 07: Lecture 7, Pg Pge
Phsics 07 Lecture 7 Accelerted Reference Frmes: The Accelerometer Your first job is with Ford. You re working on project to design n ccelerometer. The inner workings of this gdget consist of weight of mss m tht is hung inside bo tht is ttched to the ceiling of cr. You design the device with ver light string so tht ou cn mthemticll ignore it. The ide is tht the ngle the string mkes with the verticl, θ, is determined b the cr s ccelertion. Your preliminr tsk is to think bout clibrtion of the ccelerometer when the cr trvels on flt rod. Wht is the cr s ccelertion when the hnging mss mkes n ngle θ with the verticl? i 1 Accelerted Reference Frmes: The Accelerometer θ We need to solve for the ngle the plum bob mkes with respect to verticl. We will solve b using Newton s Second Lw nd checking nd components. Phsics 07: Lecture 7, Pg 5 Phsics 07: Lecture 7, Pg 6 i Accelerted Reference Frmes: The Accelerometer θ T mg -dir F = -m = -T sin θ -dir F = 0 = T cos θ mg T = mg / cos θ = T sin θ / m = g tn θ Phsics 07: Lecture 7, Pg 7 Lecture 7, Eercise Accelerted Reference Frmes You re pssenger in cr nd not wering our setbelt. Without incresing or decresing speed, the cr mkes shrp left turn, nd ou find ourself colliding with the right-hnd door. Which is correct description of the sitution? (A) efore nd fter the collision there is rightwrd force pushing ou into the door. () Strting t the time of the collision, the door eerts leftwrd force on ou. () oth of the bove. (D) Neither of the bove. Phsics 07: Lecture 7, Pg 8 Lecture 7, Eercise Accelerted Reference Frmes You re pssenger in cr nd not wering our setbelt (poor judgement). Without incresing or decresing speed, the cr mkes shrp left turn, nd ou find ourself colliding with the right-hnd door. Which sttement below is correct description of the sitution? A. efore nd fter the collision there is rightwrd force pushing ou into the door.. Strting t the time of the collision, the door eerts leftwrd force on ou.. oth of the bove D. Neither of the bove Phsics 07: Lecture 7, Pg 9 Projection motion nd moving reference frme (b) How fr w from the lunch point does the shell lnd? Phsics 07: Lecture 7, Pg 30 Pge 5
Phsics 07 Lecture 7 Recp, Lecture 7 Assignment: For Wednesd red hpter 7 MP Problem Set 3 due tonight MP Problem Set vilble now MidTerm Thursd, Oct., hpters 1-7, 1 90 minutes, 7:15-8:5 PM Rooms: 10 & 130 in Vn Vleck. Phsics 07: Lecture 7, Pg 31 Pge 6