Lecture 6 Todays Goals (Ch 4.4-6) Discuss uniform and non-uniform circular motion Circular Motion Centripetal (or radial) acceleration (direction of v changes) Tangential acceleration (magnitude of v changes) Relative motion and reference frames 1st Exam (Chapters 1-4, ~10 Multiple choice, 4 short answer) Where: 2103, 2223 & 2241 Chamberlin Hall (& quiet room) When: Monday, February 20 7:15-8:45 PM Format: Closed book, one 8 x11 sheet, hand written Electronics: Any calculator is okay but no web/cell access Quiet room: Test anxiety, special accommodations, etc. Conflicts: E-mail for approval (on or before Monday, Feb 12th) , academic/UW athletic reasons only Physics 201: Lecture 6, Pg 1 Circular Motion is common so specialized terms Angular position (CCW + CW -) Radius is r Arc distance s = r & ds = r d Tangential velocity vt = ds /dt Angular velocity, d/dt (CCW + CW -) s
vt = ds/dt = r d/dt = r vt r Physics 201: Lecture 6, Pg 2 Reformulating changes with vector notation Cartesian Coordinates Polar Coordinates rf ri dr r x i yj dr dx i dyj For a very small change dr
dr r r rf ri dr r r r dr dr r r dr r dr r d dr dr r r d Physics 201: Lecture 6, Pg 3 Circular Motion (with constant |r|)
| v |r r and dr dr r r d dr dr d d r r r dt dt dt dt v r vt s r
Physics 201: Lecture 6, Pg 4 Uniform Circular Motion (with constant |r| and |v|) Time to go once around is the period T Distance once around is 2 r Tangential velocity is vt = 2r/T | v |r 2r 2r 2 T v r s vt
r Physics 201: Lecture 6, Pg 5 Uniform Circular Motion (UCM) has only radial acceleration v UCM changes only the direction of 1. Particle doesnt speed up or slow down! 2. Velocity is always tangential; acceleration perpendicular ! v v v v v
v | v | |v | v v aavg v t a Physics 201: Lecture 6, Pg 6 Uniform Circular Motion (UCM) has only radial acceleration v UCM changes only in the direction of 1. Particle doesnt speed up or slow down!
2. Velocity is always tangential, acceleration perpendicular ! v v v d a lim v t 0 t dt aradial a path 2 a ar v r r 2 ar v / r ac Physics 201: Lecture 6, Pg 7
Again Uniform circular motion involves only changes in the direction of the velocity vector Acceleration is perpendicular to the trajectory at any point, acceleration is only in the radial direction. v Centripetal/radial Acceleration ac -ac = ar = -v2/r r Circular motion involves continuous radial acceleration Physics 201: Lecture 6, Pg 8 Mass-based separation with a centrifuge Before After How many gs (1 g is ~10 m/s2)? |ar |= vt2 / r = 2 r f = 6000 rpm = 100 rev. per second is typical with r = 0.10 m
ar = (2 102)2 x 0.10 m/s2 ar = 4 x 104 m/s2 or ca. 4000 gs !!! but a neutron star surface is at 1012 m/s2 Physics 201: Lecture 6, Pg 9 bb5 Consequence of no radial accelerationa demo In this demonstration we have a ball tied to a string undergoing horizontal UCM (i.e. the ball has only radial acceleration) 1 Assuming you are looking from above, draw the orbit with the tangential velocity and the radial acceleration vectors sketched out. 2 Suddenly the string brakes. 3 Now sketch the trajectory with the velocity and acceleration vectors drawn again. Physics 201: Lecture 6, Pg 10 Concept test What does the path look like once the string is cut?
A: B: C: v ac r D: E: Physics 201: Lecture 6, Pg 11 Non UCM (with constant |r| and changing |v|) The speed of the particle increases or decreases d|v|/dt 0 | v |r d |v| atangential dt Always tangent to the path!
s vt r Physics 201: Lecture 6, Pg 12 Acceleration with both speed and direction change 1. Particle speeds up or slows down! 2. Acceleration has tangential and radial components ! | v || v '| vr 2 ar lim v / r t 0 t vt at lim t 0 t
a ar r at v v a ar at vr Physics 201: Lecture 6, Pg 13 vt Non-uniform Circular Motion For an object moving along a curved trajectory, with varying speed Vector addition: a = ar + at (radial and tangential) at a radial
a ar vt2 r a tangential | a | a d | vt | dt 2 radial a 2 tangential Physics 201: Lecture 6, Pg 14
Total acceleration A stunt plane is performing a loop-the-loop of radius 100 m while accelerating (see figure). When its nose is pointed directly down, the speed of the plane is 50 m/s and the acceleration, tangent to the path, is 2g (i.e., 20 m/s 2). What is magnitude of the total acceleration? In x,y vector notation, what is the total acceleration? j 2 at 20 m/s j 2 2 2 i 2 ar v / r m/s i 50 / 100 m/s i
2 2 a 25 m/s i 20 m/s j vT r a (252 202 )1 / 2 m/s 2 32 m/s 2 Physics 201: Lecture 6, Pg 15 Concept Check: Which answer is best E1. You drop a ball from rest, how much of the acceleration from gravity goes to changing its speed? A. All of it B. Most of it a g j C. Half of it D. None of it
v E2. A hockey puck slides off the edge of a horizontal table, just at the instant it leaves the table, how much of the acceleration from gravity goes to changing its speed? A. All of it B. Most of it a g j v a C. Half of it radial acceleration D. None of it Physics 201: Lecture 6, Pg 16 v Relative Motion and reference frames? If you are moving relative to another person do you see the same physics?
Two observers moving relative to each other generally do not agree on the outcome of an experiment (path) For example, observers A and B below see different paths for the ball Physics 201: Lecture 6, Pg 17 Reading Assignment Chapter 5.1-6 Physics 201: Lecture 6, Pg 18
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