Chapter 8, Rotational Kinematics. Angular Displacement

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1 Chapte 8, Rotational Kinematics Sections 1 3 only Rotational motion and angula displacement Angula velocity and angula acceleation Equations of otational kinematics 1 Angula Displacement! B l A The length of the ac of the cicle of adius fom A to B is: l =! whee is in adians.! (adians) = ac length adius = l One complete evolution coesponds to:! = 2" = 2" adians (ad)! adians =

2 Two geostationay satellites ae at an altitude of 4.23!10 7 m. Thei angula sepaation is 2 0. Find the ac length s. s = =( m) (2!/180 adians) = 1.48x10 6 m 2 0 = 4.23!10 7 m 3 Total eclipse of the sun s sun sun s moon moon! sun! moon The sun and moon have almost exactly the same angula size as seen fom the eath even though the moon is much smalle than the sun. 4

3 8.68/14 Moon: diamete = 3.48! 10 6 m, distance fom eath = 3.85! 10 8 m Sun: diamete = 1.39! 10 9 m, distance fom eath = 1.50! m a) Angula size of moon = (3.48! 10 6 m)/(3.85! 10 8 m) = ad Angula size of sun = (1.39! 10 9 m)/(1.50! m) = ad b) The moon has slightly smalle angula size than the sun. c) Appaent aea of moon = ( / ) 2! appaent aea of sun = 95.1% of appaent aea of sun. 5 Angula velocity motion aound a cicula path Angula velocity is:! B l A ω = θ B θ A t B t A ad/s (aveage) ω = θ t ad/s (instantaneous) The instantaneous angula velocity is measued ove a vanishingly small time inteval. If the motion fom A to B takes time t, then v = l t = θ t = ω That is, v = 6

4 Angula acceleation Angula acceleation is the ate of change of angula velocity: Aveage angula acceleation! = " " 0 = #" t t 0 #t "# Instantaneous angula acceleation! = lim "t 0 "t 7 Equations fo otational motion Analogous to the equations fo linea motion.! =! 0 + "t! = 1 2 (" 0 + ")t v = v 0 + at x = 1 2 (v 0 + v)t! = " 0 t #t2 x = v 0 t at2! 2 =! "# v 2 = v ax 8

5 Masteing Physics Assignment 4 Is due Monday, Novembe 12 at 11 pm Coves mateial fom chaptes 6 and 7 Thee ae 8 questions fo pactice and 6 fo cedit The Final Exam Schedule is Now Final! PHYS 1020: Monday, Decembe 17, 6-9 pm Fank Kennedy Bown & Gold Gyms The whole couse 30 multiple choice questions Fomula sheet povided 9 Angula Displacement! B l A The length of the ac of the cicle of adius fom A to B is: l = θ, whee θ is in adians! adians = Angula Velocity ω = θ B θ A t B t A ad/s (aveage) ω = θ t ad/s (instantaneous) And, v = 10

6 Equations fo otational motion Analogous to the equations fo linea motion.! =! 0 + "t! = 1 2 (" 0 + ")t! = " 0 t #t2! 2 =! "# v a x v = v 0 + at x = 1 2 (v 0 + v)t x = v 0 t at2 v 2 = v ax : A stoboscope is a light that flashes on and off at a constant ate to illuminate a otating object. If the flashing ate is adjusted popely, the otating object appeas to be stationay. B A thee blade popelle is otating at 16.7 evolutions pe second. A C What should be the shotest time between flashes? The popelle appeas to be stationay if it otates fom A to B between flashes. This is 1/3 of a evolution, which takes the popelle a time: t = = seconds 16.7 Fo the next shotest time, the popelle otates fom A to C in s. 12

7 8.-/20: The angula speed of the oto in a centifuge inceases fom 420 to 1420 ad/s in 5 s. a) Though what angle does the oto move in this time? Aveage angula speed = ( )/2 = 920 ad/s. In 5 s, oto tuns though 920! 5 = 4600 adians. o, π = evolutions b) What is the angula acceleation?! =! 0 + "t (think of v = v 0 + at) So,! = ( ) ad/s 5s = 200 ad/s /64: Two people stat at the same place and walk aound a cicula lake in opposite diections. One has angula speed ad/s, the othe ad/s. How long befoe they meet? ad/s ad/s A:! A = t B:! B = 2" t (stats at 360 o, walks back) # A #B! A =! B when t = 2" t t = 2"/( ) = 1232 s. 14

8 8.13/11: The speed of a bullet can be measued with the appaatus shown. The bullet passes though two disks that ae otating togethe. The disks otate as the bullet tavels fom one disk to the othe, so the holes in the disks do not line up. The angula displacement between the bullet holes is 0.24 ad. What is the speed of the bullet? $ = 95 ad/s In time t, the bullet tavels 0.85 m and the disks otate 0.24 ad at 95 ad/s. So, t = 0.24/95 = s and v = (0.85 m)/( s) = 336 m/s = 0.85 m /-: The wheels of a bicycle have an angula velocity of 20 ad/s. The bakes ae applied, binging the bicycle to a unifom stop. Duing baking, the angula displacement of the wheels is evolutions. a) How much time does it take to stop? b) What is the angula acceleation of the wheels? a) θ = ωt, ω = (ω 0 + ω)/2 = ω 0 /2 = 10 ad/s So, t = θ ω = π 10 = 10 s b) ω = ω o + αt 0 = ω 0 + αt α = ω 0 t = = 2 ad/s2 16

9 8.27: A child is unning aound a stationay mey-go-ound at 0.25 ad/s. At the moment he sees his favouite hose, one quate tun away, the mey-go-ound stats to move in the same diection as the child is unning, acceleating at 0.01 ad/s 2. What is the shotest time it takes the child to catch up with the hose? Child What ae angula positions of child and hose? $ c = 0.25 ad/s Child: #c = $c t = 0.25 t Hose: #h = #o + $0t + %t 2 /2 = "/ t 2 /2 Hose % = 0.01 ad/s 2 They meet when #c = #h : 0.25 t = "/ t 2 / t t + π/2 = 0 t = 0.25 ± π/ " t = 7.37 s, o 42.6 s Hose passes child : A spinning top is made to spin by pulling on a sting that is wapped aound it. The length of the sting is L = 64 cm and the adius of the top is = 2 cm. The sting is pulled so that the angula acceleation of the top is 12 ad/s 2. What is the final angula velocity of the top? ω 2 = ω αθ (think v 2 = v o 2 + 2ax) What is? The numbe of tuns on the top is N = L/(2!) = 5.09 ω 2 = π = 768 (ad/s) 2 ω = 28 ad/s 18

10 8.14/68: A golf ball passes though a windmill, which has 8 blades and otates at = 1.25 ad/s. The opening between successive blades is equal to the width of a blade. A golf ball is of diamete d = m. What must be the minimum speed of the golf ball so that it passes though an opening between blades? Thee ae 8 blades and 8 gaps between blades. The angula width of each gap is 2!/16 ad. The golf ball must tavel at least a distance equal to its diamete while the blade moves the width of one gap. That is, t = d v = 2π/16 ω v = m/s v = 8dω π t = θ/ω v v 19 Equations fo otational motion Analogous to the equations fo linea motion.! =! 0 + "t! = 1 2 (" 0 + ")t! = " 0 t #t2 v a x v = v 0 + at x = 1 2 (v 0 + v)t x = v 0 t at2! 2 =! "# v 2 = v ax ac length, l = speed, v =! adians = 180 o 20

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