Physics 201, Lectue 6 Today s Topics q Unifom Cicula Motion (Section 4.4, 4.5) n Cicula Motion n Centipetal Acceleation n Tangential and Centipetal Acceleation q Relatie Motion and Refeence Fame (Sec. 4.6) Hope you hae peiewed!.
Tiial Math Reiew: Cicle q A cicle can be descibed by a cente and a adius. q The cicumfeence (i.e. linea path length along a full cicle) of a cicle of adius is 2π q A full cicula angle is 360 o o 2π tangential line q A tangential line is pependicula to the adial line fom cente to the tangential point. q Ac distance (ac length ) s = Δθ s Δθ
Reiew: Kinematical Quantities in Vecto Fom q Displacement: Δ = f i q Velocity (aeage and instantaneous): ag Δ = Δt, = Δ d lim = Δt= 0 Δt dt a q Acceleation (aeage and instantaneous): ag = Δ Δt, a = Δ lim Δt= 0 Δt = d dt
Special Notes q The mathematical teatment fo cicula motion kinematics in the next thee slides epesents some exta eadings beyond the textbook contents. It is meant to help you to hae a bette undestanding of kinematical fomulas fo the cicula motion. q In my judgment, the book teatment is oe simplified and possibly less conincing to those who want a deepe undestanding. q In any case, deiation fo those fomulas is not equied fo this couse. Please pay moe attention to the final esults that I will summaize in one slide late.
Math Pepaation: Diffeence of Two Vectos q Delta of two ectos in the same diection: - = = Δ ˆ q Delta of two ectos with the same length: Δ = f i d = f i = dθ ˆ θ 90 ο -Δθ/2 90 ο f Δθ i Δθ 0 f i dθ 0
q Fo ecto =, change can be in length and in diection. Math Pepaation: Diffeential of a Vecto ^ d keep diection but change in length: maintain length but change in diection: ˆ adial unit ecto d d ˆ d dθ θ ˆ θˆ ˆ f i dθ tangential unit ecto Togethe: d = d ˆ + d θ θ ˆ poduct ule
q Recall: Velocity in Cicula Motion d = d ˆ + dθ θ ˆ q Fo cicula motion d =0 d = dθ ˆ θ = d = dt dθ ˆ θ dt In cicula motion, elocity is always in tangential diection, i.e. always pependicula to adial ecto. q Definition: Angula elocity ω = dθ/dt Ø = d dt = d dt θ ˆ θ = ˆ ω θ and = ω
Unifom Cicula Motion q Unifom cicula motion is cicula motion with constant angula elocity (ω). q Tiial quiz: fo a unifom cicula motion with ω, how long does it take to complete a full cicle? ( 2π/ω) q Fo unifom cicula motion, peiod (T) is defined as the time the moing object takes in one full cicle. T = 2π/ω = 2π/ q Note: A elated quantity: fequency f is defined as f = 1/T
Quick Quizzes: Unifom Cicula Motion q As shown a paticle in unifom cicula motion has a peiod T and a adius R. (assume it uns in counte-clockwise.) Ø What is the magnitude of its instantaneous elocity when it passes point A? 2πR/T, 2R/T, zeo, othe Ø What is the magnitude of its aeage elocity in a time inteal when it completes a full cicle? 2πR/T, 2R/T, zeo, othe Ø What is the magnitude of its aeage speed in a time inteal when it completes a full cicle? 2πR/T, 2R/T, zeo, othe Ø Afte class execises: Answe the same questions fo time inteal fom point A to point B.
Acceleation in Unifom Cicula Motion q ecall: q Fo unifom cicula motion, and ω ae both constants. = ω ˆ θ d d ˆ θ 2 a = = ω = ω ( ˆ) dt dt hee we used: d ˆ θ = ω( ˆ) dt (why: see boad) q In unifom cicula motion, a is always pointing towads the cente Centipetal Acceleation (a c ) q Popeties of centipetal acceleation Always points to the cente a c = ω 2 = 2 / a c
Summay of Kinematics fo Unifom Cicula Motion q Instantaneous elocity is always in tangential diection = ω ˆ θ, i.e. =ω (The aboe is tue een fo non-unifom cicula motion) a c q Angula elocity ω is a constant: ω = 2π/T = 2πf q Instantaneous acceleation is always centipetal a = ω 2 ( ˆ), i.e. a c = ω 2 = 2 q Fo cicula motions, and a ae nee constant! q Note: ae ω, and a ae ω 2!
Execise: Spin of the Eath q The adius of eath is 6.37x10 6 m. To a good appoximation, the spin of the eath is unifom with a peiod T. Quick Quiz: How long is T? Answe: T= 24 h = 24x3600 = 86400 s! Conside a peson standing on the Equato: What is angula speed of the peson? ( ω = 2π /T = 7.27x10-5 ad/s ) What is the linea speed of that peson? ( =ω = 463.1 m/s ) How much is his acceleation? ( a c = ω 2 = 0.034 m/s 2 )
Non-Unifom Cicula Motion q In a geneic (non-unifom) cicula motion, acceleation usually has both centipetal and tangential components è Total acceleation: a = a c + a t Conceptual undestanding only fo this couse
Afte Class Quiz q We hae just leant that fo a paticle in unifom cicula motion, the diection of its acceleation is always centipetal. Howee, fo a geneic cicula motion, the acceleation can hae a centipetal and a tangential component. Ø what can we say about the elocity in cicula motion? A: Fo unifom cicula motion, the elocity is always pependicula to adial ecto. (i.e. tangential). But fo a geneic cicula motion, the elocity can hae both tangential and centipetal components. B: Fo any cicula motion, the elocity is always tangential.
Relatie Motion q All motions ae measued in a efeence fame. Same motion can be measued to be diffeently in diffeent efeence fame. e.g. A passenge sits in a moing bus. w..t bus, the passenge is stationay (=0) w..t Eath, the passenge is moing at bus q Conesion between efeence fames = + obj _ wt _ FameB obj _ wt _ FameA FameA _ wt _ FameB
Relatie Motion in 1-D q On a staight oad, a bus is moing fowad at a speed of 10 m/s (i.e. bus_eath = +10 m/s). in the meanwhile, a man is walking inside the bus. Quiz 1: If the man is walking fowad at 1 m/s w..t the bus (i.e. man_bus = +1.0 m/s), what is the man s elocity w..t. the Eath? Answe: man_eath = 11 m/s = 10 + 1 = man_bus + bus_eath Quiz 2: If the man is walking backwad at 1 m/s instead (i.e. man_bus = -1.0 m/s), what is the man s elocity w..t. the Eath? Answe: man_eath = 9 m/s = 10 + (-1 )= man_bus + bus_eath man _ wt _ Eath = man _ wt _ Bus + Bus _ wt _ Eath obj _ wt _ FameB = obj _ wt _ FameA + FameA _ wt _ FameB
Relatie Motion: Galilean Tansfomation q Conesion between efeence fames (Galilean Tansfomation) = + obj _ wt _ FameB obj _ wt _ FameA FameA _ wt _ FameB A_B o_a o_b o_a A_B One example Same pinciple but a diffeent configuation isualization example : A=bus, B=eath, o=ain dops
Relatie Velocity Example: Rain Tace as Seen Inside a Bus Rain seen on Eath E be E = b + be i.e. b = E be E be b E : elocity ain w..t Eath be : elocity bus w..t Eath b : elocity ain w..t. bus
Relatie Velocity Example: Coss a Rie = + be b E E : elocity ie w..t Eath be : elocity boat w..t Eath b : elocity boat w..t. ie Wate flow
Execise: Aiplane in Wind q A jet ailine moing at 590 mph due east moes into a egion whee the wind is blowing at 140 mph in a diection 60 noth of east. What is the speed and diection of the aicaft (w..t. Eath)? q Solution: ( i = east, j = noth, J=jet, E=Eath, W= wind) use (ecto) elationship JE = JW + WE JE = JW + WE WE = 140xcos(60 o )i + 140xsin(60 o )j = 70i + 121.24j JW = 590 i, JE = (590+70)i + 121.24j= 660i + 121.24j JE = 671mph =sqt(660 2 +121.21 2 ), at 10.41 o NofE =atan(121.21/660)
Exta Reading: Acceleation on a Cued Path q At eey point along the path, the total acceleation is made of by its centipetal and tangential components. Conceptual undestanding only fo this couse