Physics 201, Lecture 6

Size: px

Start display at page:

Download "Physics 201, Lecture 6"

Della Cooper
5 years ago
Views:

1 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!.

2 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

3 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

4 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.

5 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

6 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

7 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

8 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

9 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.

10 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

11 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!

12 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 = 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 = m/s ) How much is his acceleation? ( a c = ω 2 = m/s 2 )

13 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

14 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.

15 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

16 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 = = 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

17 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

18 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

19 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

20 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 j JW = 590 i, JE = (590+70)i j= 660i j JE = 671mph =sqt( ), at o NofE =atan(121.21/660)

21 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

6.4 Period and Frequency for Uniform Circular Motion

6.4 Period and Frequency for Uniform Circular Motion 6.4 Peiod and Fequency fo Unifom Cicula Motion If the object is constained to move in a cicle and the total tangential foce acting on the total object is zeo, F θ = 0, then (Newton s Second Law), the tangential