Circular motionT - periodf =1/T - frequency

-angular frequency, or angular speed

r

dt

rd

dt

dlv

r

)( 2tv

)( 1tv

r

l

rrdt

d

dt

dva

tan

4222222tan

tan

rrvdrdvaaa

aaa

rad

rad

rr

varad

22

dt

d

.22

then If fT

const

dt

d - angular acceleration

dt

vdvv

dt

dv

dt

vvd

dt

vda

ˆˆ

ˆ

Circular motion and vectors

sin

cos

ˆˆˆ

ry

rx

jyixrrr

r

x

y

raa radrad ˆ

z ;

;

ˆ

dt

ddt

d

k

zz

zz

z

radaa

dt

vdvv

dt

dv

dt

vvd

dt

vda

tan

ˆˆ

ˆ

vdt

dvrsignv

dt

dvsignav

dt

dva ˆˆˆ tantan

ra

tan

Comparison of Linear and Angular motion with Constant Acceleration

Straight-line motion Fixed-axis rotation

tvvxx

xxavv

tatvxx

tavv

consta

xx

xxx

xxx

xxx

x

)(

)(2

021

0

02

02

221

00

0

t

tt

t

const

zz

zz

zz

zzz

z

)(

)(2

021

0

020

20

221

00

0

Example: At t = 0, a grinding wheel has an angular velocity of 24.0 rad/s. It has an constant angular acceleration of 30.0 rad/s2 until a circuit breaker trips at t = 2.00 s. From then on, it turns through 432 rad as it coasts to a stop at constant angular acceleration. What was its acceleration as it slowed down?

1) Angular speed at 2 s :

sradssradsradtzz /8400.2/0.30/0.24 20

222

02

/17.84322

/840

2srad

rad

sradzzz

)(2 020

20 zz

2) Angular acceleration:

Relative motionGalilean transformations:

relation between the description of a particle in two frames which are moving with respect to each other with constant velocity.

tvrrconstv BAPBPABA

If

P

rB,A

rP,B

y

x

zB

rP,A y

x z

ABAPBPA

BAPBPA

BA

BAPBPA

aaa

vvv

tt

rrr

Clearly velocity is a reference-frame dependent quantity!

bg: backgrounds: moving sidewalk

What are some frame independent quantities?

A person walking on moving sidewalk: You can have vperson,background = 0 (not moving relative to a picture on the back wall):

Vs,bg = +v i

vp,s = -v iPicture on the background

Mass, time, temperature…

Example: Moving Sidewalk

, , ,ˆ ˆ 0p bg p s s bgv v v vi vi

Example: Two kids decide to race. Both kids walk with speed vw. One kid (A) will walk on the ground while the other (B) will walk on the “moving sidewalk” that moves with speed v0. The race is roundtrip. Which kid wins the race?

Time for roundtrip, kid A: A 2w

dt

v Let d = length of

the moving sidewalk.

Time for roundtrip, kid B: B against SW with SWt t t

B0 0w w

d dt

v v v v

0kid B relative to ground wv v v 0kid B relative to ground wv v v

2 20

2 w

w

v dv v

22

02

2 1

1

w

w

w

v dvvv

A 202

1

1w

tvv

0 A B202

1I f , then 1, so (answer A)

1w

w

v v t tvv

A) Kid A. B) Kid B. C) Tie. D) Depends on the ratio vw/v0. E)Depends on the sign of v0

A. 20 s

B. 25 s

C. 33 s

Example: A boat can make it move at 5 m/s relative to the water and is trying to go across a 100-m wide river to a point on the opposite shore and right North of its starting position. The river flows due West at 3 m/s. How long does the trip take?

wgv

bwvbgv

E

N

W

S

?

?

/3

/5

100

t

v

smv

smv

mx

bg

wg

bw

g

ssm

m

v

xt

smsmsmvvv

vvv

bg

g

wgbwbg

wgbwbg

25/4

100

/4/3/5 2222