Circular (Rotational) Motion

Angular Speed andAngular Acceleration

Centripetal (Radial) AccelerationRadial Forces

Arc Length, s, and Angular Position, • s subtends angle at

___________ radius r• measured in _______

counterclockwise from x-axis

1.7r

s

Fig. 7.1, p. 191

Angular Displacement

• Difference between final and initial angular positions

• Units are radians (rad)• Positive if

__________________

2.7if

Fig. 7.3, p. 191

Angular Speed

• Divide angular displacement by time interval

• Units are _________

3.7av ttt if

if

Fig. 7.3, p. 191 4.7lim

0 tt

Angular Acceleration

• Divide angular speed by time interval

• Units are _________

5.7av ttt if

if

Fig. 7.4, p. 193

6.7lim0 tt

Angular Kinematics w/ = const

• Draw analogy with linear motion (a = const)

xavv

attvx

atvv

220

2

221

0

0 9.72

8.7

7.7

20

2

221

0

0

tt

tRotational MotionLinear Motion

Relating Linear to Angular• _________ of v related to • Called _________ velocity vt

• Also have tangential accleration at

10.7rvt

Fig. 7.5, p. 196

11.7rat

All points have ______ angular speed & acceleration; points further from origin have ______ linear speed & acceleration

Centripetal Acceleration

• v always _____________ to r• If Δt small then Δs and Δ small

• v points toward _______• Use similar triangles (a) and (b):

Fig. 7.7, p. 200

sr

vv

r

s

v

v

r

v

t

s

r

v

t

va

2

av

Centripetal Acceleration

r

vac

2

rvt

17.72rac

Put it all together:

Total acceleration: 18.722ct aaa

Centripetal Forces• Can be any of our familiar forces• Tension, friction, normal, gravitational

19.72

r

vmmaF cc

• Apply Newton’s 2nd Law to radial, tangential, and perpendicular directions

• Net centripetal force is:

rc FF

Typical Applications

• Vehicle making a turn on an unbanked curve (friction only)

• Vehicle making a turn on a banked curve (no friction)

• The Gravitron amusement ride• Vertical circular motion

– Ferris Wheels– Loop-the-loops (roller coasters)