Circular Rotational Motion

1
Circular (Rotational) Motion

• Angular Speed and
• Angular Acceleration
• Centripetal (Radial) Acceleration
• Radial Forces

2
Arc Length, s, and Angular Position, ?

• s subtends angle ? at constant radius r
• ? measured in radians
• Counterclockwise from x-axis

Fig. 7.1, p. 191
3
Angular Displacement

• Difference between final and initial angular
  positions
• Units are radians (rad)
• Positive if counterclockwise

Fig. 7.3, p. 191
4
Angular Speed

• Divide angular displacement by time interval
• Units are rad/s

Fig. 7.3, p. 191
5
Angular Acceleration

• Divide angular speed by time interval
• Units are rad/s2

Fig. 7.4, p. 193
6
Angular Kinematics w/ ? const

• Draw analogy with linear motion (a const)

Rotational Motion
Linear Motion
7
Relating Linear to Angular

• Magnitude of v related to ?
• Called tangential velocity vt
• Also have tangential accleration at

Fig. 7.5, p. 196
All points have same angular speed
acceleration points further from origin have
greater linear speed acceleration
8
Centripetal Acceleration

• v always perpendicular to r
• If ?t small then ?s and ?? small
• ?v points toward center
• Use similar triangles

Fig. 7.7, p. 200
9
Centripetal Acceleration
Put it all together
Total acceleration
10
Centripetal Forces

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

• Apply Newtons 2nd Law to radial, tangential, and
  normal directions
• Net centripetal force is

11
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)