1. Chapter 8 Rotational Motion
© 2014 Pearson Education, Inc.

2. Chapter 8: Rotational Motion
If you ride near the outside of a merry-go-round, do you go faster or slower than if you ride near the middle?
It depends on whether “faster” means
a faster linear speed (= speed), ie more distance covered per second, or
- a faster rotational speed (=angular speed, w), i.e. more rotations or revolutions per second.
The linear speed of a rotating object is greater on the outside, further from the axis (center), but the rotational speed is the same for any point on the object – all parts make the same # of rotations in the same time interval.

3. 8-1 Angular Quantities
In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is r. The angle θ in radians is defined: where l is the arc length.
(8-1a)
© 2014 Pearson Education, Inc.

4. More on rotational vs tangential speed
For motion in a circle, linear speed is often called tangential speed
The faster the w, the faster the v in the same way (e.g. merry-go-round), i.e. v ~ w.
- w doesn’t depend on where you are on the merry-go-round, but v does: i.e. v ~ r
directly proportional to
He’s got twice the linear speed than this guy.
Same RPM (w) for all these people, but different tangential speeds.

5. 8-1 Angular Quantities
Angular displacement: Δθ = θ2 – θ1 The average angular velocity is defined as the total angular displacement divided by time: The instantaneous angular velocity:
(8-2a)
(8-2b)
© 2014 Pearson Education, Inc.

6. 8-1 Angular Quantities
The angular acceleration is the rate at which the angular velocity changes with time: The instantaneous acceleration:
(8-3a)
(8-3b)
© 2014 Pearson Education, Inc.

7. A carnival has a Ferris wheel where the seats are located halfway between the center and outside rim. Compared with a Ferris wheel with seats on the outside rim, your angular speed while riding on this Ferris wheel would be
A) more and your tangential speed less.
B) the same and your tangential speed less.
C) less and your tangential speed more.
D) None of the above
Answer: B
Same # cycles per second i.e. same angular speed, but less distance covered each second, so less tangential speed

8. 8-1 Angular Quantities
Every point on a rotating body has an angular velocity ω and a linear velocity v. They are related:
(8-4)
© 2014 Pearson Education, Inc.

9. 8-1 Angular Quantities
Therefore, objects farther from the axis of rotation will move faster.
© 2014 Pearson Education, Inc.

10. 8-1 Angular Quantities
If the angular velocity of a rotating object changes, it has a tangential acceleration: Even if the angular velocity is constant, each point on the object has a centripetal acceleration:
(8-5)
(8-6)
© 2014 Pearson Education, Inc.

11. 8-1 Angular Quantities
Here is the correspondence between linear and rotational quantities:
© 2014 Pearson Education, Inc.

12. 8-1 Angular Quantities
The frequency is the number of complete revolutions per second: f = ω/2π Frequencies are measured in hertz. 1 Hz = 1 s−1 The period is the time one revolution takes:
(8-8)
© 2014 Pearson Education, Inc.

13. 8-2 Constant Angular Acceleration
The equations of motion for constant angular acceleration are the same as those for linear motion, with the substitution of the angular quantities for the linear ones.
© 2014 Pearson Education, Inc.

14. 8-3 Rolling Motion (Without Slipping)
In (a), a wheel is rolling without slipping. The point P, touching the ground, is instantaneously at rest, and the center moves with velocity v. In (b) the same wheel is seen from a reference frame where C is at rest. Now point P is moving with velocity –v. Relationship between linear and angular speeds: v = rω
© 2014 Pearson Education, Inc.

15. Example and Demo: Railroad train wheels
Model: two tapered cups stuck together and rolling along meter sticks
First note that if you roll a tapered cup along a table, it follows a circle because the wide part moves faster than the narrow part. (Larger v so more distance covered by the wide end).
Now, if you tape two together, at their wide ends, and let them roll along two meter sticks (“tracks”), they will stay stably on the tracks.
Why? When they roll off center, they self-correct: say they roll to the left, then the wider part of the left cup and the narrower part of the right cup are on the tracks, causing rolling back to the right, since the left cup has larger tangential speed.
DEMO – need two paper cups and meter sticks?
Railroad wheels act on this same principle!