1. Chapter 8: Rotational Motion

2. translated into rotational language.
Topics of Chapter: Objects Rotating
Newton’s Laws of Motion applied to rotating objects,
translated into rotational language.
First, rotating, without translating.
Then, rotating AND translating together.
Assumption: Rigid Bodies
Definite shape. Does not deform or change shape.
Rigid Body Motion = Translational motion of center of mass (everything done up to now) + Rotational motion about an axis through center of mass. Can treat the two parts of motion separately.

3. Course Theme: Newton’s Laws of Motion!
Everything up to now:
Methods to analyze the dynamics of objects in TRANSLATIONAL MOTION.
Newton’s Laws!
Chs. 3, 4, 5: Newton’s Laws using Forces
Ch. 6: Newton’s Laws using Energy & Work
Ch. 7: Newton’s Laws using Momentum.
NOW
Ch. 8: Methods to analyze dynamics of objects in ROTATIONAL LANGUAGE.

4. Newton’s Laws of Motion! Newton’s Laws in Rotational Language!
Course Theme:
Newton’s Laws of Motion!
Ch. 8: Methods to analyze dynamics of objects in ROTATIONAL LANGUAGE.
Newton’s Laws in Rotational Language!
First, introduction to Rotational Language.
Then, Rotational Analogues of each translational concept we already know!
Then, Newton’s Laws in
Rotational Language.

5. Rigid Body Rotation A rigid body is an extended
object whose size, shape, &
distribution of mass don’t
change as the object moves
& rotates. Example: a CD

6. There are 3 Basic Types of
Rigid Body Motion

7. Pure Rotational Motion
All points in the
object move in
circles about the
rotation axis,
through the
Center of Mass. r
Reference Line
The axis of rotation is
through O & is  to the picture.
All points move in circles about O

8. Pure Rotational Motion
All points on object move
in circles around the
axis of rotation (“O”).
The circle radius is r.
All points on a straight
line drawn through the
axis move through the
same angle in the
same time. r
Figure Caption: Looking at a wheel that is rotating counterclockwise about an axis through the wheel’s center at O (axis perpendicular to the page). Each point, such as point P, moves in a circular path; l is the distance P travels as the wheel rotates through the angle θ. r

9. Angular Quantities To describe rotational motion, we need
Positive
Rotation!
To describe rotational motion, we need
Rotational Concepts: Angular Displacement, Angular Velocity, Angular Acceleration
These are defined in direct analogy to linear quantities & they obey similar relationships!

10. θ  Angular Displacement
Rigid Body Rotation
Each point (P)
moves in a circle
with the same center!
Look at OP:
When P (at radius
r) sweeps out angle θ.
θ  Angular Displacement
of the object r

11. θ  Angular Displacement
Commonly, we measure θ in degrees.
Math of rotation: Easier if
θ is measured in Radians
1 Radian  Angle swept out
when the arc length = radius
When   r, θ  1 Radian
θ in Radians is defined as:
θ = ratio of 2 lengths (dimensionless)
θ MUST be in radians for this to be valid! r 
Reference Line
  (/r)

12. θ for a full circle = 360º = (/r) radians
θ in Radians for a circle of radius r, arc length  is defined as: θ  (/r)
Conversion between radians & degrees:
θ for a full circle = 360º = (/r) radians
Arc length  for a full circle = 2πr
 θ for a full circle = 360º = 2π radians
Or 1 radian (rad) = (360/2π)º  57.3º
Or 1º = (2π/360) rad  rad
In doing problems in this chapter, put your calculators in RADIAN MODE!!!!

13. Angular Displacement
Figure Caption: A wheel rotates from (a) initial position θ1 to (b) final position θ2. The angular displacement is Δθ = θ2 – θ1.

14. Angular Velocity (Analogous to linear velocity!)
Average Angular Velocity =
angular displacement θ = θ2 – θ1
(rad) divided by time t:
(Lower case Greek omega, NOT w!)
Instantaneous Angular Velocity
(Units = rad/s) The SAME for all points
in the object! Valid ONLY if θ is in rad!

15. Angular Acceleration (Analogous to linear acceleration!)
Average Angular Acceleration = change in angular velocity ω = ω2 – ω1 divided by time t:
(Lower case Greek alpha!)
Instantaneous Angular Acceleration = limit of α as t, ω 0
(Units = rad/s2)
The SAME for all points in body!
Valid ONLY for θ in rad & ω in rad/s!

16. Relations Between Angular & Linear Quantities
Ch. 5 (circular motion):
A mass moving in a circle
has a linear velocity v & a
linear acceleration a.
We’ve just seen that it also
has an angular velocity &
an angular acceleration. Δ Δθ r
 There MUST be relationships between
the linear & the angular quantities!

17. Connection Between Angular & Linear Quantities
Radians! 
v = (/t),  = rθ
 v = r(θ/t) = rω
v = rω
 Depends on r
(ω is the same for all points!)
vB = rBωB, vA = rAωA
vB > vA since rB > rA

18. Summary: Every point on a rotating body has an angular velocity ω and a linear velocity v. They are related as:

19. Relation Between Angular & Linear Acceleration
In direction of motion:
(Tangential acceleration!)
atan= (v/t), v = rω
 atan= r (ω/t)
atan= rα
atan : depends on r
α : the same for all points
_____________

20. Angular & Linear Acceleration
From Ch. 5: there is also
an acceleration  to the
motion direction
(radial or centripetal
acceleration)
aR = (v2/r)
But v = rω
 aR= rω2
aR: depends on r
ω: the same for all points
_____________

21. Total Acceleration Tangential: atan= rα Radial: aR= rω2
 Two  vector
components of
acceleration
Tangential:
atan= rα
Radial:
aR= rω2
Total acceleration
= vector sum:
a = aR+ atan
_____________
a ---

22. Relation Between Angular Velocity & Rotation Frequency
f = # revolutions / second (rev/s)
1 rev = 2π rad
 f = (ω/2π) or ω = 2π f
= angular frequency
1 rev/s  1 Hz (Hertz)
Period: Time for one revolution.
 T = (1/f) = (2π/ω)

23. Translational-Rotational Analogues & Connections
Translation Rotation
Displacement x θ
Velocity v ω
Acceleration a α
CONNECTIONS
 = rθ, v = rω
atan= r α
aR = (v2/r) = ω2 r

24. Correspondence between Linear & Rotational quantities

25. Conceptual Example: Is the lion faster than the horse?
On a rotating merry-go-round, one child sits on a horse near the outer edge & another child sits on a lion halfway out from the center.
a. Which child has the greater translational velocity v?
b. Which child has the greater angular velocity ω?
Answer: The horse has a greater linear velocity; the angular velocities are the same.

26. Example: Angular & Linear Velocities & Accelerations
A merry-go-round is initially at rest (ω0 = 0). At t = 0 it is given a constant angular acceleration α = 0.06 rad/s At t = 8 s, calculate the following:
a. The angular velocity ω b. The linear velocity v of a child located r = 2.5 m from the center.
Figure Caption: Example 10–3. The total acceleration vector a = atan + aR, at t = 8.0 s.
Solution: a. The angular velocity increases linearly; at 8.0 s it is 0.48 rad/s.
b. The linear velocity is 1.2 m/s.
c. The tangential acceleration is 0.15 m/s2.
d. The centripetal acceleration at 8.0 s is 0.58 m/s2.
e. The total acceleration is 0.60 m/s2, at an angle of 15° to the radius.
c. The tangential (linear) acceleration atan of that child.
d. The centripetal acceleration aR of the child.
e. The total linear acceleration a of the child.

27. Example: Hard Drive
The platter of the hard drive of a computer rotates at frequency f = 7200 rpm (rpm = revolutions per minute = rev/min)
a. Calculate the angular velocity ω (rad/s) of the platter.
b. The reading head of the drive r = 3 cm (= 0.03 m) from the rotation axis. Calculate the linear speed v of the point on the platter just below it.
c. If a single bit requires 0.5 μm of length along the direction of motion, how many bits per second can the writing head write when it is r = 3 cm from the axis?
Solution: a. F = 120 Hz, so the angular velocity is 754 rad/s.
b. V = 22.6 m/s.
c. 45 megabits/s