1. Chapter 10: Rigid Object Rotation

2. Topic of Chapter: Bodies rotating
First, rotating, without translating.
Then, rotating AND translating together.
Assumption: Rigid Body
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 axis through center of mass. Can treat two parts of motion separately.

3. COURSE THEME: NEWTON’S LAWS OF MOTION!
Chs : Methods to analyze dynamics of objects in TRANSLATIONAL MOTION. Newton’s Laws!
Chs. 5 & 6: Newton’s Laws using Forces
Chs. 7 & 8: Newton’s Laws using Energy & Work
Ch. 9: Newton’s Laws using Momentum.
NOW
Chs. 10 & 11: Methods to analyze dynamics of objects in ROTATIONAL LANGUAGE. Newton’s Laws in Rotational Language!
First, Rotational Language. Analogues of each translational concept we already know!
Then, Newton’s Laws in Rotational Language.

4. Rigid Body Rotation
A rigid body is an extended object whose size, shape, & distribution of mass do not change as the object moves and rotates. Example: CD 

5. Three Basic Types of Rigid Body Motion

6. Pure Rotational Motion
All points in the body move
in circles about the rotation axis
(through the CM)
Reference Line 
Axis of rotation through O &
 to picture. All points
move in circles about O

7. Sect. 10.1: Angular Quantities
Description of rotational
motion: Need concepts:
Angular Displacement
Angular Velocity, Angular Acceleration
Defined in direct analogy to linear quantities.
Obey similar relationships!
Positive Rotation! 

8. (at radius r) travels an arc length , OP sweeps out an angle θ.
Rigid body rotation:
Each point (P) moves
moves in circle with
the same center!
Look at OP: When P
(at radius r) travels an
arc length , OP
sweeps out an angle θ.
θ  Angular Displacement of the object 
Reference Line
NOTE: Your text calls the arc length s instead of !

9. θ  Angular Displacement Math of rotation: Easier if
Commonly, 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  (/r)
θ = ratio of 2 lengths (dimensionless)
θ MUST be in radians for this to be valid!
 here is text’s s! 
Reference Line

10. 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

11. Angular Velocity (Analogous to linear velocity!)
Ave. angular velocity =
angular displacement
θ = θf - θi (rad) divided by time t:
ωavg  (θ/t)
(Lower case Greek omega, NOT w!)
Instantaneous angular
velocity = limit ω as t, θ 0
ω  limt 0 (θ/t) = (dθ/dt)
(Units = rad/s)
The SAME for all points in the body!
Valid ONLY if θ is in rad!

12. Angular Acceleration (Analogous to linear acceleration!)
Average angular acceleration = change in angular velocity ω = ωf - ωi divided by time t:
αavg  (ω/t)
(Lower case Greek alpha!)
Instantaneous angular accel. = limit of α as t, ω 0
α  limt 0 (ω/t) = (dω/dt)
(Units = rad/s2)
The SAME for all points in the body!
Valid ONLY if θ is in rad & ω is in rad/s!

13. Sect. 10.2: Kinematic Equations
Recall: 1 dimensional kinematic equations for uniform (constant) acceleration (Ch. 2).
We’ve just seen analogies between linear & angular quantities:
Displacement & Angular Displacement:
x  θ
Velocity & Angular Velocity:
v  ω
Acceleration & Angular Acceleration:
a  α
For α = constant, we can use the same kinematic equations from Ch. 2 with these replacements!

14. NOTE: These are ONLY VALID if all angular
For α = constant, & using the replacements, x  θ, v  ω
a  α we get these equations:
NOTE: These are ONLY VALID if all angular
quantities are in radian units!!

15. Example 10.1: Rotating Wheel
A wheel rotates with constant angular acceleration α = 3.5 rad/s2. It’s angular speed at time t = 0 is ωi = 2.0 rad/s.
(A) Find the angular displacement Δθ it makes after t = 2 s.
Use: Δθ = ωit + (½)αt2
= (2)(2) + (½)(3)(2)2 = 11.0 rad (630º)
(B) Find the number of revolutions it makes in this time.
Convert Δθ from radians to revolutions:
A full circle = 360º = 2π radians = 1 revolution
11.0 rad = 630º = 1.75 rev
(C) Find the angular speed ωf after t = 2 s. Use:
ωf = ωi + αt = 2 + (3.5)(2) = 9 rad/s