Chapter 10: Rigid Object Rotation
• 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.
COURSE THEME: NEWTON’S LAWS OF MOTION!
• Chs. 5 - 9: 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.
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 
Three Basic Types of Rigid Body Motion
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
Sect. 10.1: Angular Quantities
Positive Rotation!

• Description of rotational
motion: Need concepts:
Angular Displacement
Angular Velocity, Angular Acceleration
• Defined in direct analogy to linear quantities.
• Obey similar relationships!
• 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.

Reference Line
θ  Angular Displacement of the object
NOTE: Your text calls the arc length s instead of !
• θ  Angular Displacement
• Commonly, measure θ in degrees.
 here is text’s s!
• 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!

Reference Line
• 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  0.017 rad
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!
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!
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!
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!!
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