Chapter 10: Rotational Motion
• Topic of Chapter: Objects 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 an axis through center
of mass. Can treat the two parts of motion separately.
COURSE THEME: NEWTON’S LAWS OF MOTION!
• Chs. 4 - 9: Methods to analyze the dynamics of objects in
TRANSLATIONAL MOTION. Newton’s Laws!
– Chs. 4 - 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 don’t
change as the object moves
and rotates. Example: CD 
Three Basic Types of Rigid Body Motion
Pure Rotational Motion
All points in the object move
in circles about the rotation
axis (through the
Center of Mass)
Reference Line
The axis of rotation
is through O & is
 to the picture. All
points move in
circles about O
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. All
points on a straight line
drawn through the axis
move through the same
angle in the same time.
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 object rotation:
– Each point (P) moves
in a circle with the
same center!
• Look at OP: When P
(at radius R) travels an
arc length ℓ, OP sweeps
out angle θ.

Reference Line
θ  Angular Displacement of the object
• θ  Angular Displacement
• 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 ℓ  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!

Reference Line
• θ in Radians for a circle of radius R, arc
length ℓ is defined as: θ  (ℓ/r)
• Conversion between radians & degrees:

Or
Or
θ for a full circle = 360º = (ℓ/R) radians
Arc length ℓ for a full circle = 2πR
θ for a full circle = 360º = 2π radians
1 radian (rad) = (360/2π)º  57.3º
1º = (2π/360) rad  0.017 rad
Angular Displacement: Ex. 10-1: Birds of prey—in radians
A particular bird’s eye can
just distinguish objects that
subtend an angle no smaller
than about θ = 3  10-4 rad.
a. How many degrees is this?
b. How small an object can the
bird just distinguish when
flying at a height of 100 m?
Angular Displacement
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!
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!