Section 8-2: 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!
The equations of motion for constant angular acceleration are the same
as those for linear motion, substituting the angular quantities for the
linear ones. For α = constant, & using the replacements
x  θ, v  ω, a  α we get the equations:
NOTE
These are ONLY VALID if all angular quantities are
in radian units!!
Example 8-6: Centrifuge Acceleration
A centrifuge rotor is accelerated from rest to
frequency f = 20,000 rpm in 30 s.
a. Calculate its average angular acceleration.
b. Through how many revolutions has the
centrifuge rotor turned during its
acceleration period, assuming constant
angular acceleration?
Example: Rotating Wheel
• A wheel rotates with constant angular acceleration α = 3.5 rad/s2.
It’s angular speed at time t = 0 is ω0 = 2.0 rad/s.
(A) Calculate the angular displacement Δθ it makes after t = 2 s.
Use:
Δθ = ω0t + (½)αt2
= (2)(2) + (½)(3)(2)2 = 11.0 rad (630º)
(B) Calculate 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 ω after t = 2 s. Use:
ω = ω0 + αt = 2 + (3.5)(2) = 9 rad/s
Example: CD Player
• Consider a CD player playing a CD. For the player to
read a CD, the angular speed ω must vary to keep the
tangential speed constant (v = ωr). A CD has inner
radius ri = 23 mm = 2.3  10-2 m & outer radius
ro = 58 mm = 5.8  10-2 m. The tangential speed
at the outer radius is v = 1.3 m/s.
(A) Find angular speed in rev/min at the inner & outer radii:
ωi = (v/ri) = (1.3)/(2.3  10-2) = 57 rad/s = 5.4  102 rev/min
ωo = (v/ro) = (1.3)/(5.8  10-2) = 22 rad/s = 2.1  102 rev/min
• (B) Standard playing time for a CD is 74 min, 33 s (= 4,473 s). How many
revolutions does the disk make in that time?
θ = (½)(ωi + ωf)t = (½)(57 + 22)(4,473 s)
= 1.8  105 radians = 2.8  104 revolutions
Section 8-3: Rolling Motion
• Without friction, there would be no rolling motion.
• Assume: Rolling motion with no slipping

Can use static friction
• Rolling (of a wheel) involves:
– Rotation about the Center of Mass (CM)
PLUS
– Translation of the CM
• Wheel, moving on ground with axle velocity v.
Rolls
with no
slipping!
ω
Relation between axle speed v &
angular speed ω of the wheel:
v = rω
Example 8-7
Bicycle: v0 = 8.4 m/s. Comes
to rest after 115 m. Diameter =
0.68 m (r = 0.34m)
r = 0.34m
v0 = 8.4 m/s

a) ω0 = (v0/r) = 24.7rad/s
b) total θ = (/r) = (115m)/(0.34m)
= 338.2 rad = 53.8 rev
c) α = (ω2 - ω02)/(2θ). Stopped
 ω = 0  α = 0.902 rad/s2
d) t = (ω - ω0)/α. Stopped
 ω = 0  t = 27.4 s
v=0
d = 115m

 vg = 8.4 m/s