1. Kinematic Equations

2. Kinematic Equations Displacement & Angular Displacement: x  θ
Recall: 1 dimensional kinematic equations for uniform (constant) acceleration (Ch. 2).
We’ve jseen 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!

3. These are ONLY VALID if all angular quantities are
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!!

4. Example: 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?
Solution: a. The final angular velocity is 2100 rad/s, so the acceleration is 70 rad/s2.
b. The total angle is 3.15 x 104 rad, which is 5000 rev.

5. 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

6. Example: CD Player = 1.8  105 radians = 2.8  104 revolutions
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)( )(4,473 s)
= 1.8  105 radians = 2.8  104 revolutions

7.  Can use static friction
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

8. Wheel, moving on the ground with axle velocity v.
Relation between axle speed v & angular speed ω of the wheel:
v = rω
Rolls
with no
slipping! ω

9. Example Bicycle: v0 = 8.4 m/s. Comes to rest after 115 m.
Diameter = 0.68 m
(r = 0.34m)
a) ω0 = (v0/r) = 24.7rad/s
b) total θ = (/r) = (115m)/(0.34m)
= rad = 53.8 rev
c) α = (ω2 - ω02)/(2θ).
Stopped
 ω = 0  α = rad/s2
d) t = (ω - ω0)/α. Stopped
 ω =  t = 27.4 s
v = 0
d = 115m 
 vg = 8.4 m/s