Rotational Motion

1. Rotational Motion
Rotational Motion is exactly like Linear Motion!
Each variable has an equilavent.
Linear Angular Relationship
x θ x = r θ
v ω v = r ω
a α a = r α
F τ F = τ/r
Linear Angular
x = vt + ½ at2
θ = ωt + ½αt2
vf = vi + at ωf = ωi + αt
vf
2
= vi
2
+ 2ax ωf
2
= ωi
2
+ 2αθ
The equations are also the same.

2. ω vtan
θ
To summarize:
θ is how far
around the circle it
goes. (similar to
distance)
ω is how fast
around the circle it
goes (similar to
velocity)
α is how quickly it
changes how fast
it is going (similar
to acceleration)
Frequency is the number of revolutions
(times around the circle) divided by
how long it takes.
F=Rev/s

3. Summary
To
convert
Distance
(x)
m
Angle
(θ)
radians
x = θr Frequency
(f)
f = 1/T = ω/2π
Usually written as ω = 2πf
Unit: 1 Hertz = 1 Hz = 1
rev/s = s-1
velocity
(v)
m/s
angular
velocity
(ω)
rad/s
v = ωr
Centripetal
(radial) acceleration
(a)
m/s2
(a = v2
/r)
angular
acceleration
(α)
rad/s2
atan = αr Period
(T)
s or s/rev
T=1/f
Tangent
Acceleration
(atan)
m/s2

4. Sample Problems
• A CD has a playing time of 74 minutes. When the music starts, the CD is
rotating at an angular speed of 480 revolutions per minute (rpm). At the
end of the music, the CD is rotating at 210 rpm. Find the magnitude of
the average angular acceleration of the CD. Express your answer in
rad/s2
.
ωf = 210 rpm
ωi = 480 rpm
t = 74 min
α = ?
2 1min
50.3 /
1 60
rad
x x rad s
rev s
π
=
2 1min
22.0 /
1 60
rad
x x rad s
rev s
π
=
60
4440
1min
s
x s= ωf = ωi + αt
22 = 50.3 + α (4440)
α = -0.00637 rad/s2

5. Sample
Problem 2
• A daddy takes his children to the
playground. They run over to the
merry-go-round and beg to be spun to
death. The daddy gets the merry-go-
round traveling at a tangent velocity of
11 m/s. If the merry-go-round takes
124 s to come to a stop, how many
revolutions did the merry-go-round
make?
ωf = 0 rad/s
vtan = 11 m/s
t = 124 s
θ = ? (in rev)
11 2.5
4.4 /i
v r
rad s
ω
ω
ω
=
=
=
ωf = ωi + αt
0 = 4.4 + α (124)
α = -0.0354 rad/s2
ωf
2
= ωi
2
+ 2αθ
0 = 4.42
+ 2 (-0.0354) (θ)
θ = 273 rad
1
273 43.5
2
rev
rad rev
rad
θ
π
= =

6. Sample
Problem 2
• A daddy takes his children to the
playground. They run over to the
merry-go-round and beg to be spun to
death. The daddy gets the merry-go-
round traveling at a tangent velocity of
11 m/s. If the merry-go-round takes
124 s to come to a stop, how many
revolutions did the merry-go-round
make?
ωf = 0 rad/s
vtan = 11 m/s
t = 124 s
θ = ? (in rev)
11 2.5
4.4 /i
v r
rad s
ω
ω
ω
=
=
=
ωf = ωi + αt
0 = 4.4 + α (124)
α = -0.0354 rad/s2
ωf
2
= ωi
2
+ 2αθ
0 = 4.42
+ 2 (-0.0354) (θ)
θ = 273 rad
1
273 43.5
2
rev
rad rev
rad
θ
π
= =