http://www.businessinsider.com.a
u/chinas-three-gorges-damreally-will-slow-the-earthsrotation-2010-6#ixzz2gLFLVmuJ
Translation (Linear motion)All points move through
the same distance in the
same time.
Rotation (Angular motion)All points move through
the same angle in the
same time.
Axis of Rotation
Angular Position ()
d
𝒅
𝜽=
𝒓
  is measured in radians
 d = arc length
 r = distance of a point from the axis
Notes on radians (rad):
 1 revolution = 360 = 2 rad
 Radians are a dimensionless unit
 Don’t reset  to zero
after one revolution:
1 revolution = 2 rad,
1 ½ revolutions = 3 rad,
2 revolutions = 4 rad, etc…
d
Angular Displacement ()
 = f - i
Counterclockwise (CCW)
Clockwise (CW)
+
-
Recall:
Now:
When acceleration (a)
is constant……
When angular acceleration ()
is constant……
1)
2)
3)
4)
vf = vi + at
x = vit + ½ at2
vf2 = vi2 + 2ax
x = ½ (vi+vf )t
1)
2)
3)
4)
f = i + t
 = it + ½ t2
f2 = i2 + 2
 = ½ (i + f )t
Notice that these four equations are exactly the same
as the previous four, with these substitutions:
x

v

a

Use them in exactly the same way!
• A Centrifuge can accelerate from rest at a
constant angular acceleration of 7.0 rad s-2,
taking 3 minutes to reach top speed
– A) What is the final angular velocity?
– B) How many rotations does it turn through during
this time?
Answers:
• 𝜔𝑓 = 𝜔𝑖 + α𝑡
• 𝜔𝑓 = 0 + 7.0 × 180
• 𝜔𝑓 = 1300 𝑟𝑎𝑑 𝑠 −1
• Θ = 𝜔𝑖 +
• Θ=0
1
𝛼𝑡 2
2
1
+ 7.0
2
× 1802
• Θ = 110 000 rad
• # of turns = 110000/2 = 18000 rev
What is the angular acceleration of a compact disc
that begins at rest and accelerates to 36 rads-1 in
1.5 s?
ω0 = 0
ω = 36 rads-1
α=?
t = 1.5s
θ = ignore
Audio CD players read their discs at a constant 150 kB/s and
thus must vary the disc's rotational speed from around 500 rpm
(actually 8 Hz), when reading at the innermost edge, to 200 rpm
(actually 3.5 Hz) at the outer edge.
http://en.wikipedia.org/wiki/Revolutions_per_minute
A tire on car has an angular acceleration of –5 rad/s2
applied to it. If the wheel was originally turning at 80
rad/s and turned through 500 rad during the
deceleration, what was the final speed of the wheel?
θ = 500 rad
ω0 = 80 rad/s
ω=?
α = –5 rad/s2
t = ignore
ω2 = ω02+ 2αθ
ω= 37 rad/s
Note for one revolution:
1 rev
2

 2f
T
period
Frequency
So ω can also be called angular frequency
Angular to UCM Connection

v
r
ar 

r
r
2
2
 r
2
Angular motion example:
Bubba has been given an old record player for Christmas. He
also has a bug collection. One day while he's bored he puts 2
of his best bugs onto the record player. He places a yellow
bug half-way from the centre and a pink bug at the edge of
the 29cm diameter record player. Then he sets the speed to
33rpm and plays the player at a constant speed.
(a) convert 33 rpm to suitable ‘Physics’ units
33 rpm means 33 revolutions per minute.
Convert to radians per second by:
= 3.4557… ≈ 3.5 rad s-1 (2 SF)
2
33 
60
(b) Calculate the frequency and period of the bugs' motion
(c) Bubba's sister Betty-Sue explains that the bugs have one type
of speed that is the same while another type of speed is
different. Explain what she means and which speeds she is
talking about.
•The linear speed (tangent to the circular path) is different for each
bug.
•The bug with the larger radius (at the edge of the player) will have
a larger linear speed since it makes a larger circle in the same time
as the bug closer to the centre.
v  r
•But both bugs have the same angular speed since they complete
the circle in the same time period, T, and have the same frequency,
f, and the equation: gives the same answer for each bug.
•see ‘Torque’ PhET applet
(d) What is the angular velocity of each bug.
(e) Calculate the linear velocity of each bug.
See (a)
After a while he changes the speed on the player to 45rpm. It
takes the record player 0.85 seconds to adjust to this new
speed.
(f) Calculate the angular acceleration while the record player
adjusts to its new speed.
(g) Calculate the angular displacement while the record player
is adjusting to this new speed.
(h) Draw a graph of angular velocity vs time while the player
adjusts to this new speed. Label all important parts of your
graphs.