How many degrees in one radian?
CCW: +
A)  1 rad = 2π degrees
θ0 = 0
B)  1 rad = 180°
CW: C)  1 rad = 10°
D)  1 rad = 57.3°
E)  Radian is not a measure of angle, so the question is
nonsense.
In 180°: θ = s/r = half of circumference of a circle/radius
= πr/r = π radians.
Therefore: 1 radian x (180°/π rad) ≈ 57.3°. Note: The arc length of 1 radian is equal to 1 radius.
L32 F 11/7/14 a+er lecture 1 Assignments
Announcements:
•  CAPA 11 is now live. HW11 is due next week.
•  You should read Ch. 10 now.
Today:
•  Continuing with rotational motion, found in Ch. 10. Today rotational
kinematics.
•  On Monday we’ll move on to rotational dynamics. Also we’ll
discuss center-of-mass on Monday, found in Ch. 9.
L32 F 11/7/14 a+er lecture 2 Rotational Motion
position velocity acceleration -- rotation about a fixed axis.
As distinct from “translational motion” which we
have studied so far.
Translational Motion
Rotational Motion
x (m) v (m/s) = dx/dt a (m/s2) = dv/dt Constant acceleration a:
θ (rad)
ω (rad/s) = dθ/dt
α (rad/s2) = dω/dt
Constant angular acceleration α:
v = v0 + at
ω = ω 0 + αt
1 2
x = x0 + vot + at
2
v 2 = v02 + 2aΔx
1 2
θ = θ 0 + ω ot + α t
2
ω 2 = ω 02 + 2αΔθ
L32 F 11/7/14 a+er lecture 3 Rotational Kinematics
s = r θ
arc-length
formula
Rotational velocity:
Δs
dθ
ω=
dt
[ω] = rad/s = s-1
rate of change of the angle θ
ω +: CCW
ω -: CW
L32 F 11/7/14 a+er lecture 4 Rotational Kinematics
s = r θ
arc-length
formula
Relating v and ω:
Δs
v = rω
L32 F 11/7/14 a+er lecture or
ds
v=
dt
d
= (rθ )
dt
dθ
=r
dt
= rω
dθ v
ω=
=
dt r
tangential velocity (m/s)
use arc-length formula
r constant for circular motion
definition of angular velocity ω
Rate at which θ is changing
5 s = rθ v = rω
A pocket watch and Big Ben are both keeping
perfect time. Which minute hand has the larger angular
velocity ω?
A)  Pocket watch.
B)  Big Ben.
C)  Same ω. The angular speed is the same: Δθ complete rotation 2π rad 1hour
ω=
=
=
⋅
= 0.00174rad / s
Δt
1hour
1hour 3600s
L32 F 11/7/14 a+er lecture 6 s = rθ v = rω
A pocket watch and Big Ben are both keeping
perfect time. Which minute hand’s tip has the larger
tangential velocity v?
A)  Pocket watch.
B)  Big Ben.
v = rω
C)  Same v. L32 F 11/7/14 a+er lecture 7 s = rθ v = rω
A small wheel and a large wheel are connected by a belt. The small
wheel turns at a constant angular velocity ωS.
How does the magnitude of the angular velocity of the large wheel
ωL compare to that of the small wheel ωS?
A)  ωS = ωL
B)  ωS > ωL C)  ωS < ωL Check this by experiment.
L32 F 11/7/14 a+er lecture 8 s = rθ v = rω
A small wheel and a large wheel are connected by a belt. The small
wheel turns at a constant angular velocity ωS.
There is a bug S on the rim of the small wheel and a bug L on the
rim of the big wheel? How do their speeds compare?
A)  vS = vL
B)  vS > vL C)  vS < vL They both travel with the speed of the belt.
L32 F 11/7/14 a+er lecture 9 Rotational Kinematics
Δs
s = r θ
arc-length
formula
dθ v
ω=
=
dt r
angular
velocity
Units:
Note: rpm = 60 f = 60/τ
L32 F 11/7/14 a+er lecture # revolutions
1 rev
1
f =
=
=
(Hz)
sec
period τ
# radians
2π
2π
ω=
=
=
= 2π f
sec
period
τ
2π r
v = ωr =
= 2π rf
τ
10 A ladybug is clinging to the rim of a spinning wheel which
is spinning CCW and is speeding up. At the moment
shown, when the bug is at the far right, what is the
approximate direction of the ladybug's acceleration? 
v2

Δv

v1
L32 F 11/7/14 a+er lecture 
v2

v1
11 Rotational Kinematics
s = r θ
arc-length
formula
Δs
ar is due to change in direction of

velocity vector v (centripetal accel)
atan is due to change in the magnitude

(speed) of v

2
| a | = a = atan
+ ar2
L32 F 11/7/14 a+er lecture ω=
dθ v
=
dt r
Rate at which θ is changing
dv
tangential acceleration (m/s )
atan =
dt
d(rω )
use equation for ω
=
dt
dω r constant for circular motion
=r
dt
definition of α
= rα
dω atan
Rate at which ω is
α=
=
changing
dt
r
2
12 Rotational Kinematics
Describing rotational motion
angle of rotation (rads)
=
v 2π
=
= 2π f
r
τ
angular velocity (rad/s)
=
atan
r
angular acceleration (rad/s2)
Recall that centripetal acceleration is expressed in terms of
tangential velocity as: ar = v2/r. How is it expressed in terms of
angular velocity ω?
A) ar = ω2/r B) ar = rω C) ar = rω2
D) ar = r2ω2
ar = v2/r = (rω)2/r = rω2 L32 F 11/7/14 a+er lecture 13 Example 1. A vinyl record has radius r = 6 in = 15.2 cm and spins at 33.3 rpm (revolutions per minute).
1. What is its period?
A) 33.3 sec
B) 3 sec C) 1.8 sec s = r θ
dθ v 2π
ω=
= =
= 2π f
dt r τ
dω atan
α=
=
dt
r
D) 0.55 sec
rev
33.3
→
1 min
L32 F 11/7/14 a+er lecture 1 min ⎛ 60 sec ⎞
sec
=τ
⎜⎝
⎟⎠ = 1.8
33.3 rev 1min
rev
14 Example 1. A vinyl record has radius r = 6 in = 15.2 cm and spins at 33.3 rpm (revolutions per minute).
1. What is its period?
τ = 1.8 sec
s = r θ
dθ v 2π
ω=
= =
= 2π f
dt r τ
dω atan
α=
=
dt
r
2. What is its frequency?
A) 33.3 Hz
B) 3 Hz
C) 1.8 Hz D) 0.55 Hz
1
f =1/τ =
= 0.55 Hz
1.8 sec
L32 F 11/7/14 a+er lecture 15 Example 1. A vinyl record has radius r = 6 in = 15.2 cm and spins at 33.3 rpm (revolutions per minute).
1. What is its period?
τ = 1.8 sec
2. What is its frequency?
f
s = r θ
dθ v 2π
ω=
= =
= 2π f
dt r τ
dω atan
α=
=
dt
r
= 0.55 Hz
3. What is its angular velocity?
A) 0.55 Hz
B) 3.49 rad/sec
C) 1.8 rad/sec D) 0.88 rad/sec
ω = 2π f = (2)(3.14)(0.55) rad / sec = 3.49 rad / sec
L32 F 11/7/14 a+er lecture 16 Example 1. A vinyl record has radius r = 6 in = 15.2 cm and spins at 33.3 rpm (revolutions per minute).
1. What is its period?
τ = 1.8 sec
2. What is its frequency?
f
s = r θ
dθ v 2π
ω=
= =
= 2π f
dt r τ
dω atan
α=
=
dt
r
= 0.55 Hz
3. What is it angular velocity?
ω = 3.49 rad / sec
4. What is the speed v of a bug on its rim?
A) 53 cm/sec B) 3.49 cm/sec C) 4.35 cm/sec D) 8.44 cm/sec
v = rω
= (15.2cm)(3.49 rad / sec)
= 53.0 cm / sec
L32 F 11/7/14 a+er lecture 17