Chapter 10: Rotational Motion
Thursday February 26th
• Discussion of the mid-term exam
• Review of Mini Exam III
• Intro. to rotational motion (not on mid-term)
• Definition of rotational variables
• Rotational Kinematics
• Examples and iclicker
Mid-term Exam on Tuesday:
•  Full class period – 1hr 15 mins
•  Cumulative – will cover everything up to and incl. LONCAPA #12
•  Monday recitation for review; see online review material
•  No labs next week; next labs right after spring break
•  Wednesday back to normal schedule with LONCAPA #13
Reading: up to page 158 + page 167 in Ch. 10
Translation and Rotation
Translation
Rotation
Translation + rotation
Translation and Rotation
Translation
Rotation
Rigid
body
Translation + rotation
Fixed
axis
The rotational variables (scalar notation)
Reference line
Angular position:
s
θ=
r
(in radians )
•  s is the length of the arc from the x-axis (θ = 0 rad) to the
reference line (at angle θ), at constant radius r.
•  The angle θ is measured in radians (rad), which is a ratio of
arc length to radius; it is, therefore, a dimensionless
2π r
quantity.
o
1 revolution = 360 =
= 2π rad
r
o
360
1rad =
= 57.3o = 0.159revolutions
2π
The rotational variables (scalar notation)
Angular displacement:
Δθ = θ 2 − θ1
An Angular displacement in the counterclockwise direction
about an axis (usually the z-axis) is positive, and one in
the clockwise direction is negative.
The rotational variables (scalar notation)
ω
Average angular velocity:
ω avg
Instantaneous angular velocity:
θ 2 − θ1 Δθ
=
=
t2 − t1
Δt
Δθ dθ
ω = Lim
=
Δt→0 Δt
dt
Units of angular velocity are rad.s-1 [strictly speaking a frequency, s-1]
The rotational variables (scalar notation)
α
Average angular acceleration:
α avg
Instantaneous angular acceleration:
ω 2 − ω 1 Δω
=
=
t2 − t1
Δt
Δω dω
α = Lim
=
Δt→0 Δt
dt
Units of angular acceleration are rad.s-2 [i.e., s-2]
Rotation at constant angular acceleration
Equation
number
10.7
10.8
Missing
quantity
Equation
θ − θ0
ω = ω0 + αt
θ − θ 0 = ω 0t + α t
1
2
2
ω
10.9
ω = ω + 2α (θ − θ 0 )
t
10.6
θ − θ 0 = ω t = 12 (ω 0 + ω )t
α
θ − θ0 = ω t − α t
ω0
2
2
0
1
2
2
Important: equations apply ONLY if angular acceleration is constant.
Transformation between linear & rotational variables
r
s
θ
v depends on r
s
Angular position: θ =
r
(in radians )
ds dθ
=
r = ωr
Tangential velocity: vt =
dt dt
circumference 2π r 2π
Time period for rotation: T =
=
=
velocity
v
ω
Rolling motion
s =θ R
The wheel moves with speed ds/dt
R
⇒ vcom = ω R
Transformation between linear & rotational variables
r
s
θ
v depends on r
s
Angular position: θ =
r
(in radians )
ds dθ
=
r = ωr
Tangential velocity: vt =
dt dt
dvt dω
Tangential acceleration: at =
=
r = αr
dt
dt