Rotational Kinematics
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Angular Position
•
Angular Velocity
•
Angular Acceleration
•
Rotation with Constant Angular Acceleration
•
Homework
1
Angular Position
•
•
Consider an object rotating about a fixed axis through
O perpendicular to the plane as shown below
A particle at point P has an angular position
s
θ=
r
where s is the linear distance through which the point
moves and r is the distance of the point P from the
axis
•
The unit of angular position is the radian (rad)
•
1 rev. = 360◦= 2π rad
2
Angular Velocity
•
•
Average angular velocity
θf − θi ∆θ
ω=
=
tf − t i
∆t
Instantaneous angular velocity
∆θ dθ
ω = lim
=
∆t→0 ∆t
dt
3
Relation Between Angular and Linear
Speed
s = θr
ds dθ
= r
dt
dt
v = ωr
4
Angular Acceleration
•
Average angular acceleration
α=
•
ωf − ωi ∆ω
=
tf − t i
∆t
Instantaneous angular acceleration
∆ω dω
=
∆t→0 ∆t
dt
α = lim
5
Relation Between Angular and Linear
Acceleration
v = ωr
dv dω
=
r
dt
dt
at = αr
v2
ar =
= ω 2r
r
6
Direction of Angular Velocity Vector
•
•
•
The direction of the angular velocity vector is given
by the right hand rule.
The four fingers of the right hand are wrapped in the
direction of the rotation.
The extended right thumb points in the direction of ω.
7
Rotation with Constant Angular
Acceleration
8
Example
A grindstone wheel has a constant angular acceleration of
0.35 rad/s2. (a) If it starts from rest, what is the angular
displacement after 18 s? (b) What is the angular speed of
the wheel at t = 18 s?
9
Example Solution
A grindstone wheel has a constant angular acceleration of
0.35 rad/s2. (a) If it starts from rest, what is the angular
displacement after 18 s?
ωi = 0 α = 0.35 rad/s2
t = 18 s
1 2
θf − θi = ωit + αt
2
1
2
θf − θi = 0 + 0.35 rad/s (18 s)2 = 57 rad
2


1
rev

 ' 9 rev
θf − θi = 57 rad 
2π rad
(b) What is the angular speed of the wheel at t = 18 s?
ωf = ωi + αt
2
ωf = 0 + 0.35 rad/s (18 s) = 6.3 rad/s
10
Homework Set 18 - Due Mon. Oct. 25
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Read Sections 10.1-10.3
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Answer Questions 10.2 & 10.3
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Do Problems 10.1, 10.4, 10.6, 10.9 & 10.12
11