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Rotational Kinematics
Chapter 10
Herriman High AP Physics C
Section 10.2
The Rotational Variables
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θ (theta) – angular displacement – radians
ω (omega) – angular velocity – radians/sec
α (alpha) = angular acceleration – radians/sec2
t – still time and still in seconds
Herriman High AP Physics C
Measuring Angular
Displacement
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θ = s/r
360° = 2π radians
= 1 revolution
v= ωr =
Linear velocity =
angular velocity x radius
a = αr
Linear acceleration =
angular acceleration x radius
Herriman High AP Physics C
r
θ
s
Sample Problems

If an exploding fireworks shell makes a
10° angle in the sky and you know it is
2000 meters above your head, how
many meters wide is the arc of the
explosion?
Herriman High AP Physics C
Solution
θ = L/r
θ = 10° x π rad/180° = 0.17 rad
0.17 rad = L/2000 meters
L = 2000 meters x 0.17 rad = 349 meters
Herriman High AP Physics C
Sample Problems
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Convert the following measures from
Radians to degrees:
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3.14 rad
150 rad
24 rad
Herriman High AP Physics C
Solution
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3.14 rad x 180°/π rad = 180°
150 rad x 180°/π rad = 8594°
24 rad x 180°/π rad = 1375°
Herriman High AP Physics C
Sample Problems
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What is the linear speed of a child
seated 1.2 meters from the center of a
merry-go-round if the ride makes one
revolution in 4 seconds?
What is the child’s acceleration?
Herriman High AP Physics C
Solution
a)v = ωr and ω=2π rad/4 sec = 1.6 rad/s
since r = 1.2 m then
v = 1.58 rad/s x 1.2 m = 1.9 m/s
b) Since the linear velocity is not changing there is
no linear or tangential acceleration, but the child is
moving in a circle so there is centripetal or radial
acceleration which you will recall fits the
equation:
a c = v2/r and since v = ωr ac = ω2r
so a c = (1.9 m/s)2/1.2 m = 3.0 m/s2 or
ac = ω2r = (1.6 rad/s)2(1.2 m) = 3.0 m/s2
Herriman High AP Physics C
Section 10.3 - 10.5
Rotation with Constant Angular Acceleration & the
Relationship Between Linear and Angular Acceleration
Linear
Angular
V = v0+at
x = v0t+½at2
V2 = v02+2ax
Vavg = (v0+vf)/2
ω=ω0+άt
θ=ω0t+½άt2
ω2=ω02+2άθ
ωavg=(ω0+ωf)/2
Herriman High AP Physics C
Sample Problem
A angular velocity of wheel changes from
10 rad/s to 30 rad/s in 5 seconds.
a)
What is its angular acceleration?
b)
What is its angular displacement while
it is accelerating?
Herriman High AP Physics C
Solution
α = (ωf – ω0)/t =
=(30 rad/s – 10 rad/s)/5 sec = 4 rad/s2
θ = ω0t = ½αt2
= 10 rad/s + ½(4 rad/s2)(5 sec)2
= 60 rad
Herriman High AP Physics C
Section 10.6
Kinetic Energy of Rotation
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K = ½ Iω2
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I = Rotational Inertia
ω = Angular Velocity
Rotational Inertia is a measure of how
difficult it is to rotate an object on a given
axis. This is determined by some ratio of the
mass of the object and the radius of the
object along the axis of rotation.
Herriman High AP Physics C
Section 10.7
Calculating the Rotational Inertia
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I = ∫ r2 dm
For various objects the results of these
calculations can be found on page 253 in
table 10.2
Parallel axis Theorem
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If you know the I value for a body through the
center of mass then a shortcut to finding its I
value for a parallel axis is given as:
I = Icm + Mh2
Where h is the perpendicular distance between
the axes.
Herriman High AP Physics C
Section 10.8
Torque
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Torque = Fr = force x radius
Torque is measured in newton•meters
which means that it has the same units
as work in a linear system.
Herriman
Alta High
HighAPAPPhysics
Physics C
Section 10.9
Newton’s Second Law for Rotation
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Torque = Iα
= rotational inertia x angular acceleration
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Hence Fr = Iα
Herriman
Alta High
HighAPAPPhysics
Physics C
Sample Problem
A force of 10 newtons is applied to the edge
of a bicycle wheel (a thin ring mass 1 kg
and radius of 0.5 meters). What is the
resulting angular acceleration of the
wheel?
If the wheel was at rest when the force was
applied and the force is applied for 0.4
seconds what is the angular velocity of the
wheel immediately after it is applied?
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Alta High
HighAPAPPhysics
Physics C
Solution
Since Fr = Iα then α = Fr/I and since the wheel
is a thin ring:
I = mr2 = (1 kg)(0.5 m)2 = 0.25 kg m2
So α = (10 N)(0.5 m)/0.25 kg m2
= 20 rad/s2
ωf = ω0 + αt
= 0 rad/s + (20 rad/s2)(0.4 sec)
= 8 rad/s
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Alta High
HighAPAPPhysics
Physics C
Section 10.10
Work and Rotational Kinetic Energy
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By Conservation of energy:
∆K = Kf – Ki = ½ Iωf2 – ½ Iωi2 = W
W = ∫ r dθ
W = Τ∆θ
P = dθ/dt = Tω
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For a table of corresponding relationships
Translational:Rotational see table 10.3 on
page 261
Herriman High AP Physics C
Problems Types
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Finding angles when distance and
length of arc are known
Converting from revolutions to radians
and radians to degrees
Finding angular velocity and angular
acceleration if radius and their linear
counterparts are known
Using angular kinematic equations
Herriman High AP Physics C
Problems Types



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Finding angles when distance and length of
arc are known
Converting from revolutions to radians and
radians to degrees
Finding angular velocity and angular
acceleration if radius and their linear
counterparts are known
Using angular kinematic equations
Calculating Torque
Herriman
Alta High
HighAPAPPhysics
Physics C