101 Angular Position θ In linear or translational kinematics we looked at the position of an object Δx Δy Δd We started at a reference point position x i and our definition of position relied on how far away from that position we are ID: 528879
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Slide1
Chapter 10 – Rotational Kinematics & EnergySlide2
10.1 – Angular Position (θ)
In linear (or translational) kinematics we looked at the position of an object (
Δx
,
Δy
,
Δd
…)
We started at a reference point position (x
i
) and our definition of position relied on how far away from that position we are.
Likewise, our
angular position
relies on
how
far we’ve rotated
(
Δθ
)
from a reference line.Slide3
10.1 - Angular Position (
θ
)Slide4
10.1 – Angular Position (
θ
)
Degrees and revolutions:Slide5
10.1 – Angular Position (
θ
)
Arc length
s
, measured in radians:
Arc length is how far (length) we’ve moved around the circle (arc).Slide6
10.1 – Angular Velocity (ω
)
Change in linear position of an objet over time is
velocity
.
How quickly we change position.
Linear Velocity
Rotational Velocity
Change in angular position
of an object over time is
angular velocity
.
How quickly angle changes.Slide7
10.1 – Angular Velocity (
ω
)
Sign Convention
:Slide8
10.1 – Angular Velocity (
ω
)Slide9
A drill bit in a hand drill is turning at 1200 revolutions per minute (1200 rpm). Express this angular speed in radians per second (
rad
/s)
2.1
rad
/s19 rad/s125
rad
/s
39
rad
/s
0.67
rad/sSlide10
Question 10.1a
Bonnie and
Klyde
I
w
Bonnie
Klyde
Bonnie
sits on the outer rim of a merry-go-round, and
Klyde
sits midway between the center and the rim. The merry-go-round makes one complete revolution every
2 seconds.
Klyde’s angular velocity is:
a)
same as Bonnie’s
b) twice Bonnie’s
c) half of Bonnie’s
d) one-quarter of Bonnie’s
e) four times Bonnie’sSlide11
10.1 – Angular Velocity (
ω
)Slide12
10.1 – Angular Acceleration (α)
Linear Acceleration
Defined as how quickly our velocity is changing per unit time.
When we speed up or slow down.
Angular Acceleration
Defined as how our
angular velocity (
ω
)
changes per unit time.
How fast we rotate, does that speed up or slow down?
Ex: airplane propellers
Really, really, REALLY dumb idea…Slide13
10.1 – Angular Acceleration (α)Slide14
10.1 – Angular Acceleration (
α
)
Sign Convention
:Slide15
10.2 – Rotational Kinematics
Analogies between linear and rotational kinematics:Slide16
Example 10.2 (pg. 304)
If the wheel is given an initial angular speed of 3.40
rad
/s and rotates through 1.25 revolutions and comes to rest on the BANKRUPT space, what is the angular acceleration of the wheel (assuming it’s constant)?Slide17Slide18
10.3 – Tangential Speed
What is tangential speed?
Imagine riding a merry-go-round, and suddenly letting go before the ride stops. With what velocity will you fly off the merry-go-round? Slide19
Question 10.1b
Bonnie and Klyde II
w
Bonnie
Klyde
a)
Klyde
b) Bonnie
c) both the same
d) linear velocity is zero for both of them
Bonnie
sits on the outer rim of a
merry-go-round, and
Klyde
sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds.
Who has the larger linear (tangential) velocity?Slide20
10.3 – Centripetal Acceleration of Rotating ObjectSlide21
10.3 – Tangential AccelerationSlide22
10.3 – Tangential & Centripetal Acceleration
This merry-go-round has
BOTH
tangential
and
centripetal
acceleration.Slide23
10.1 – 10.3 Summary
Arch Length
Average Angular Velocity
Instantaneous Angular Velocity
Period of Rotation
Average Angular Acceleration
Instantaneous Angular AccelerationSlide24
10.1 – 10.3 Summary
Linear Kinematics
(a = constant)
Rotational Kinematics
(
α
= constant)Slide25
10.4 -
Rolling Motion
If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds:Slide26
10.4 – Rolling Motion
We may also consider
rolling motion
to be a
combination
of
pure rotational
AND pure
translational motion
:Slide27
10.5 – Rotational Kinetic Energy
Linear Kinetic Energy
Depends on an objects
linear speed
.
NOT
valid for a rotating object because
v
is different for points of various distances from the axis of rotation.
Rotational Kinetic Energy
Depends on an objects
angular speed
.Slide28
10.5 – Moment of Inertia
Rotational Kinetic Energy depends on
ω
2
and r
2. AKA the distribution of mass of the rotating object.Moment of Inertia (I) –
Rotational Kinetic Energy can be rewritten asSlide29
10.5 – Moment of Inertia
Moment of Inertia is the distribution of mass throughout the rotating object.Slide30
10.5 – Moment of Inertia
Calculate the Moment of Inertia of this object.Slide31
Conceptual Checkpoint 10-2Slide32Slide33
10.5 – Moment of Inertia of Various Objects
Moments of inertia of various regular objects can be
calculated (pg. 314):
M = total mass
R = radius
L = LengthSlide34
10.5 – Kinetic Energy Comparison
Kinetic Energy
Linear Quantity
Angular Quantity
Speed Variable
v
ω
Mass
Variable
m
I
Final Equation
½
mv
2
½
I
ω
2Slide35
10.6 – Conservation of Energy
The
total kinetic energy
of a
rolling object
is the
sum of its linear and rotational kinetic energies
: Slide36
Example 10.5 (pg 316)
What’s the total Kinetic Energy of this 1.20 kg rolling object?Slide37
What’s the speed of this object when it reaches the bottom of the ramp?