remainder of classcommunity.wvu.edu/~miholcomb/chapter 07a, rotational motion.pdf · tires will...
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Remainder of Class
• This week: Circular Motion and Fluids
• Next week: Thermo and Oscillatory Motion
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As we did for linear motion, first we
developed the equations of motion.
Then we can discuss the forces that
cause this motion.
Circular Motion
Every linear quantity (Ch. 2-6) has a rotational equivalent.
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Today’s Objectives
Rotational Motion
After today, you should be able to:
• Convert Between Degrees/Radians/Revs
• Define angular velocity/acceleration
• Go between linear and angular quantities
in order to solve problems
Practice: 7.1, 7.5, 7.7, 7.9, 7.11, 7.13
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θrs =
x
r
r
s
speed
θ
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θrs =
x
r
r
s
speedWhat is s for one rotation
(circumference of a circle)?
a) πr
b) 2πr
c) πr2
d) 2πr2 Q79
θ
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Radians
r
s=θ
Circumference of a circle = 2πr
Thus, θ = 2π for one revolution of a circle
What are the units of θ?
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Radians
Notes:
θ measured in radians
2π radians = 360o
2π radians = 1 revolutionr
s=θ
Circumference of a circle = 2πr
Thus, θ = 2π for one revolution of a circle
What are the units of θ?
radiansradians
ππ
2)360
2(360 =
°°
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Radians
Notes:
θ measured in radians
2π radians = 360o
2π radians = 1 revolution
Q80
r
s=θ
Circumference of a circle = 2πr
Thus, θ = 2π for one revolution of a circle
What are the units of θ?
π/3 radians of a pizza = what is the angle?
A. 30 B. 60 C. 90 D. 120 E. 240
radiansradians
ππ
2)360
2(360 =
°°
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Angular Position vs Displacement
Just like we could define initial and final
positions to find linear displacement, we can
do the same for angular displacement
r
xif
∆=−=∆ θθθ
r
s=θ
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The tires on a car have a diameter of
2.0 ft and are warranted for 60,000
miles. Determine the angle (in
radians) through which one of these
tires will rotate during the warranty
period.
How many revolutions of the tire are
equivalent to our answer?
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The tires on a car have a diameter of
2.0 ft and are warranted for 60,000
miles. Determine the angle (in
radians) through which one of these
tires will rotate during the warranty
period.
How many revolutions of the tire are
equivalent to our answer?How would this change if the tires were bigger?
r
x∆=∆θ
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Rotational Motion
Angular Quantities
– Every point in a solid rotating about an axis moves
in a circle
– Describe motion of that point by: Angular position:
s s
axis r
x∆=∆θ
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Rotational Motion
Angular Quantities
– Every point in a solid rotating about an axis moves
in a circle
– Describe motion of that point by:
– A point closer to the center (smaller r) will
have a smaller arclength for the same angle
– Or, if ∆∆∆∆x is constant, a small wheel makes
more turns for same distance (∆θ = ∆x/r)
Angular position:
s s
axis r
x∆=∆θ
∆x = r ∆θ
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Consider two people racing
around a track.
In a backyard race, which side (inner or outer) of
the track would you rather be on?
Q81
A.Outer
B.Inner
C.Finish in same
time either way.
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Consider two people racing
around a track.
In a backyard race, which side (inner or outer) of
the track would you rather be on?
Q81
A.Outer
B.Inner
C.Finish in same
time either way.
θ∆=∆ rx
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Rotational vs. Linear Motion
Average angular velocity:
t∆
∆=
θω
units: rad/s
t
xv
∆
∆=
(m/s)
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Rotational vs. Linear Motion
Average angular velocity:
t∆
∆=
θω
units: rad/s
Average angular acceleration:
tt
o
∆
∆=
∆
−=
ωωωα
units: rad/s2
t
xv
∆
∆=
(m/s)
t
va
∆
∆=
(units: m/s2)
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Find the angular speed of Earth's
rotation about its axis. (No #s!)
t∆
∆=
θω
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BIG BEN in London and a little
alarm clock both keep perfect
time. Which minute hand has the
bigger angular velocity ω?
A) Big Ben
B) little alarm clock
C) Both have the same ω
Q82
t∆
∆=
θω
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Relations Between Angular and
Linear Quantities
Linear and angular velocities
t
r
t
xv
∆
∆=
∆
∆=
θ
ωrv =
v is linear speed of point P
Note:
Since ω is the same for all points of a
solid object undergoing rotational motion,
v is proportional to distance (r) of point
from center of rotation
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Relations Between Angular and
Linear Quantities
Linear and angular velocities
t
r
t
xv
∆
∆=
∆
∆=
θ
v is linear speed of point P
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What is the linear
speed of an object at
rest on Earth’s surface
at the equator with
respect to Earth’s
center? (Earth’s radius
is ~ 6400 km.)
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Relating Ch.2 to Rotational Motion
RotationalLinear
∆θ ∆x
t
xv
∆
∆=
t∆
∆=
θω
t
va
∆
∆=
t∆
∆=
ωα
For constant a:
atvv o += to αωω +=
For constant α:
2
21 tto αωθ +=∆
2
21 attvx o +=∆
ωrv =
θ∆=∆ rx
αra =
xavv o ∆+= 222
θαωω ∆+= 222o
Same process as Ch. 2: List what you know in variable form,
then match with a formula
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Example: Thrown for a CurveTo throw a curve ball, a pitcher
gives the ball an initial angularspeed of 36.0 rad/s. When thecatcher gloves the ball 0.595 slater, its angular speed hasdecreased (due to air resistance)to 34.2 rad/s.
(a) What is the ball’s angular acceleration, assuming it to be constant?
(b) How many revolutions does the ball makebefore being caught?
to αωω +=
2
21 tto αωθ +=∆
θαωω ∆+= 222o
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Example: Thrown for a CurveTo throw a curve ball, a pitcher
gives the ball an initial angularspeed of 36.0 rad/s. When thecatcher gloves the ball 0.595 slater, its angular speed hasdecreased (due to air resistance)to 34.2 rad/s.
(a) What is the ball’s angular acceleration, assuming it to be constant?
(b) How many revolutions does the ball makebefore being caught?
20 (34.2 rad/s) (36.0 rad/s)3.03 rad/s
(0.595 s)t
ω ωα
− −= = = −
1 2
0 2t tθ ω α∆ = +
to αωω +=
2
21 tto αωθ +=∆
θαωω ∆+= 222o
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Wheel of MisfortuneAmy is on the Wheel of Fortune and has to spin the wheel. She gives the wheel an initial angular speed of 3.40 rad/s. It then rotates through 1.25 revolutions and comes to rest on BANKRUPT.
(a) Find the wheel’s angular acceleration,assuming it is constant.
(b) How long does it take for the wheel
to stop?
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Wheel of MisfortuneAmy is on the Wheel of Fortune and has to spin the wheel. She gives the wheel an initial angular speed of 3.40 rad/s. It then rotates through 1.25 revolutions and comes to rest on BANKRUPT.
(a) Find the wheel’s angular acceleration,assuming it is constant.
(b) How long does it take for the wheel
to stop?2 2
0 2ω ω α θ= + ∆
2 2 220 0 (3.40 rad/s)
0.736 rad/s2 2(2 rad/rev)(1.25 rev)
ω ωα
θ π
− −= = = −
∆
0 tω ω α= +0
2
0 (3.40 rad/s)4.62 s
( 0.736 rad/s )t
ω ω
α
− −= = =
−
to αωω +=
2
21 tto αωθ +=∆θαωω ∆+= 222
o
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A student sees the following question on an exam:
A flywheel with mass 120 kg, and radius 0.6
m, starting at rest, has an angular acceleration of
0.1 rad/s2. How many revolutions has the wheel
undergone after 10 s?
Which formula should the student use to answer
the question?
to αωω +=
2
21 tto αωθ +=∆
2 2
0 2ω ω α θ= + ∆
A.
B.
C.Q83
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A. 0.40 rad
B. 0.80 rad
C. 1.0 rad
D. 2.0 rad
A DVD is initially at rest so that the
line PQ on the disc’s surface is along
the +x-axis. The disc begins to turn
with a constant α = 5.0 rad/s2.
At t = 0.40 s, what is the angle
between the line PQ and the +x-axis?
Q84t∆
∆=
θω
t∆
∆=
ωα
r
x=θ
to αωω +=
2
21 tto αωθ +=∆
θαωω ∆+= 222o
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A. a faster linear speed and a faster angular speed.
B. the same linear speed and a faster angular speed.
C. a slower linear speed and the same angular speed.
D. the same linear speed and a slower angular speed.
E. none of the above
Compared to a gear tooth on the
rear sprocket (on the left, of
small radius) of a bicycle, a gear
tooth on the front sprocket (on
the right, of large radius) has
Q85
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My kids’ tricycle Predict
I put stickers on the
bottom of the front and
back wheels of
different sizes. As I
roll the bike (without
slipping), the stickers
complete a circle (360
degrees) at:
a) The same time
b) Different times
c) Depends on the
speed of the bikeQ86stickers
Why is this different than the clock example?
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Clicker Answers
79=B, 80=B, 81=B, 82=C, 83=B, 84=A, 85=D, 86=B