Two Dimensional Motion - Santa Rosa Junior Collegelwillia2/40/40ch4p2_s14.pdf · In this case, as...
Transcript of Two Dimensional Motion - Santa Rosa Junior Collegelwillia2/40/40ch4p2_s14.pdf · In this case, as...
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Circular Motion
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Milky Way Galaxy
Orbital Speed of Solar System: 220 km/s Orbital Period: 225 Million Years
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Mercury: 48 km/s
Venus: 35 km/s
Earth: 30 km/s
Mars: 24 km/s
Jupiter: 13 km/s
Neptune: 5 km/s
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Equatorial Rotational Period: 27.5 days
Differential Solar Rotation Entangled Magentic Field Lines makes Sun Spots
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464m/s
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Precession causes the position of the North Pole to change over a period of 26,000 years.
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Orbital Speed of Earth: ~ 30 km/s
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Although the Moon is always lit from the Sun, we see
different amounts of the lit portion from
Earth depending on
where the Moon is
located in its month-long
orbit. Orbital Speed of Moon: ~ 1 km/s
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155mph~70m/s
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RADAR: RAdio Detecting And Ranging
200mph~90m/s
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Electrons in Bohr Orbit: 2 million m/s
(Speed of Light: 200 million m/s)
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Maximal Kerr Black Hole
Rotational Speed = Speed of Light
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Translational vs Rotational
//
mx
v dx dta dv dt
∆==
//
I
d dtd dt
θω θα ω
∆==
2 /c
t
s rv ra v ra r
θω
α
∆ = ∆=
==
Connection
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Translational and Rotational Kinematics
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© 2013 Pearson Education, Inc.
When angular velocity ω is constant, this is uniform circular motion.
In this case, as the particle goes around a circle one time, its angular displacement is ∆θ = 2π during one period ∆t = T.
Angular Velocity in Uniform Circular Motion
Slide 4-87
: angular velocity (rad/s)ω
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Angular Position: Rotation Angle
∆θ = rotation angle
∆s = arc length
r = radius
sr
θ ∆∆ =
The rotation angle is the ratio of arc length to radius of curvature. For a given angle, the greater the radius, the greater the arc length.
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r
s = ? s = C = 2πr
θ = s/r θ = 2 πr /r
θ = 2 π
How many degrees in 1 radian?
How many radians in 3600 ? Ans: 3600 = 2 π radians
Consider a circle with radius r.
2 π rad = 3600
1 rad = 3600 /2 π = 57.30
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d
s θ
r = 4 m
A box is fastened to a string that is wrapped around a pulley. The pulley turns through an angle of 430. What is the distance, d, that the box moves?
43 43 1x=
243360
x π=
.75rad=
First: How many radians is 430 ?
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d
s θ
r = 4 m
A box is fastened to a string that is wrapped around a pulley. The pulley turns through an angle of .75 rad. What is the distance, d, that the box moves?
s = r θ d = r θ d = 4m (0.75 rad) d = 3 m rad d = 3 m
430 = .75rad
Radian is dimensionless and is dropped!
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Angular Velocity : angular velocity (rad/s)ω
θω =ddt
( / )=
d s rdt
1=
dsr dt
t∆
vr
=
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: angular velocity (rad/s)v: tangential velocity (m/s)ω
θω = =d vdt r
v rω=
Angular & Tangential Velocity
is the same for every point & v varies with r. ω
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Problem Calculate the angular velocity of a 0.500 m radius car tire when the car travels at a constant speed of 25.0 m/s.
ω =vr
25 /.5m sm
=
50 /rad s=
Insert rad for angular speed. Keep it to define units.
v rω=
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Tangential and Angular Acceleration changes speed
ωα =ddt
=tdvadt
( )ω=
d rdt
α=ta r
=drdtω
v rω=
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Angular –Tangential Bike A bike wheel with a radius of 0.25 m undergoes a constant
angular acceleration of 2.50 rad/s2. The initial angular speed of the wheel is 5.00 rad/s. After 4.00 s
a) What angle has the wheel turned through? b) What is the final angular speed? c) What is the final tangential speed of the bike? d) How far did the bike travel?
2
012
t tθ ω α∆ = + 2 215 / 4 2.5 / (4 ) 402
rad s s rad s s rad= + =
0f tω ω α= + 25 / 2.5 / 4 15 /rad s rad s s rad s= + =
15 / .25 3.75 /v r rad s m m sω= = =
40 .25 10d s r rad m mθ= ∆ = ∆ = =
Givens:
02
5 /
2.5 /4
rad srad s
t s
ω
α
=
==
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© 2013 Pearson Education, Inc.
Centripetal Acceleration
Slid 4 94
2
( / )dv vd vadt rd v r
θθ
= = =
( )ds d rvdt dt
θ= =
v rω= s rθ=
( )d rdtvθ
=
Combining:
Centripetal Acceleration Only changes Direction!!!
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Centripetal and Tangential Accelerations
2
=cvar
α=ta r2( )ω
=rr
2ω=ca r
(r constant)
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Total Acceleration
2 2r ta a a= +
ˆˆ θθ= − +
ra a r a
/θ α= = =ta a r dv dt2 2 /ω= = =r ca a r v r
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The Earth rotates once per day around its axis as shown. Assuming the Earth is a sphere, is the rotational speed at Santa
Rosa greater or less than the speed at the equator?
366 m/s
464 m/s
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Is the centripetal acceleration greater at the Equator or at Santa
Rosa?
2
cvar
= 20.034 /m s
20.027 /m s
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The Earth rotates once per day around its axis. Assuming the Earth is a sphere with radius 6.38 x 106m, find the tangential speed of a person at the equator and at 38 degrees latitude (Santa Rosa!) and their centripetal accelerations.
At the equator, r = 6.38 x 106m: 62 2 (6.38 10 ) 464 /
86,400r x mv m st sπ π
= = =∆
( )222
6
464 /.034 /
6.38 10= = =c
m sva m sr x m
At Santa Rosa, r = 6.38 x 106m cos38: 62 2 (6.38 10 )cos38 366 /
86,400r x mv m st sπ π
= = =∆
( )222
6
366 /.027 /
cos38 6.38 10 cos38= = =c
m sva m sr x m
ca
ca
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What is the total acceleration acting on a person in Santa Rosa?
cag
The vector sum.
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Is your apparent weight as measured on a spring scale more at the Equator or at Santa Rosa?
cag
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Since you are standing on the Earth (and not in the can) the centrifugal force tends
to throw you off the Earth. You weigh less where the centripetal force is greatest because
that is also where the centrifugal force is greatest – the force that tends to throw you
out of a rotating reference frame.
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Centrifugal Force & Acceleration “Center FLEEing”
Factual = Centripetal Force Ffictitious = Centrifugal Force
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Centrifugal Force & Acceleration
Suppose there is a lady bug in the can. There is a centripetal force acting on the bug, transmitted to her feet by the can. Her feet push back on the can producing a centrifugal “fictitious” force, that acts
like artificial gravity.
“Center FLEEing”
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Centripetal & Centrifugal Force Depends on Your Reference Frame
Inside Observer (rotating reference frame) feels Centrifugal Force pushing them against the can.
Outside Observer (non-rotating frame) sees Centripetal Force pulling can in a circle.
Center-seeking
Center-fleeing
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Centrifugal Force is Fictitious? The centrifugal force is a real effect. Objects in a rotating frame feel a centrifugal force acting on them, trying to push them out. This is due to your inertia – the fact that your mass does not want to go in a circle. The centrifugal force is called ‘fictitious’ because it isn’t due to any real force – it is only due to the fact that you are rotating. The centripetal force is ‘real’ because it is due to something acting on you like a string or a car.
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In Physics, we use ONLY
CentriPETAL acceleration NOT
CentriFUGAL acceleration!
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Artificial Gravity How fast would the space station segments A and B have to rotate in
order to produce an artificial gravity of 1 g?
56 / ~ 115Av m s mph=
104 / ~ 210Bv m s mph=
Can the two segments be connected?
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Space Station Rotation
A space station of diameter 80 m is turning about its axis at a constant rate. If the acceleration of the outer rim of the station is 2.5 m/s2, what is the period of revolution of the space station? a. 22 s b. 19 s c. 25 s d. 28 s e. 40 s
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td
adt
=v
2
r Cva ar
= − = −
2 2r ta a a= +
Nonuniform Circular Motion
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Circular Motion Total Acceleration • The tangential
acceleration causes the change in the speed of the particle
• The radial acceleration
comes from a change in the direction of the velocity vector
td
adt
=v
2
r Cva ar
= − = −2 2r ta a a= +
Total Acceleration:
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Total Acceleration The total acceleration of a particle moving clockwise in a circle of radius 2.50 m at a certain of time is 15.0m/s2. At this instant, find (a) the radial acceleration, (b) the speed of the particle, and (c) its tangential acceleration.
( ) ( )o 2 2cos30.0 15.0 m s cos30 13.0 m sca a= = ° =
2
cvar
= ( )2 2 2 22.50 m 13.0 m s 32.5 m scv ra= = =
32.5 m s 5.70 m sv = =2 2 2t ra a a= +
( ) ( )2 22 2 2 2 215.0 m s 13.0m s 7.50 m s= − = − =t ra a a
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“Coriolis Force”
• This is an apparent force caused by changing the radial position of an object in a rotating coordinate system
• The result of the rotation is the curved path of the ball
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“Coriolis Force”
• This is an apparent force caused by changing the radial position of an object in a rotating coordinate system
• The result of the rotation is the curved path of the ball
![Page 46: Two Dimensional Motion - Santa Rosa Junior Collegelwillia2/40/40ch4p2_s14.pdf · In this case, as the particle goes ... θ = 2 πr /r . θ = 2 ... The fan blade is slowing down. What](https://reader034.fdocument.pub/reader034/viewer/2022051600/5aa3fb567f8b9a7c1a8b84b2/html5/thumbnails/46.jpg)
Coriolis Effect
![Page 47: Two Dimensional Motion - Santa Rosa Junior Collegelwillia2/40/40ch4p2_s14.pdf · In this case, as the particle goes ... θ = 2 πr /r . θ = 2 ... The fan blade is slowing down. What](https://reader034.fdocument.pub/reader034/viewer/2022051600/5aa3fb567f8b9a7c1a8b84b2/html5/thumbnails/47.jpg)
![Page 48: Two Dimensional Motion - Santa Rosa Junior Collegelwillia2/40/40ch4p2_s14.pdf · In this case, as the particle goes ... θ = 2 πr /r . θ = 2 ... The fan blade is slowing down. What](https://reader034.fdocument.pub/reader034/viewer/2022051600/5aa3fb567f8b9a7c1a8b84b2/html5/thumbnails/48.jpg)
© 2013 Pearson Education, Inc.
A ball rolls around a circular track with an angular velocity of 4π rad/s. What is the period of the motion?
A. s
B. 1 s
C. 2 s
D. s
E. s
QuickCheck 4.11
Slide 4-88
12
1
2π
1
4π
![Page 49: Two Dimensional Motion - Santa Rosa Junior Collegelwillia2/40/40ch4p2_s14.pdf · In this case, as the particle goes ... θ = 2 πr /r . θ = 2 ... The fan blade is slowing down. What](https://reader034.fdocument.pub/reader034/viewer/2022051600/5aa3fb567f8b9a7c1a8b84b2/html5/thumbnails/49.jpg)
© 2013 Pearson Education, Inc.
A ball rolls around a circular track with an angular velocity of 4π rad/s. What is the period of the motion?
A. s
B. 1 s
C. 2 s
D. s
E. s
QuickCheck 4.11
Slide 4-89
12
1
2π
1
4π
T =
2πω
![Page 50: Two Dimensional Motion - Santa Rosa Junior Collegelwillia2/40/40ch4p2_s14.pdf · In this case, as the particle goes ... θ = 2 πr /r . θ = 2 ... The fan blade is slowing down. What](https://reader034.fdocument.pub/reader034/viewer/2022051600/5aa3fb567f8b9a7c1a8b84b2/html5/thumbnails/50.jpg)
© 2013 Pearson Education, Inc.
The fan blade is slowing down. What are the signs of ω and α? A. ω is positive and α is positive.
B. ω is positive and α is negative.
C. ω is negative and α is positive.
D. ω is negative and α is negative.
E. ω is positive and α is zero.
QuickCheck 4.15
Slide 4-105
![Page 51: Two Dimensional Motion - Santa Rosa Junior Collegelwillia2/40/40ch4p2_s14.pdf · In this case, as the particle goes ... θ = 2 πr /r . θ = 2 ... The fan blade is slowing down. What](https://reader034.fdocument.pub/reader034/viewer/2022051600/5aa3fb567f8b9a7c1a8b84b2/html5/thumbnails/51.jpg)
© 2013 Pearson Education, Inc.
The fan blade is slowing down. What are the signs of ω and α? A. ω is positive and α is positive.
B. ω is positive and α is negative.
C. ω is negative and α is positive.
D. ω is negative and α is negative.
E. ω is positive and α is zero.
QuickCheck 4.15
Slide 4-106
“Slowing down” means that ω and α have opposite signs, not that α is negative
![Page 52: Two Dimensional Motion - Santa Rosa Junior Collegelwillia2/40/40ch4p2_s14.pdf · In this case, as the particle goes ... θ = 2 πr /r . θ = 2 ... The fan blade is slowing down. What](https://reader034.fdocument.pub/reader034/viewer/2022051600/5aa3fb567f8b9a7c1a8b84b2/html5/thumbnails/52.jpg)
© 2013 Pearson Education, Inc.
Rasheed and Sofia are riding a merry-go-round that is spinning steadily. Sofia is twice as far from the axis as is Rasheed. Sofia’s angular velocity is ______ that of Rasheed.
A. half
B. the same as
C. twice
D. four times
E. We can’t say without knowing their radii.
QuickCheck 4.12
Slide 4-95
![Page 53: Two Dimensional Motion - Santa Rosa Junior Collegelwillia2/40/40ch4p2_s14.pdf · In this case, as the particle goes ... θ = 2 πr /r . θ = 2 ... The fan blade is slowing down. What](https://reader034.fdocument.pub/reader034/viewer/2022051600/5aa3fb567f8b9a7c1a8b84b2/html5/thumbnails/53.jpg)
© 2013 Pearson Education, Inc.
Rasheed and Sofia are riding a merry-go-round that is spinning steadily. Sofia is twice as far from the axis as is Rasheed. Sofia’s angular velocity is ______ that of Rasheed.
A. half
B. the same as
C. twice
D. four times
E. We can’t say without knowing their radii.
QuickCheck 4.12
Slide 4-96
![Page 54: Two Dimensional Motion - Santa Rosa Junior Collegelwillia2/40/40ch4p2_s14.pdf · In this case, as the particle goes ... θ = 2 πr /r . θ = 2 ... The fan blade is slowing down. What](https://reader034.fdocument.pub/reader034/viewer/2022051600/5aa3fb567f8b9a7c1a8b84b2/html5/thumbnails/54.jpg)
© 2013 Pearson Education, Inc.
Rasheed and Sofia are riding a merry-go-round that is spinning steadily. Sofia is twice as far from the axis as is Rasheed. Sofia’s speed is ______ that of Rasheed.
A. half
B. the same as
C. twice
D. four times
E. We can’t say without knowing their radii.
QuickCheck 4.13
Slide 4-97
![Page 55: Two Dimensional Motion - Santa Rosa Junior Collegelwillia2/40/40ch4p2_s14.pdf · In this case, as the particle goes ... θ = 2 πr /r . θ = 2 ... The fan blade is slowing down. What](https://reader034.fdocument.pub/reader034/viewer/2022051600/5aa3fb567f8b9a7c1a8b84b2/html5/thumbnails/55.jpg)
© 2013 Pearson Education, Inc.
Rasheed and Sofia are riding a merry-go-round that is spinning steadily. Sofia is twice as far from the axis as is Rasheed. Sofia’s speed is ______ that of Rasheed.
A. half
B. the same as
C. twice
D. four times
E. We can’t say without knowing their radii.
QuickCheck 4.13
Slide 4-98
v = ωr
![Page 56: Two Dimensional Motion - Santa Rosa Junior Collegelwillia2/40/40ch4p2_s14.pdf · In this case, as the particle goes ... θ = 2 πr /r . θ = 2 ... The fan blade is slowing down. What](https://reader034.fdocument.pub/reader034/viewer/2022051600/5aa3fb567f8b9a7c1a8b84b2/html5/thumbnails/56.jpg)
© 2013 Pearson Education, Inc.
Rasheed and Sofia are riding a merry-go-round that is spinning steadily. Sofia is twice as far from the axis as is Rasheed. Sofia’s acceleration is ______ that of Rasheed.
A. half
B. the same as
C. twice
D. four times
E. We can’t say without knowing their radii.
QuickCheck 4.14
Slide 4-99
![Page 57: Two Dimensional Motion - Santa Rosa Junior Collegelwillia2/40/40ch4p2_s14.pdf · In this case, as the particle goes ... θ = 2 πr /r . θ = 2 ... The fan blade is slowing down. What](https://reader034.fdocument.pub/reader034/viewer/2022051600/5aa3fb567f8b9a7c1a8b84b2/html5/thumbnails/57.jpg)
© 2013 Pearson Education, Inc.
Rasheed and Sofia are riding a merry-go-round that is spinning steadily. Sofia is twice as far from the axis as is Rasheed. Sofia’s acceleration is ______ that of Rasheed.
A. half
B. the same as
C. twice
D. four times
E. We can’t say without knowing their radii.
QuickCheck 4.14
Slide 4-100
Centripetal acceleration a = v2
r= ω 2r