Download 05_Circular_orbits_2012Sep27

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Physics: Mechanics
Dr. Bill Pezzaglia
Circular Motion
& Gravity
Updated: 2012Sep27
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Circular Motion & Gravity
A. Angular Kinematics
B. Central Force and Orbits
C. Ficticious Forces
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A. Angular Kinematics
1. Angles
2. Angular Velocity
3. Angular Acceleration
1. Angles
Perhaps 5000 BC people changed from a nomadic
culture to agrarian, settling in Sumer. Sumerians
needed a calendar to tell them when to plant food.
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Surviving Babylonian “Cuneiform” Clay Tablets of
astronomical positions of sun & planets
x
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The Ecliptic Circle
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The Babylonians determined the exact path
of the sun through the zodiac constellations
Hammurabi [1810-1750 BC]
•
school of scribes defines
sexagessimal numbers
(base 60).
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(a) Angles in sexagesimal
• Degrees: 360=circle = 24 hours
• Arcminutes (minutes of arc): 60’=1
• Arcseconds (seconds of arc): 60”=1’
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(b) Angle in Radians
• Circle: 360=2 radians
• Or
1 radian =57.3
• Arc length formula is easy
in radians: s=R
• Its messier in degrees:

sR

57.3
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(c) Angular Displacement
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• Analogous to linear displacement
• Definition: how far its rotated: =f- I
• There is a “wrap around” ambiguity, e.g. if you
look at the clock and it says 3, and later it says
5, is it 2 hours later or 14 hours later?
• Technically angular displacement is a pseudovector
(not a “vector”), it represents a plane, not a line.
2. Angular Velocity
(a) Definition: (analogous
to average velocity)
• Expressed in terms of
period “T” for one
complete revolution:
• For constant angular
velocity we have simple
equation


t
2

T
  0  t
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(b) Units of Angular Velocity
• SI units: radians per second


t
Other systems used:
• Degrees/second
• Rpm: revolutions per minute
(example, convert 1 rpm to rads/sec):
1
rev
min
 1min  2 rads 
rads




0
.
105

sec

 60 sec  1rev 
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(c) Tangential Velocity
• The (tangential) speed at a spot
distance “r” from axis can be
calculated as the total distance
traveled (circumference of circle)
in one period:
2 r
vt 
T
vt  r
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3. Angular Acceleration
(a) Definition


t
• SI Units: rads/sec2
• Kinematic equations for
rotation under constant
angular acceleration are
completely analogous to
those for linear acceleration:
  t
  12  t 2
  2
2
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B. Central Force & Orbits
1. Centripetal Force
2. Orbits
3. Newton’s Law of Gravity
1. Centripetal Force
(a) Newton (1684) introduces concept of the force
towards the center needed to make an object go in a
circular path.
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v
2
F  m  mR
R
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(b) Turning the Curve
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•What is the maximum speed a car can go around a curve
of radius “R” given friction coefficient . Does the result
depend upon the mass of the car (should truck go slower
than a car)?
v  Rg
(c) Banked Curves
•What banking angle will
make a car turn a curve
even if no friction?
v  Rg tan 
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2. Orbitals
(a) Galileo deduces that if the cause of the centripetal
acceleration is gravity (centripetal force) then we can
calculate the orbital speed
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v
ac   g
R
v  gR
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Tycho (1546-1601)
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•Tycho Brahe
measuring star positions
(without a telescope)
•Measurements of
position of Mars
showed deviations from
circular orbits!
•Asks Kepler to come
and interpret data.
b. Johannes Kepler (1571-1630)
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•Tycho at first invited Kepler
to help in analysis of his data,
but then jealously wouldn’t
let him have the information.
•On his deathbed he gave
Kepler the data.
•Kepler used it (particular
data on Mars), to develop
three laws of planetary
motion.
2c.1 Kepler’s 1st Law (1605)
• Law No. 1. Each planet moves around the Sun
in an orbit that is an ellipse, with the Sun at
one focus.
– This is contrary to the earlier belief that the orbits
were perfect circles or combinations of circles.
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• Ellipses, circles
(parabolas and
hyperbolas) are “conic
sections”, studied first
by the greeks.
• But it would NEVER
occur to the greeks that
an orbit is an ellipse.
(why?)
Fig 2-3, p.45
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Drawing an ellipse
Focus
Focus
Fig 2-4, p.45
The Ellipse
Do you remember any of this from high
school geometry?
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Highly eccentric
Focus
Focus
Not very eccentric
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Planet orbits tend to have
low eccentricity (nearly
circular).
Comet orbits tend to
be highly eccentric.
Fig 2-10, p.53
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2c.2 Kepler’s
2nd
Law (1609)
Kepler also noticed that when
Mars is closest to the Sun in
its elliptical orbit, it moves
faster than when it is farther
away.
This led him to formulate his
Second Law of Planetary Motion.
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2c.2 Kepler’s 2nd Law (Equal areas in Equal Times)
According to his second law, a planet moves fastest when closest to
the Sun (at perihelion) and slowest when farthest from the Sun (at
aphelion). As the planet moves, an imaginary line joining the planet
and the Sun sweeps out equal amounts of area (shown as colored
wedges in the animation) in equal intervals of time.
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2c.2 Kepler’s 2nd Law
Kepler’s 2nd law is actually a form of conservation of angular
momentum
A 1
L
1 mrv
 2 rv  2

t
m
2m
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2c.3 Kepler’s 3rd Law: “Harmonic Law”
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Planets closer to
the sun move
faster.
This is consistent
with his 2nd law,
that showed a
planet will move
faster at perihelion.
He searched for a
relationship
between orbital
period and
distance to the
sun.
2c.3 Kepler’s
3rd
Law (1618)
• The square of the orbital period (P) is directly
proportional to the cube of the semimajor axis
of the orbit (a).
P2 = a3
This law explains the proportions of the sizes of the
orbits of the planets and the time that it takes them to
make one complete circuit around the Sun.
[Note: in physics, the symbol “a” is also used to represent
“acceleration”. Confused?]
Why is it called the “harmonic law”? Kepler thought the spacing
between planets was related to musical intervals.
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An example of Kepler’s third law: The orbit of Mars
(Recall: P2 = a3)
Mars’ orbit period (P) is 1.88 years. P2 = 3.53
Kepler’s law says that P2 = a3, so 3.53 = a3.
So then a = (3.53)1/3 (the cube root of 3.53), or 1.52.
Thus, the semimajor axis (average distance of Mars from
the Sun) is 1.52 Astronomical Units.
But how big is an Astronomical Unit?
Kepler didn’t know.
2c Inverse Square Law
Edmond Halley (1656-1742) deduced that Kepler’s
3rd law implies that gravity (of sun) must decrease
with square of distance.
2R
v
P
2
v
2 R
ac   4 2
R
P
2
3
P R
1
a 2
R
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3. Newton’s
4th
Law: Gravity
• In 1679 Hooke writes Newton, proposing inverse square law.
• In 1684 Wren, Hooke and Halley discussed, at the Royal Society,
whether the elliptical shape of planetary orbits was a consequence
of an inverse square law of force depending on the distance from
the Sun.
• Later in the same year in August, Halley visited Newton in
Cambridge and asked him what orbit a body would follow under an
inverse square law of force
• Sr Isaac replied immediately that it would be an Ellipsis, the Doctor
struck with joy and amasement asked him how he knew it, why, said
he I have calculated it, whereupon Dr Halley asked him for his
calculation without any farther delay, Sr Isaac looked among his
papers but could not find it, but he promised him to renew it, and
then to send it him.
• Actually, he had apparently derived it in 1680, after correspondence
from Hooke. Later Hooke claims Newton stole the idea and did not
give him credit!
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(a) The Law of Gravity
• Newton’s 4th law: The mutual force between two
bodies is proportional to their masses, and inversely
proportional to square of distance.
• Gravitation Constant “G” measured 100 years later
by Cavendish.
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b. Newton’s Orbit
• Combining Newton’s 2nd and
4th laws, we see that the mass
of the test body cancels out!
• Orbital speed depends only on
size of orbit (and mass of
planet)
GmM
ma  F 
2
R
2
v
GM
a 2
R
R
GM
v
R
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c. Newton-Kepler Law
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•Newton derived Kepler’s 3rd law from scratch. More
important, he showed that the mass of the central body
can be determined from the orbital period and radius of a
moon:
2
v
mM
m G 2
R
R
2R
v
P
2
2
3
v R 4 R
M

2
G
G P
The Newton-Kepler Law
• In the Principia Newton also deduced Kepler's
third law, but in an important new form
• Mass of central body: M = a3/P2
– Orbital Radius “a” (in astronomical units)
– Period “P” (in years)
– Mass “M” in units of “solar masses”
• To measure mass of
–
–
–
–
Earth, use moon’s orbit
Jupiter, use Galilean moons
Sun, use orbits of planets
Galaxy, use orbits of stars around galaxy
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C. Fictitious Forces
Aka d’Alembert or “pseudo” force
1. Accelerated Frames
2. Centrifugal Force
3. Applications
4. (Coriolis & Euler force?)
1. Accelerated Frames
(a) The elevator
•
•
You feel heavier in an elevator accelerating upwards
You feel lighter in an elevator accelerating downward
Add graphic here
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1b. The Einstein Equivalence Principle
Reference at rest with Gravity is indistinguishable to a reference
frame which is accelerating upward in gravity free environment.
The apple accelerating downward due to gravity looks the same as an
apple at rest in space, with the floor accelerating upward towards it.
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1c. Sideways acceleration
•In an accelerated reference frame (e.g. accelerating
truck) you “feel” forces that mimic gravity.
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2. Centrifugal Force
•Huygen (1659?): If you are in a
rotating frame of reference, you
will “feel” a force outward
•At equator of earth it will make
you “feel” less heavy by 0.5%
2
v
2
F  m  mR
R
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3. Applications
•Textbook Knight (page 179) makes the rather “poor”
statement: “A centrifugal force will never appear on a freebody diagram and never be included in Newton’s Laws”.
•I strongly disagree with this overly conservative statement.
It implies that “physics” should only be done in inertial
frames. This is silly. You live on a rotating earth which is
revolving around the sun, so not a single experiment ever
done has been in an inertial frame.
•There are many problems that are more easily solved by
allowing yourself to “sit” on the object that is accelerating.
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Example: loop the loop
•How fast must roller coaster go
so it won’t fall off the track?
2
v
m  mg
R
v  gR
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