Announcements

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Lecture 19

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Announcements

• HW 10 due on TODAY at the end of lecture
• HW 11 due on Wednesday, Nov 28.

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(No Transcript)
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• A 10 kg diving board has a 2 kg box near the
  right end of the board. The board is supported by
  two supports. The support on the right is
  movable. If that support is moved a little to the
  left what will happen to the two support forces?
• Nothing
• Both support forces increase
• Both support forces decrease
• The force from the left support increases and the
  force from the right support decreases
• The force from the left support decreases and the
  force from the right support increases

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Example Wine Butler

• Torques must balance
• Take origin to be support point

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Slipping Ladder

• Ladder leans against wall
• Assume no friction between ladder and wall
• Must have friction between ladder and ground!
• Will ladder slip?
• Take origin to be point of contact with ground
• For ladder to not slip

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Equilibrium Example - Bookshelf

• Bookshelf books has a mass M
• Supported by the wall, block, and rod
• What are the forces?
• Take origin to be point where bookshelf meets wall

T
N
q
Fs
Mg
L
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Extending Boards

• Can extend stack of boards beyond edge of table
• Farthest we can extend a board is when its COM is
  at the edge of the board underneath
• COM of 1st board is L/2 beyond 2nd board
• COM of boards 12 relative to edge of board 2 is
• 123 relative to edge of board 3 is

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a stack of N blocks that (just barely) balances
on the table yet will overhang the edge of the
table by
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Summary of Equilibrium

• Static equilibrium occurs for a rigid body at
  rest when net force and torque is zero
• For uniform gravitational force, effect of
  gravitational force on an object is equivalent to
  putting entire gravitational force acting acting
  on the center of mass position
• We will skip topic of elasticity
• Objects are not perfectly rigid when a force acts
  on them
• Detailed analysis of statics requires taking into
  account these deformations

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Gravitation
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Gravitational Force

• Gravity is by far the most common force that we
  deal with in our lives
• Its the one force that we always feel acting on
  us (unless we are in free-fall!)
• We dont really understand gravity
• No established quantum field theory such as we
  have for the other forces
• Our best mathematical description of gravity
  Einsteins General Theory of Relativity is a
  classical theory with all sorts of oddities
• Expansion of the universe
• Warping of space-time
• Black holes
• but we know enough to get by in Physics 5
• For classical mechanics, sufficient to describe
  force due to gravity
• F mg downward cant be the whole story this
  is a very earth-centric formulation!!
• Newton was the first earthling to propose a
  reasonably accurate theory of gravity

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Planetary Motion

• Many different civilizations show evidence of
  having studied the motions of the stars and
  planets
• Motion of the stars was observed to be relatively
  straight-forward
• Relative positions were fixed
• Movement could be ascribed to the rotation of the
  Heavenly Sphere
• Motion of the planets was long a mystery
• Position with respect to stars was constantly
  changing
• No clear pattern to the motion
• We even see retrograde motion, where planet
  reverses course for a while
• Early ideas didnt pan out
• Spheres rotating within spheres, etc.
• Understanding planetary motion was a major
  breakthrough in the early days of physics

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Tycho Brahe (1546-1601)

• devised the most precise astronomical instruments
  available before the invention of the telescope
• compiled extensive data on the planet Mars
• made his observations from Uraniborg, on the
  island Hveen in the sound between Denmark and
  Sweden

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Johannes Kepler (1571-1630)

• assistant and successor to Tycho Brahe
• sun emits a force that diminishes inversely with
  distance and pushes the planets around in their
  orbits
• Cosmographic Mystery, 1596
• Brahes data of Mars ? elliptical orbit
• 1st and 2nd law of planetary motion
• New Astronomy, 1609
• 3rd law of planetary motion
• Harmony of the World, 1619

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Retrograde motion of Mars

• One phenomenon that ancient astronomers had
  difficulty explaining was the retrograde motion
  of the planets.
• Over the course of a single night, a planet will
  move from East to West across the sky, like any
  other celestial object near the ecliptic.
• If observed from one night to the next, however,
  a planet appears to move from West to East
  against the background stars most of the time.
  Occasionally, however, the planet's motion will
  appear to reverse direction, and the planet will,
  for a short time, move from East to West against
  the background constellations.
• This reversal is known as retrograde motion.
• The retrograde motion of Mars occurs when the
  Earth passes by the slower moving Mars.
• The two planets are like race cars on an oval
  track. Earth has the inside lane and moves faster
  than Mars -- so much faster, in fact, that it
  makes two laps around the course in about as much
  time as it takes Mars to go around once.

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• About every 26 months, Earth comes up from behind
  and overtakes Mars. While we're passing by the
  red planet, it will look to us as though Mars is
  moving up and down. Then, as we move farther
  along our curved orbit and see the planet from a
  different angle, the illusion will disappear and
  we will once again see Mars move in a straight
  line.
• Just to make things a little more odd, the orbits
  that Earth and Mars follow don't quite lie in the
  same plane. It's as if the two planets were on
  separate tracks that are a little tilted with
  respect to each other. (loop vs open zigzag)

2005
2003
http//www.astro.uiuc.edu/projects/data/Retrograde
/ http//mars.jpl.nasa.gov/allabout/nightsky/night
sky04.html
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Keplers Laws

• Kepler published his three Laws that described
  the motion of the planets in 1610
• Laws were developed empirically from Tycho
  Brahes careful measurements
• Keplers laws matched the data, but no deeper
  understanding of their origin was possible until
  Newton developed his theory of gravitation and
  his laws of mechanics
• This is often the way science works!
• Acquire data probing an important problem
• Empirical formulations are found that describe
  the data
• New insights lead to a deeper understanding of
  the underlying science

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Keplers First Law

• All planets move in elliptical orbits
• a is half the length of the major axis
• e is the eccentricity of the ellipse
• e 0 Circle
• e lt 1 Ellipse
• e 1 Parabola
• e gt 1 Hyperbola
• An ellipse has two foci separated by a distance
  2ea
• Sum of distances from foci to any point on the
  ellipse is constant
• Example drawing an ellipse using chalk and
  string
• Center of mass of Sun-planet system lies at one
  of the ellipses foci
• The above solution can be derived from Newtons
  laws only for forces that vary as r-2

r
q
Major axis (length 2a)
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Keplers Second Law

• A planets orbit sweeps out equal areas in equal
  time
• In a time dt, angle theta changes by dq w dt
• Distance traveled in this time is ds r dq
  rw dt
• Area swept out is dA ½ r ds ½ r2w dt
• 2nd Law is equivalent to angular momentum
  conservation

ds
r
q
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Keplers Third Law

• Law of Periods
• With some effort, the Law of Periods can be
  derived for the general case of elliptical orbits
• We will simply check that it works for circular
  orbits
• The semi-major axis a in this case is just the
  radius r of the circle

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Isaac Newton (1642-1727)

• one of the most important scientists of all time
• Principia Mathematica 1687
• laws of motion
• law of universal gravitation

G 6.67 ? 10-11 Nm2/kg2
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Gravitational Force

• Essential elements
• All objects with mass have an attractive force
  between themselves and all other massive objects
• The magnitude of the gravitational force varies
  as 1/r2, where r is the separation between two
  masses
• The magnitude of the gravitational force also
  varies as the product of the masses
• The direction of the force is along the line
  connecting the two masses
• G is a constant that specifies the strength of
  the gravitational force

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Falling Apples

• Newtons basic idea was that the force causing an
  apple to fall to earth was the same force that
  caused the moon, planets, and stars to follow
  their orbits through space
• Just as the earth pulls the apple towards the
  earth, the apple pulls the earth towards the
  apple
• In principle, to calculate the force between the
  apple and the earth, we would need to sum up the
  force vectors between the apple and each part of
  the entire earth
• Nifty Theorem A uniform spherical shell of
  matter of mass M has the same gravitational
  attraction to an object outside the shell as a
  point mass M at the center of the shell

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Superposition

• If an object experiences multiple forces, we can
  use
• The principle of superposition - the net force
  acting on an object is the vector sum of the
  individual forces acting on that object.

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Rank these situations
A small blue ball has one or more large red balls
placed near it. The red balls are all the same
mass and the same distance from the blue one.
Rank the different cases based on the net
gravitational force experienced by the blue ball
due to the neighboring red ball(s). 1.
13gt2gt4 2. 3gt2gt1gt4 3. 4gt3gt2gt1 4. 2gt13gt4
5. 4gt123
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Ranking the situations

• Does case 4 give the largest net force or the
  smallest?
• Is there another case with the same magnitude net
  force as case 1?
• Remember this kind of situation when we look at
  charged objects next semester. There are many
  similarities between gravitational interactions
  and interactions between charged objects.

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Ranking the situations

• Does case 4 give the largest net force or the
  smallest?
• The smallest there is no net force in case 4.
• Is there another case with the same magnitude net
  force as case 1? Yes, case 3.
• Remember this kind of situation when we look at
  charged objects next semester. There are many
  similarities between gravitational interactions
  and interactions between charged objects.

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Cavendish Experiment

• The universal nature of the gravitational force
  isnt obvious
• The attractive force is generally small unless
  one mass is extremely large
• Example consider two 1 kg objects with their
  centers 1 cm apart
• It was Cavendish who verified Newtons prediction
  that all objects felt an attractive gravitational
  force
• Experiment two masses on a torsion spring
• When external masses are brought close by, the
  gravitational force produces a torque on the
  torsion spring and causing the spring to rotate
  slightly

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Origin of little g

• We have seen that near the surface of the earth,
  objects in free-fall are accelerated at a rate of
  g 9.8 m/s2
• We now have the tools to actually calculate what
  we expect g to be
• Consider the force between the earth and an
  object of mass m sitting on the earths surface

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Measuring g to Probe the Earth

• Geologists often find it useful to measure the
  gravitational force in different areas
• Gravitational force is sensitive to composition
  of the earth near the surface
• High density ? larger g
• Need to also account for more mundane effects
• Earth isnt perfectly spherical ? radius changes
• Earth is rotating ? need to account for
  centripetal forces
• Example measuring g at the equator

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• The differences of Earth's gravity around the
  Antarctic continent
• This gravity field was computed from sea-surface
  height measurements collected by the US Navy
  GEOSAT altimeter between March, 1985, and
  January, 1990. The high density GEOSAT Geodetic
  Mission data that lie south of 30 deg. S were
  declassified by the Navy in May of 1992 and
  contribute most of the fine-scale gravity
  information.

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• At scales between entire mountain ranges and ore
  bodies, Bouguer gravity anomalies may indicate
  rock types.
• For example, the northeast-southwest trending
  high across central New Jersey represents a
  graben of Triassic age largely filled with dense
  basalts. Salt domes are typically expressed in
  gravity maps as lows, because salt has a low
  density compared to the rocks the dome intrudes.
  (from USGS)

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At the center of the Earth

• If we bring an object from far away toward the
  Earth, the gravitational force increases. The
  closer it gets, the bigger the force. This is
  certainly true when the mass is outside the Earth
  - what happens if we bring it right to the
  surface and then keep going, tunneling into the
  Earth?
• What is the force of gravity on an object if is
  right at the center of the Earth?
• zero
• infinite

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Inside the Earth

• The net gravitational force on an object at the
  center of the Earth is zero forces from
  opposite sides of the Earth cancel out.
• This is a consequence of Gauss' Law for Gravity.
  One implication of Gauss Law is that inside a
  uniform spherical shell, the force of gravity due
  to the shell is zero. Outside the shell, the
  force is exactly the same as that from a point
  object of the same mass as the shell, placed at
  the center of the shell.
• Newtons Law of Universal Gravitation applies as
  long as one object is not overlapping the other.

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• Imagine a hole drilled all the way through the
  earth right through its center. You release a
  ball at one end of the hole. After the ball fell
  half way to the center of the earth, the force of
  gravity on the ball is

• larger than
• equal to
• smaller than
• the force on the ball at the top of the hole

earth
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Midterm 2

• Average76.7
• Standard Deviation18.15