Announcements

1
Announcements

• Today is the last day of new material
• Review session on Sunday 8PM in BH166
• Homework 11 is due on Sunday at end of class
• Conference sections will be held Thursday and
  next Tuesday
• Final Exam will be Wednesday, December 14
• Note date listed in course information was
  inadvertently listed as Monday, December 14
• Will cover all material in course
• Will focus more heavily on material in HRW
  Chapters 10-13, 15
• Usual rules apply can bring a single 8.5x11
  sheet of paper and basic calculator

2
Gravitational Potential Energy

• Work required to lift an object in the presence
  of gravity
• We previously saw that this work increased the
  potential energy of the object
• For a constant gravitational force, we had U
  mgh
• If we continue to lift the object (say, a
  spaceship), the gravitational force will diminish
  as we get further from the earths surface
• We can find the potential energy for this case by
  calculating the work done in lifting an object

3
Gravitational Potential Energy

• Suppose we start at the earths surface
• Take r1 to infinity, then the potential energy is
• i.e., potential energy set to 0 infinitely far
  from earth

4
Work Required for Deep Space Travel

• If we want to leave the earths neighborhood, we
  need to do work to bring gravitational potential
  energy up to 0
• Example work required to send 1 kg mass into
  deep space
• Neglect gravitational force of sun, other planets
• Escape velocity velocity needed to escape
  earths gravity

5
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

6
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

7
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)
8
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
9
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

10
Energy of an Orbiting Object

• What is the kinetic and potential energy of an
  object in orbit?
• Energy is negative because it is bound to the
  object it is orbiting cant escape without
  additional energy

11
General Relativity

• In case you are interestedwill not be on exam!
• Consider the motion of an object of mass m in the
  vicinity of a fixed object of mass M
• The acceleration/motion of the object doesnt
  depend on the objects mass!
• For a given gravitational field, the motion is
  effectively a matter of geometry it doesnt
  depend in any way on the properties of the object
  itself

12
Equivalence Principle

• Einstein hypothesized that uniform acceleration
  and being acted on by gravity are
  indistinguishable
• This is known as the equivalence principle
• Thought experiment imagine you are traveling in
  a space ship with no windows or instruments
• If you are accelerating with an acceleration g,
  you will feel a force of mg
• If you are not accelerating, but are near the
  surface of earth, you will also feel a force of
  mg
• How would you tell these two effects apart?
• Perhaps you could shoot a light beam across the
  space ship if you were accelerating, it
  wouldnt travel in a straight line
• Equivalence principle says that light will also
  bend due to gravity!
• Bending of light by gravity was seen a few years
  after Einstein developed his theory of general
  relativity

13
Curvature of Space-time

• In the space ship example, we talked about
  whether light traveled in a straight or curved
  line
• One problem how do you even define a straight
  line if light itself takes a curved path?
• We can get around this problem for a rigid space
  ship by taking it to a place where it is neither
  being accelerated nor acted on by gravity
• but how do we setup a Cartesian coordinate
  system for the universe as a whole in the
  presence of gravity?
• Einsteins solution light always travels in a
  straight line, its just that space is curved by
  objects with mass

14
Black Holes

• If light bends near a large mass, what happens if
  we make the mass really big?
• Light is bent more strongly as we increase the
  mass
• At some point, light emitted by the object is
  bent so sharply that it cant emerge from the
  object
• We call this a Black Hole
• Black Holes can be seen by the effect of their
  gravity on other objects
• Star orbiting a black hole will show small
  position changes, but no other star is visible
• This can happen with either a black hole or a
  neutron star by looking at the rotational
  velocity and period, you can estimate the mass of
  the unseen object
• Neutron stars with masses more than 2x the suns
  mass should form black holes
• Some evidence for very large black holes in the
  center of galaxies from motion of stars in the
  galaxy

15
Expanding Universe

• If you start letting gravity alter the fabric of
  space and time, you might expect some surprises
• One surprise when Einstein developed relativity
  was that the universe should be either expanding
  or contracting
• This isnt just a matter of objects moving
  around, but space itself was expanding/contracting
  
• Einstein wasnt happy with this, so he added a
  Cosmological Constant to the theory to force
  the universe to be static
• This seemed like a big mistake when we started to
  see evidence for the big bang, during which the
  universe expanded rapidly
• More recent studies show the universes expansion
  is accelerating perhaps we do need a
  cosmological constant!

16
Gravitation Summary

• Gravitational force
• Shell theorem gravitational force from a
  uniform spherical shell is the same as if all
  mass was at the center
• Gravitational potential energy
• Keplers Laws elliptic orbits, Law of areas, Law
  of Periods
• Energy of orbiting object
• General Relativity is pretty bizarre!!