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Homework/Quiz etc
Problematic email addresses:
[email protected]
[email protected]
[email protected] (this one ok, I have an alternative)
-Mailbox full errors for these
A class email noted L1 + L2 and answer to homework on
the webpage
http://www.jca.umbc.edu/~turner/2003_phys316.html
Homework will not be graded, but same and similar questions will
appear in exams and quizzes.
QUIZ next Tuesday -at start of lecture, 45 minutes. Bring a
calculator. Will cover last weeks, todays and some of Thursdays
material.
Closed book!
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Review of Kepler
Kepler’s First Law: The orbit of each planet about the Sun
is an ellipse with the Sun at one focus
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Orbits
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Review of Kepler
Kepler’s Second Law: As a planet moves in its orbit around
the Sun, it sweeps out equal areas in equal times
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Review of Kepler
Kepler’s Third Law: P2=R3 where P is the period in years &
R is the semi-major axis in AU (1 AU is Earth-Sun dist)
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Review of Kepler
So, Kepler parameterized the planetary orbits, without
understanding why things were as they are…
and P2=R3 only true using certain units (years and AU,
all Earth-related measures)
-in general P2= k R3 and the physics hidden in the
constant, k, was not understood.
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Galileo Galilei - The Telescope
At the same time as Kepler proposed his laws of planetary motion,
Galileo was proving planets really did move around the Sun,
supporting Copernicus model
Circa 1600, - still 3 objections to a heliocentric solar system with
noncircular orbits. All were rooted in the beliefs of Aristotle.
-The Earth could not be moving because if it were, all the objects
on it would be left behind as it moved.
-The heavens were thought to be perfect and unchanging
-If the Earth orbits the sun, we ought to be able to see stellar
parallax.
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Galileo Galilei - The Telescope
In 1609 Galileo Galilei starts using the telescope for
astronomy-took the basic telescope used for terrestrial
viewing and turned it into a scientific instrument
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Galileo Galilei - The Telescope
Letter from Galileo
reporting the discovery
of Jupiter’s moons…
The fact they orbit
Jupiter and not Earth
challenged the idea of
a favored position for
the Earth
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Galileo Galilei - The Telescope
Observations showed phases for Venus, proving it must
orbit the Sun and not the Earth
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Galileo Galilei - The Telescope
Showed the Sun had
changing sunspots-adding
weight against the view of a
perfect and unchanging
heavens
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Galileo Galilei - The Telescope
Summary, 1609: Galileo starts using the telescope for
astronomy
Discovers:
-phases of Venus (it orbits the Sun)
-satellites of Jupiter (they orbit Jupiter)
-mountains on the moon
-sunspots
Celestial bodies clearly seen to be complex, imperfect and
changing-supporting Copernican/heliocentric models
in 1632 Galileo published “Dialogue on Two World Systems”
strongly supporting the Copernican system
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Galileo Galilei - Relativity
Galileo also added to scientific understanding with some
new ideas, realized force is not responsible for motion, but for
changes in motion:He also postulated
- velocity is not absolute,
- speed of a falling body is independent of weight,
i.e “Galileo's Principle of Equivalence” (later)
- a theory of relativity
all (mechanical) experiments performed in
inertial frames will give the same results
These are cornerstones of Einstein's theories of relativity.
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
A Few Reminders/ Simple Concepts
Speed conveys how fast one is going
Velocity conveys speed and direction -can change velocity
without changing speed
Acceleration/deceleration describes the change in velocity
(g=9.8m/s/s)
Momentum describes the product mass x velocity
(note mass is amount of matter comprising a body,
weight is the sum of accelerating forces on a body)
Force describes a change in momentum
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Simple Concepts of Motion
Angular Momentum the product mass x velocity x radius
for an object moving in a circle (or orbit)
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Mass always the same, weight depends on force
acting on mass
Jane Turner
[4246] PHY 316 (2003 Spring)
Inertial Frames - Reminder
An inertial frame is one in which
under the influence of no forces,
an object will remain at rest or in uniform motion (elevator scenario 1)
Accelerating (or rotating) frames not inertial frames.
Given the universal force of gravity, no body can actually be under the
influence of no forces. However the concept of inertial frames is still
useful since:
- effect of gravity can be very small.
Thus physical insight can be obtained
(e.g. Einstein's Special Theory of Relativity).
-valid under Einstein's General Theory of Relativity:
a frame in free-fall under the influence of gravity
is an inertial frame.
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Galileo’s Principle of Equivalence
Inertial and gravitational mass are equivalent
Gravitational Mass
Inertial Mass
measure of how strongly
is measure of how
a body is affected by the
strongly the body is force of Gravity
accelerated (by A) It is the mg in Newton's
universal law of
by a given force.
gravitation when the
It is the mi in
Newton's 2nd-law: body is a distance R from
another body of mass M:
Force = mi A
Jane Turner
Force = G mg M R-2
[4246] PHY 316 (2003 Spring)
Lecture 3
Galileo’s Principle of Equivalence
The inertial mass mi determines how the body accelerates
as a results of the application of any force.
The gravitational mass mg determines how the body "feels"
a gravitational force (and how much of a gravitational
force it generates).
The fact that one can equate the above two forces:
mi A = G mg M R-2
Then if mi equals mg then one sees that the acceleration (due
to the force of gravity) is independent of mass.
Galileo: If all forces apart from gravity can be
ignored, all objects fall at the same rate
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Galileo’s Principle of Equivalence
This equivalence is a mathematical confirmation of
Galileo's experiments (cannonball & feather etc) and can be
used to derive Kepler's 3rd law of planetary motion
The fact that mi and mg are equal is often referred to as
Galileo's Principle of Equivalence
- used and extended by Albert Einstein (1915) as he
formulated General Relativity. Indeed the whole of
General Relativity rests on Einstein's Principle of
Equivalence
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Physical Cosmology Arrives
Isaac Newton's discovery that it is gravity which plays the vital role of
determining the motion of the planets
"signifies the the arrival of the first physical cosmology"
Newton's universal theory of gravity contains
the concept of action at a distance
Many scientists did not accept this concept.
Newton's law of gravity was not immediately
accepted universally.
Instead many scientists preferred to stick with
the prevailing idea of the time (e.g. Rene Descartes)
that forces work through contact.
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Isaac Newton (1643-1727)
Formulated a theory of mechanics &
gravity that explained the solar system
with remarkable accuracy!
Realized that gravity responsible for the
motion of the Moon and planets.
Newton’s law of universal gravitation
Every mass attracts every other mass
Force drops off with square of distance
Kepler’s laws are a direct consequence of
Newton’s law of gravity
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Orbits under gravity
Gravity also allows us
open hyperbolic or
parabolic orbits (like
those of comets)
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Newton Universal Law of Gravitation
Every mass attracts every other
mass through the force called
gravity
Fg=GM1M2/r2
r -dist between
centers
G grav const
Fg force of grav.
attraction
Jane Turner
The force of attraction between 2
objects is directly proportional to
the product of their masses.
Doubling the mass of one body
doubles the force of gravity
between the two
[4246] PHY 316 (2003 Spring)
Lecture 3
Isaac Newton
In 1687, Isaac Newton publishes
Philosophiae Naturalis Principia Mathematica
Stephen Hawking, in A Brief History of Time notes
"probably the most important single work ever published in the physical sciences.”
Newtons 3 laws of motion:
1:Every body continues in its state of rest or (straight-line)
motion until compelled to do otherwise
by an external force (ie. intertial frames)
So, if a body is not acted on by any forces, its vel remains
const- conservation of momentum
2:Force equals Mass times Acceleration
-defines inertial mass as the degree by which a body resists
being accelerated by a force
3:To every Force there is an equal (& opposite) Reaction
Remember these!
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Isaac Newton
N1:Every body continues in its state of rest or (straight-line)
motion until compelled to do otherwise
by an external force (inertial frames)
Comes from Galilean Relativity- Galileo was the first to state
this in his ‘’law of inertia’’
Concept: Frames of references at rest or moving with constant
velocity are called inertial frames. Newtons laws hold within
these frames of reference
In non-inertial frames you may be fooled into thinking there
are forces acting on freely moving bodies
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Isaac Newton
N2:Net force is proportional to mass x acceleration
An object subject to n forces experiences a net force
Fnet =  Fi = ma (summed over i=1…n)
However, a=dv/dt - giving us
Fnet = m dv/dt = d (mv)/dt = dp/dt
Vectors, quantities having magnitude and direction, are in boldface
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Isaac Newton
N2:Net force is proportional to mass x acceleration
Thus N2 may be expressed as - the net
force on an object equals its rate of change
of momentum
Fnet = dp/dt
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Isaac Newton
N3:For every action there is an equal and opposite reaction
action and reaction are forces
F12 = -F21
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Law of Universal Gravitation
Using his 3 laws of motion+ Keplers 3rd law - Newton
was able to find an expression describing the force
which holds the planets in their orbits.
Consider a circular orbit of mass m about much larger
mass M (M >> m) . Recall K3:P2= kr3
Remember this one!
r is dist. between the objects and k is a constant, in
circular orbit P is
P = 2 r/v
(v is velocity of the mass m)
substitute 2nd eqn into first to get
4  2r2/v2 = kr3
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Law of Universal Gravitation
4  2r2/v2 = kr3
-rearrange terms and mult. through by m
m v2/r = 4 2m/ kr2
left hand side is recognizable as the centripetal force for
circular motion; thus
F = 4 2m/ kr2
must be the grav force keeping m in orbit around M
Now, the force exerted by m on M equal the mag of that
exerted by M on m
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Law of Universal Gravitation
F = 4 2m/ kr2
Now, the force exerted by m on M equal the mag of that
exerted by M on m -thus the form of the eqn should be
symmetric w.r.t. exchange of m and M
Expressing the symmetry explicitly and grouping the
constants into a new one, G we arrive at the law of
universal gravitation found by Newton
F = GMm/r2
G=6.67 x 10-11 m3 kg-1 s-2
Jane Turner
SI units
[4246] PHY 316 (2003 Spring)
Lecture 3
Law of Universal Gravitation
Law of universal gravitation
F = GMm/r2
Remember this one!
Gravitational force follows an inverse square lawdoubling separation between two objects, grav
attraction
drops x 4
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Gravity due to Earth
Law of universal gravitation
F = GMm/r2
Remember this one!
Consider mass m falling from height h above Earths
surface, earths radius denoted R and mass M
F = G M m/(R + h)2
h << R so
F = G M m/R2
However, F = ma = mg thus
g = G M /R2
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Gravity due to Earth
g = G M /R2
M = 5.974 x 1024 kg & R = 6.378x 106 m gives
acceleration due to Earths gravity
g = 9.8 m s-2
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Gravitational Potential Energy
Consider now the energy (work) req. to raise mass m to
height h against grav. force, ie the change in potential
energy of the system
Uf - Ui = U = - F. dr
(between ri & rf)
where F is the force vector and ri & rf are initial and
final position vectors dr is the infinitesimal change in
posn vector
If gravitational force on m is due to mass M then
U =  G Mm/r2 dr
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Gravitational Potential Energy
If gravitational force on m is due to mass M then
U =  G Mm/r2 dr
evaluate the integral to give
Uf - Ui = - G Mm(1/rf - 1/ri) dr
since we are interested in relative change in potential we
can choose to define p.e going to zero at infinity , let rf
approach infinity so Uf = 0 then (dropping subscripts)
U = - G Mm/r - gravitational p.e.
Now, total mechanical energy of a particle is
E = 1/2 mv2 - G Mm/r
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Escape velocity
E = 1/2 mv2 - G Mm/r
can be used to calculate the escape velocity around
mass M (>>m) by equating kinetic energy and grav.
force
1/2 mv2 = G Mm/r
which may be solved for the velocity
vesc = √(2GM/r) Remember this one!
mass of escaping object does not appear! For earth
vesc = 11.2 km/s
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Generalization of Keplers work
Newton explained Keplers laws by solving the law of
universal gravitation together w/ the laws of motion.
Solved a pair of algebraic equations w/ use of calculus
Newtons work showed Keplers first two laws apply to
any object orbiting another under the force of gravity, or
objects orbiting each other with their center of mass at
one focus.
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Newtons form of Keplers Third Law
Newton also generalized Kepler’s third law as
P2=42 R3 /G(M1+M2)
Remember this one!
Allowed Kepler’s Laws to be applied to moons and (much later) binary
stars and extrasolar planets.
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Very good, but not perfect..
Isaac Newton also contributed to science with
- invention of the telescopes using mirrors
- theories of light (colors & corpuscles)
- development of calculus
But Isaac Newton did not get it all right..
e.g. He incorrectly believed
space & time were absolute
and unaffected by the presence of objects
His character was not perfect either
-he had huge academic (& personal) rows with
Robert Hooke (re: who 1st discovered the 1/r2 law)
Gottfried Wilhelm von Leibnitz (re: who 1st discovered calculus)
(e.g. see Steve Hawking “A Brief History of Time”)
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Einstein's Principle of Equivalence
Albert Einstein extended Galileo's Principle of Equivalence
(inertial & gravitational masses are equal)
i.e. acceleration and gravity cannot be distinguished.
This led to Einstein's Principle of Equivalence
The Laws of Physics are the same in a uniformly accelerated
reference frame as in a uniform gravitational field
Albert Einstein “The happiest thought of my life”
The whole of General Relativity rests on this (testable) principle
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Einstein's Principle of Equivalence
The consequences of the Principle of Equivalence include
- the effects of a gravitational force are indistinguishable
from those present in an accelerated reference frame
- there is an (accelerating) reference frame in which the effects
of gravity are not experienced (falling elevator)
- the path of light is "bent" by gravity
- clocks run "slow" under the influence of gravity
gravitational "redshift" of light waves
- gravity affects anything carrying energy (E=mc2)
...more later on GR
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Overview
Background reading -Chapter 3
… things to know (for a quiz next week)
-Contributions to astronomy from the
ancient Greeks
-Ptolemy and his model
-Copernicus, how did he change our view
and what was his first model
-What is meant by the Copernican & Perfect
Cosmological Principles
-Kepler, how did he progress from the
heliocentric model using circular orbits
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3
Overview
-What are Keplers 3 laws, what factors does the
orbital period depend on?
-What were the major achievements of Galileo?
-What was Newtons contribution to astronomy?
-Should be able to remember & use Newtons
Law of Universal Gravitation and Keplers Laws
(esp. newtons general version of K3)
-& calc escape velocities
Jane Turner
[4246] PHY 316 (2003 Spring)
Lecture 3