Physics 55 Monday, September 26, 2005

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Physics 55Monday, September 26, 2005

1. Newtons Law of Gravitation With Examples
2. Connection of Newtons Laws to Fundamental
   Conservation Laws

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Announcements

• Answers to Assignment 3 posted.
• Quiz this Friday at 115 pm sharp.
• Homework 4 due this Friday.
• Short Assignment 5 on Friday and due Wednesday,
  Oct 5. Will give detailed answers on Oct. 5.
• Review session for midterm next week (to be
  announced).
• Midterm on Friday, Oct 7.

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Questions from Previous Lecture?

• Keplers three laws
• Newtons three laws
• Circular motion radial force causes
• acceleration

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Simplest Motion Uniform Motion
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Nonuniform Motion Acceleration
Speed is not constant or direction of motion is
not constant (but speed can be constant in case
of circular motion).
Where are speeds large in this picture if
stroboscope samples at equal times? Where are
speeds small in this picture? a (1/m) F
A
B
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Newtons Great InsightNonuniform Motion Caused
by Forces
Something from one object like Sun somehow
influences motion of other object like Earth.
That something is still not understood in any
fundamental sense but Newton discovered could be
described by an astonishingly simple and precise
mathematical rule now known as the universal law
of gravitation. The gravitational force becomes
weaker with distance but has an effect no matter
how far one object is from other object. Total
force on object is sum of forces from all other
objects so depends on relative positions of all
the other objects. Mathematics can be hard,
computers have helped to obtain insight.
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Newtons Third Law is Built intoThe Law of
Gravity Since M1 M2 M2 M1
As strange as it may sound, you pull as hard on
the Earth via gravity as the Earth pulls on you
(you get the same value of F since value does not
depend on order of product of masses M1 and
M2). Does not seem this way because mass of
Earth is enormous and so its acceleration
a(1/m)F due to your force is not noticeable.
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Henry Cavendish 1730-1810Measured G and So
Weighed the Earth
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Newton Described GravityHe Did Not Explain
Gravity!

• An important point Newton was successful because
  he gave up trying to explain what is gravity
  but instead tried to describe gravity
  mathematically and economically.
• This was huge change in philosophy, say compared
  to people like Descartes who tried to come up
  with intuitive mechanisms.
• Modern science has followed the path of Newton
  we know how to describe and predict many
  phenomena with great accuracy but we dont have
  an intuitive mechanistic explanation for why the
  description holds. This is especially frustrating
  with quantum mechanics, the theory of atomic
  particles and light.

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Difference Between Mass and Weight of Object
Mass m of object is a number (units of kg) that
measures resistance to motion of object caused by
force acting on object. It is an intrinsic
property of an object and does not change.
Example Kicking a bowling ball in outer space
can break your toe because it has a mass and so
inertia and resists force from your toe. Weight
of an object is total force (a vector or arrow)
that arises from gravitational pull and
accelerations (for example, from elevator or
amusement ride). Weight has units of
newtons. Example If you stand on a bathroom
scale in outer space far from any object, you
will measure zero weight even though you
certainly have a mass.
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Example of Gravitational Force Causing
Acceleration
Letter to Einstein From Jerry, Richmond, VA
1952 Dear Sir, I am a high school student and
have a problem. My teacher and I were talking
about Satan. Of course you know that when he fell
from heaven, he fell for nine days, and nine
nights, at 32 feet a second and was increasing
his speed every second. I was told there was a
foluma formula to it. I know you don't have
time for such little things, but if possible
please send me the foluma. Thank
you, Jerry
Challenge for class given formula v g t, with
what speed did Satan hit the Earth given
gravitational acceleration near surface of Earth
of g 9.8 m/s2? Given formula d ½ g t2, how far
did Satan fall? Note 1 AU 1.5 108 km
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Solution to the Einstein Letter
Answer to the letter Lets assume as Jerry does
that Satan falls from heaven with a constant
acceleration equal to the acceleration g at the
surface of the Earth and see where our thinking
takes us. We dont know if Satan was thrown from
heaven with some initial speed so lets assume
the simplest case that he was pushed out of
heaven and so starts falling with zero initial
speed. Assuming nine full days of 24 hours each
(should we use solar or sidereal days for a
heavenly problem?), the final speed would be v
g t ( 9.8 m/s2)(9 d x 24 h/d x 60 m/h x 60 s/m)
8 106 m/s 0.025c, i.e., 2.5 percent the
speed of light which is really cruising (nothing
orbiting within the solar system moves so
quickly!). The distance traveled would be given
by the formula d (1/2) g t2 (1/2)(9.8
m/s2)(9 d x 24 h/d x 3600 s/h)2 3 1012 m 3
109 km 20 AU or halfway to the planet Pluto,
whose elliptical orbit has a semimajor axis of 40
AU. Two punch lines whoever originated the
Satan story was not thinking big enough since it
seems unlikely that heaven would lie between
Neptune and Pluto. Second, a constant
acceleration of one g leads to impressive speeds,
about 3 the speed of light after just 9 days of
travel. As you might guess, it takes a powerful
rocket and a lot of fuel to maintain such an
acceleration for so long. The fact that Satan
falls from a distance of 20 AU means that the
assumption of constant acceleration g is wrong
for an object so far from Earth, gravity is much
weaker and the acceleration GMEarth/d2 starts
small and becomes bigger, slowly increasing to
the value g as Satan gets close to the Earth. So
the above calculation overestimates the final
speed and overestimates the distance traveled. To
get a more accurate estimate of Satans final
speed and the distance Satan traveled, we need to
use calculus (or a clever trick without calculus
using Newtons version of Keplers third law, see
the extra credit problem of Assignment 4). We
also need to make some further scientific
assumptions, such as whether Earth is the only
source of gravity acting on Satan or whether we
need to include the Sun. Lets continue to work
with the assumption that Earth is the source of
gravity and that hell lies within Earth (the
magma?). Then I get a value for the distance
fallen of 6 105 km 1.5 x distance to Moon, much
closer to Earth. The final speed of Satan at the
Earths surface is obtained by energy
conservation (decrease in potential energy must
equal Satans kinetic energy) which I find to
give a value of v 11 km/s or about 34 times the
speed of sound. This is fast but not beyond human
technology to achieve.
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Simple Worked Example of Gravity and Forces
Which ball has no weight? Which ball wont move
at all? Which ball has the largest acceleration?
Which two balls will collide first? Speed of
balls after 1 hour? Note For each ball, you
need to calculate total force from two forces
arising from the other two to determine a balls
acceleration. Assume balls start at rest and pull
on each other via gravity.
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Gravitational Acceleration Near Earths Surface
Gravitational acceleration depends on mass M of
object and on distance R from center of spherical
object to surface (radius). What is acceleration
on surface of Moon, for which mMoon 0.012
mEarth and RMoon 0.27 REarth? Answer
gMoon/gEarth 0.012/.272 0.16. So your weight
mg on the Moon is about 1/6 your weight on Earth.
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Conservation Laws and Newtons Laws
From our modern perspective, we now understand
Newtons first, second, and third laws to be
statements of conservation of momentum. In
absence of force, momentum of object is constant
so its velocity (direction and speed) is
constant. This is Newtons first law, but should
have been called Galileos law of inertia. Force
causes momentum of object to change either speed
or direction changes. This is Newtons second
law d/dt(mv) sum of forces. Since momentum is
conserved, whenever two objects interact
(asteroid colliding with Earth, astronaut pushing
off from International Space Station), the force
of first object on a second must be exactly
opposite in direction and strength of the force
of second object on first.
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Emmy Noether (1882-1935)Conservation is Related
to Symmetry
Discovered deep connection between conservation
laws and symmetries of physical equations

• Translational invariance in time implies energy
  conservation.
• Translational invariance in space implies
  momentum conservation.
• Rotational invariance implies angular momentum
  conservation.
• Other possible symmetries time reversal, mirror
  reflection, inversion, discrete rotations, etc.

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Points Covered at White Board

• Conservation of momentum whenever there is a
  collision of objects or some object separates
  into pieces, the total momentum has the same
  value. Use to predict speeds, explains how
  rockets work.

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Las Vegas CSI ProblemUsing Conservation Laws to
Solve a Crime
After finishing Physics 55 and graduating from
Duke, you follow your childhood dream and become
a member of the Las Vegas Crime Scene
Investigation (CSI) team. One day, you are called
to a crime scene where you are asked to determine
whether a certain rifle at the scene was used as
a weapon. You can do this by determining whether
the rifle is capable of firing a high-velocity
bullet (mass 4 grams) in excess of 900 m/s (about
2000 mph). This is the minimum speed that could
explain the substantial damage done by the bullet
to a wall when the bullet fortunately missed the
victim. Not having time to get back to the lab
but having some rope, a scale, and a measuring
tape, you set up the following experiment you
suspend a soft block of wood (mass 0.5 kg) from a
long rope so the block can swing freely back and
forth. You then clear the room, make sure that
the block is hanging motionless, then fire a
bullet from the rifle horizontally into the wood
block. (The bullet gets trapped in the soft
wood.) You then observe that, from the impact of
the bullet, the block swings upward along an arc
to a maximum height that is 0.8 m above the
initial position of the block. I will show in
class how using conservation of momentum followed
by conservation of energy allows us to determine
the speed of the bullet and determine whether the
rifle was the weapon.
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Solution to the CSI Rifle Problem