Isaac Newton

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AST101 Lecture 4 Feb 3, 2003
Isaac Newton
Isaac Newton was born in 1642, the year Galileo
died and 12 years after Keplers death. From
age 27, Newton was Lucasian Professor of
Mathematics at Cambridge University. Newton was
a shy and solitary person who shunned controversy
his major discoveries went unpublished for
nearly 20 years.
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• Newtons achievement span a huge array of topics
  and include
• Discovering the binomial theorem
• Inventing calculus
• Experimenting with light and developing the
  theory of optics
• The Law of Gravitation
• Enunciating the laws of mechanics governing all
  motion
• Inventing the first reflecting telescope
  (mirrors instead of lenses)
• Experiments in alchemy (probably getting mercury
  poisoning, and leading to episodes of insanity)

No one would argue with a claim for Newton as one
of the most productive and influential scientists
ever. Many would nominate him as Man of the
Millenium in Science
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Newton knew of course of Galileos and Keplers
work, and understood that what was lacking was
understanding of the cause behind the observed
motions of the planets. Keplers suggested that
the Sun somehow reached out with invisible
paddles to guide the planets around their
orbits. Galileo thought that the natural state of
motion was for bodies to travel in
circles. Newton sought a more comprehensive
theory that could explain planets orbits,
falling apples and cannonball trajectories from a
single set of principles. This theory was
worked out in 1665 1667 while Cambridge classes
were suspended due to the bubonic plague. But
only in 1687 were his results published in
Philosophiae Naturalis Principia Mathematica
the cornerstone of physics and astronomy ever
since. In this work, he introduced the essential
ingredient of using mathematics to describe the
physical world, and he provided the explanation
for Keplers Laws

• The planets move in ellipses, with the Sun at one
  focus.
• The line from the Sun to the moving planet sweeps
  out equal areas in equal times.
• The square of the planets orbital period (P) is
  proportional to the cube of the semi-major axis
  of the ellipse.

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Summary of Newtons laws of motion and law of
gravitation
1. Vectors and scalars

• Many quantities we use to describe the world are
  simple numbers the size of the number matters,
  but there is no sense of direction associated.
• Examples are Mass (or weight), Time,
  Temperature, or Energy (for example heat,
  kinetic energy of motion, potential energy due to
  position)
• These quantities are called SCALARS
• Other quantities called VECTORS have an
  essential component of direction. For example
  Displacement refers to the location of an object
  relative to some origin or starting place.

The direction of a displacement is essential to
know if you travel for 10 miles north from
campus, you have a quite different (wetter)
experience than if you travel 10 miles west!
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For VECTORS, both magnitude and direction must be
specified.
The location of the tree relative to the origin
(x0, y0) is given by the red DISPLACEMENT
vector. Both the magnitude or distance (200 m)
and the direction (30O north of east) are
essential to telling where to find the tree.
North
200 m
30O
East
position at time t1
difference in position Dr . The time difference
is Dt t1-t2. The velocity is v Dr/Dt
position at time t2
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Acceleration
Case (a) is like the car accelerating from the
stop light Case (b) is what planets in circular
orbit do (A more complex combination of the two
ways is possible) Define speed as magnitude of
velocity (speed is a scalar)
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Motion in a circle with constant speed (for
example, moon around Earth)
v1
R
v1
v2
Dv change in velocity
R
v2
Even though magnitude of velocity does not change
here, the direction does. Dv is in direction
toward center of circle. Can show with geometry
that the magnitude of the acceleration is
a v2/R (Centripetal acceleration) A planet
or moon in circular orbit is accelerating
(continually falling toward center) !
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Newtons Laws of Motion

• If no force is applied to a body, its velocity
  does not change. (If originally at rest, stays
  at rest if moving it continues at same magnitude
  and direction of velocity.)
• If a force is applied to a body of mass m, it is
  accelerated with
• F m a
• For any pair of objects, the force exerted on the
  second by the first is equal in magnitude but
  opposite in direction to the force exerted by the
  first on the second.

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• If no force is applied to a body, its velocity
  does not change. (If originally at rest, stays
  at rest if moving it continues at same velocity.)

Newton understood what Aristotle and Galileo did
not that the natural state of motion is to
continue in a straight path if there are no
external influences (forces). Not to come to
rest (Aristotle), and not to move in circles
(Galileo). The first law is sometimes called
the law of inertia.
2. If a force is applied to a body of mass m, it
is accelerated with
F m a .
In the 2nd law, we should consider a Force to be
the cause an acceleration to be the effect or
result and the mass the body experiencing the
force to determine the size of the effect.
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Mass a property inherent in every object
(including planets, stars and us). For objects
on earth, the mass is related to the weight. It
basically determines the resistance of a body to
acceleration. Force a push or pull, just as in
common language usage. A force is a vector
since it matters whether the push is to the right
or left. There are different types of forces (a
push with your arm, friction, electrical,
gravitational etc.). Although some forces occur
when two objects are in contact, others like
gravity or the electric force act even when the
objects are at a distance from each other.
The 2nd law says that the direction of an
acceleration is always in the direction of the
force. Thus a forward force on a car due to
the action of the engine on the wheels produces a
forward acceleration.
For centripetal acceleration of a planet in
orbit, the force must be toward the center of the
circle. The planet is in continual free fall
around the sun.
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F m a
Consequences of the Second Law

• For a fixed mass M, a force F2 2 F1 will
  produce twice the acceleration as that of F1

a1
a2 2 a1
F1
F2 2F1
Mass M
Mass M

• For a fixed force F, a mass 2M will receive half
  the acceleration as a mass M

a1
a2 ½ a1
F
F
Mass M
Mass 2M

• Show that the second law implies the first
  (remember that zero acceleration means the
  velocity is not changing)

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3. For any pair of objects, the force exerted
on the second by the first is equal in magnitude
but opposite in direction to the force exerted by
the first on the second.
F1
F2
F1 is the force that the triangle exerts on the
star F2 is the force that the star exerts on the
triangle Newtons 3rd law tells us that the
magnitudes of F1 and F2 are the same, but that
they are opposite (directed along the same line
but in opposite directions). Thus, while the Sun
attracts the Earth, the Earth also attracts the
Sun with an equal but opposite force.
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In astronomy, the gravitational force is of
primary importance. Newton also determined the
character of the GRAVITATIONAL FORCE. The
gravitational force, Fgrav , between two bodies
of masses M1 and M2 depends also on the distance,
r, separating them (actually the distances
between their centers).
Fgrav G M1M2 /r2
The direction of the gravitational force is
attractive, directed along the line separating
the two masses.
Forces on M1 and on M2 are equal in magnitude and
opposite in direction (Newtons 3rd law)
Force on M1 due to M2
Force on M2 due to M1
M1
M2
r
G is a constant that is determined only by
experiment. It is called Newtons constant of
Gravitation. Its value is found to be G 6.67 x
10-11 Nm2/kg2 but you will not need to remember
it for this course. (It is a small number.)
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No contact of the two masses is needed gravity
force operates at a distance. The strength of the
force is proportional to the product of the
masses of the two bodies.
Fgrav
The strength of the force diminishes as the
separation increases (falls off like 1/r2).
The Newton constant G is small, so to get
appreciable force, we need very large masses
(e.g. the Sun, the Earth etc.). The force
between two people standing 1 m apart is only 3 x
10-10 times the force exerted on either person by
the Earth (their weight).
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• Example
• A stone falling toward Earths surface is
  responding to the Gravitational force between
  Earth and stone.
• F (on stone) G MEMS /R2
• ME is mass of earth MS is mass of stone R is
  the distance from the center of Earth to the
  stone. For heights not far above the surface of
  the earth, R is approximately the radius of the
  Earth. (The radius of Earth is 6.4 x 106 m, so
  raising a stone to 6 m is a negligible distance
  compared to radius.
• For any such stone, G, ME and R are constants and
  we can lump them into one constant called g .
• F MS g , with g G ME /R2
• Compare this with Newtons 2nd law F MSa and
  we learn that g is just the acceleration
  experienced by any falling body near the surface
  of the Earth.
• Measuring g in the lab, knowing G, and knowing R
  from mapping the Earth we can turn the equation
  for g around to get
• ME g R2/G

• Thus experiments on falling bodies tell us the
  mass of the Earth!

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• Example Derive Keplers 3rd Law
• Consider a planet in orbit around the sun
  (approximate orbit as circle for simplicity).
  Call the mass of the Sun M1, and mass of planet
  M2.

F GM1M2/r2 M2 a M2 v2/r Dont measure v
directly, but v (orbit circumference) / period
v 2pr/P. Substitute GM1M2/r2 M2 (2pr)2 /
(P2 r). Solve this equation for P2 to get
M2
r
M1

• Do the algebra to show this ?

This looks like Keplers 3rd law since r for a
circle is the semi-major axis. Newton made one
essential improvement In fact, by Newtons 3rd
law, planet also exerts force on Sun, so the Sun
also moves. Both planet and Sun orbit a common
point called the center of mass. This leads to
the change in Keplers 3rd law by Newton
For earth Sun, the earths mass is negligible
and Keplers form is OK. But if the planet
were as massive as Sun, need Newtons fix.
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Newton says that
(units are seconds, meters and kilograms)
or P2 a3 /(M1M2) in any set of units.
Can simplify by choosing appropriate units for
period P, masses M and semi-major axis a For
Earth ( ) around Sun ( )
period P
1 year
semi-major axis a 1 AU

M1 M2 M So using units with P in years, a in
AU and Masses in solar masses,
P2 a3 /(M1M2)
Shorthand symbols for Earth and Sun
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Newton further showed that the Gravitational
force requires two orbiting objects to both move
in ELLIPSES of same eccentricies, with a COMMON
focus located at their fixed center of mass
(c.m.) (Circle is special case) Center of mass
like the fulcrum of a see-saw
M2
M1
d1
d2
M1/M2 d2/d1
Larger mass is closer to c.m. For Sun and
planet, c.m. is inside the Sun. Motion on
elliptical orbits is faster for the lighter
object on the larger ellipse.
The relative orbit (e.g. the Earths orbit seen
from the Sun) is also an ellipse with same
eccentricity as either absolute orbit. It is
this relative orbit for which P2 a3 /(M1M2)