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Isaac Newton
AST101
Lecture 4 – Feb 3, 2003
Isaac Newton was born in 1642, the year Galileo died and 12 years after
Kepler’s 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unpublished for nearly 20 years.
Newton’s 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
Newton knew of course of Galileo’s and Kepler’s work, and understood that
what was lacking was understanding of the cause behind the observed motions
of the planets. Kepler’s 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 Kepler’s Laws:
1.
The planets move in ellipses, with the Sun at one focus.
2.
The line from the Sun to the moving planet sweeps out equal areas
in equal times.
3.
The square of the planet’s orbital period (P) is proportional to the
cube of the semi-major axis of the ellipse.
Summary of Newton’s 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!
*
*
For VECTORS, both magnitude and direction must be specified.
North
30O
East
The location of the tree relative to
the origin (x=0, y=0) 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.
Examples of vectors:
Displacement – locating an object relative to another, or relative to the
origin of our coordinate system. Often denote by r . The → sign is used
to tell us the quantity is a vector.
Velocity – the rate at which the displacement changes with time. So, if
the change in r is Dr over a time interval Dt, we define the velocity as
v = Dr/Dt
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
Acceleration:
What happens if velocity changes? For example when your car starts
from v = 0 at a stop light and increases as you hit the gas pedal. You
ACCELERATE .
Acceleration = rate of change of velocity
But velocity is a vector, so acceleration must be vector too:
acceleration = a = Dv/Dt
There are two basic ways to get acceleration:
a) change the magnitude of velocity, but keep velocity in same direction
b) change the direction of velocity, but keep magnitude the same
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)
Motion in a circle with constant speed
(for example, moon around Earth)
v1
R
Velocity at time t1 = v1;
Velocity at time t2 = v2
Radius of circle = R
v2
v1
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) !
Newton’s Laws of Motion
1. 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.)
2. If a force is applied to a body of mass m, it is
accelerated with
F = m a
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.
1.
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.
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.
Consequences of the Second Law
F = m a
 For a fixed mass M, a force F2 = 2 F1 will produce twice the
acceleration as that of F1
a1
F1
Mass M
a2 = 2 a1
F2 = 2F1
Mass M
 For a fixed force F, a mass 2M will receive half the acceleration as
a mass M
a1
F
a2 = ½ a1
F
Mass M
Mass 2M
 Show that the second law implies the first (remember
that zero acceleration means the velocity is not changing)
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
Newton’s 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.
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.
M1
Force on M1
due to M2
Force on M2
due to M1
r
M2
Forces on M1 and on M2
are equal in magnitude
and opposite in direction
(Newton’s 3rd law)
G is a constant that is determined only by experiment. It is called
Newton’s 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.)
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).
 Example:
A stone falling toward Earth’s 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 Newton’s 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!
 Example: Derive Kepler’s 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.
M2
M1
r
F = GM1M2/r2 = M2 a = M2 v2/r
Don’t 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:
 Do the algebra to show this →
P2
4p2
=
G M1
r3
This looks like Kepler’s 3rd law since r for a circle is the semi-major axis.
Newton made one essential improvement: In fact, by Newton’s 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 Kepler’s
3rd law by Newton:
4p2
2
P =
r3
G(M1 +M2)
For earth – Sun, the earth’s mass is negligible and
Kepler’s form is OK. But if the ‘planet’ were as massive
as Sun, need Newton’s fix.
4p2
Newton says that
=
r3
G(M1 +M2)
meters and kilograms)
P2
(units are seconds,
or P2 ~ a3 /(M1+M2) in any set of units.
Can simplify by choosing appropriate units for period P, masses M and
semi-major axis a:
Shorthand symbols for Earth and Sun
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 /(M1+M2)
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 Earth’s 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 /(M1+M2)