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•  equal and opposite force for every force:
how can anything ever be moved?
• 
•  acceleration caused by total force on a
single object, i.e. forces do not cancel out
because they act on different objects.
•  Each object ‘feels’ only ONE force
examples
•  airplane?
–  propeller pushes air backwards
–  air pushes propeller (attached to aircraft)
forwards
•  motorboat?
–  same thing, except with water
–  in a perfect fluid there would be no motion
forward
examples
•  hot air balloon rising?
–  pushes cold, outside air down
–  cold outside air pushes balloon up
•  automobile accelerating?
•  automobile going around corner?
Rockets
•  Need to accelerate in space (no air)
•  need something to push against
•  push exhaust gases backwards
•  exhaust gases push rocket forwards
•  must take your “pushing stuff” with you – very $$
$$$
Newton’s Universal Law of Gravitation
•  First and most important:
–  No apples falling on anybody’s head!!
•  Surmised that an apple falls from a tree for
the same reason Moon goes around Earth
•  Newton’s Cannon
–  Astronauts in Space station
Newton’s Universal Law of Gravitation
•  Recall GG: all objects fall at same rate :
α=F/m=constant, so F should be proportional to
object’s mass(!!!)
–  force of gravity must depend on
object’s mass!
•  Apple drops from tree: is apple being pulled
towards Earth, or, is Earth being pulled towards
apple? Both!
–  force of gravity between two objects
must depend on both their masses
Newton’s Universal Law of Gravitation
•  force of gravity gets weaker as objects get
further apart
•  If you go far from Earth the weight vanishes
m⋅M
F =G 2
r
•  G: Universal Gravitation Constant
Newton’s Gravitational Law
•  For two massive objects, the
gravitational force (attractive
always) is proportional to the
product of their masses divided
by the square of the distance
between them
•  Newton’s genius was to realize
the Universality of the
Gravitational force
- The same force that
makes an apple fall keeps
Earth revolving around Sun
NEWTON
•  Newton proposed a quantitative description of
the gravitational force, a force law that we call
Newton's law of gravitation: Every particle
attracts any other particle with a gravitational
force

m1m2
F = G 2 rˆ
r
Here m1 and m2 are the masses of the particles, r is the distance
between them, and G is the gravitational constant
G = 6.67 x10−11 N − m 2 / Kg 2
As the Figure shows, a particle
m2 attracts a particle m1 with
a gravitational force F that is
directed toward particle m2,
and particle m1 attracts
particle m2 with a gravitational
force -F that is directed
toward m1.
The forces F and -F form a third-law force pair; they are opposite in
direction but equal in magnitude. They depend on the separation of the
two particles, but not on their location: the particles could be in a deep
cave or in deep space. Also, forces F and -F are not altered by the
presence of other bodies, even if those bodies lie between the two
particles we are considering.
Force Calculations
1) Force of attraction between two students:
FA = (6.67 x10−11
N − m 2 (70 Kg )(50kg )
−7
)
=
2.33
x
10
N
2
2
Kg
(1m)
2) Force to lift a book:
FL = (6.67 x10−11
N − m 2 (5.98 x1024 Kg )(0.5 Kg )
)(
= 4.9 N
2
6
2
Kg
(6.37 x10 m)
FL
6
Ratio =
= 21x10
FA
Escape Velocity
•  If you fire a projectile upward, usually
it will slow, stop momentarily, and
return to Earth.
•  There is, however, a certain minimum
initial speed that will cause it to move
upward forever, theoretically coming to
rest only at infinity.
•  This initial speed is called the (Earth’s)
escape speed or escape velocity
2GM
v=
R
2(6.67 x10−11 N − m 2 / Kg 2 )(5.98 x1024 Kg )
v=
= 11.2km / s = 6.95mi / s
6
6.37 x10 m
Newton’s Universal Law of Gravitation
•  able to mathematically prove Kepler’s Laws
•  led to the discovery of Neptune and Pluto
•  All objects are attracted to all other objects
•  Tides of the oceans -> Moon’s gravitational pull
to Earth’s oceans.
Back to Kepler’s Laws of Planetary
Motion
•  Remember: Kepler’s law’s are simple consequences of
Newton’s Laws of motion
Some Properties of Planetary Orbits
Perihelion: closest approach to Sun
Aphelion: farthest distance from Sun
Kepler’s Laws of Planetary Motion
I The orbits of the planets are ellipses (not circles!)
with the Sun at one focus
SUN
II The line joining the Sun and the planet sweeps out
equal areas in equal times
III Square of period of planet’s orbital motion (P) is
proportional to cube of semimajor axis (a)
- this means ratio P2/a3 should be constant
Newtonian Mechanics and beyond
•  Kepler’s laws are a consequence of Newton’s laws
•  Some fine-tuning was done to Kepler’s laws due to
Newton’s laws.
•  Einstein’s theory of relativity (curvature of space)
super-tuned Newtonian Celestial mechanics
•  Who knows what is next…but whatever it is … welcome!
Sir Isaac Newton:
•  changed the world
forever
•  The Age of
Reason
•  Demonstrated
how to predict
the future
•  Shook the
Universe
Summary
•  Aristotle’s framework led to unanswered questions
•  Galileo’s insight led to the essential nature of motion
•  Newton’s intellect led to a precise, mathematical
explanation
Some comments III:
•  If you have a uniform spherical shell then the
net gravitational force on an object inside the
cell is zero.
m
M
Gravitational forces are still there but they cancel out. This is an example of
dynamical equilibrium and an application of the superposition principle
Units and Standards
•  In order to perform measurements and communicate the results we need some
reference system(s) with some standard measures, units.
•  These systems only need to define the base units not the derivatives, e.g. base
unit is length, time, mass etc but velocity or force are derivative units
•  The most common system is the International System (SI). It used to be called
MKSA from the names of the first four base units, Meter, Kg, Sec, Ampere, for
length, mass, time and electrical current. There are in total 7 base quantities but
the other three are for advance physics
•  Once you have such a system you then need to define some standard measures.
LENGTH: you measure length in what? -> meters. How long is a meter? This
changes with time: it used to be the 1/10,000,000 of the arc from North Pole to
Equator, then the prototype meter bar in Paris, then the wavelength of 86Kr and
now it is expressed in term of fraction of the speed of light in vacuum.
TIME: Time has both ‘length’, duration but also moment, instant. So there are
time intervals (e.g. 1 hour, 2 days) but also instances (e.g. at 2:05:04 EST). There
is no absolute scale for instances, i.e. the beginning is arbitrary. The unit is based
on atomic clocks of 133Cs.
MASS : There are two kinds of mass, amount of matter and inertia. The inertia is
the more sensible one. The unit is the Kgr prototype in Paris or atomic masses.
Some comments:
•  Whenever you have spherical objects you can replace them but
single points at their center. This is because according to
Newton if you have a uniform spherical shell then the
gravitational force on another object outside the cell is the
same as if the whole mass of the cell was at its center. Earth
can be considered to be (like an onion) a superposition of such
shells. That is why when we calculated the force to lift the book
we put as distance the radius of Earth.
M
d
M
m
d
m
}
Both are the same
Some comments II:
•  Whenever you have to calculate the net force on a
object by several others then you can apply the
superposition principle which says that the net force
will be just the sum of the individual forces.
M
d
d’
m
FNET = FM + Fm’
m’
Motion (supplementary notes)
•  We will start with the study of motion in one
dimension (1D)
•  To begin we need to define the concept of
position and displacement of an object. For
this we need a reference axis (1D) or a
reference (coordinate) system (3D).
(Linear) Acceleration
•  Acceleration is the quantitative answer to the question: “What is the rate of
change of velocity”. It is defined as the ratio of the velocity difference in some
time interval and the time interval. Acceleration is also a vector quantity since it is
the result of the division of a vector (velocity) with a scalar (time).
Constant Acceleration
There are cases of constant acceleration, i.e. instant and average acceleration are the
same. Such a case is the free-fall acceleration. As we are going to prove soon, all
objects near the surface of Earth free-fall with the same acceleration (a) which has a
magnitude g=9.8m/s2 and is directed towards the center of Earth.
In the case of constant acceleration and ONLY in that case we can derive some
further formulas, so called equations of motion, which govern such a motion.
Velocity graphs
•  In order to find the kind of motion we also use the so called v(t) graphs. These
graphs are very useful in one dimensional cases. We use two axes, one for the
velocity (v) and the other for time (t). We then plot the velocity of the particle at
particular times and see what happens to velocity.
Uniform motion, const. velocity and
Zero acceleration
v
t
Uniform acceleration
Uniform deceleration
Example of x(t) graphs
•  The x(t) graph on the left has a curved part (ab), a straight part (bc) and again a
curved part in the opposite direction (cd). How is the corresponding v(t) graph, i.e.
the velocity as a function of time, is going to look?
x
c
v
d
?
b
a
t
t