Matter and Motion: Ancient View
*world and human race had always existed and
continue to exist indefinitely (Aristotle)
*emphasis on natural philosophy, the foundations
of SCIENCE
*geocentric theory of the universe - earth-centered
*universe divided into two domains
-celestial (eternal and perfect beings)
-terrestrial (temporal and corruptible things
*uniform circular motion (Plato)
*four elements: earth, wind, air, fire on
earth
*one element: ether in celestial world
Ptolemy’s Geocentric Model
*cosmological model of the universe
- outermost sphere - that of the stars
- immobile earth as center of the
universe
- celestial body in uniform motion on
each sphere. Rates of motion differ
* geometric model of the universe shown
in next slide
Ptolemy’s model con’t.
* model fails to explain astronomical
observations like:
- changes in shapes, brightness,
speeds of bodies, distances and
*retrograde motion of planets
- where the sun, the moon and the
planets move with respect to the
stars from west to east but at times
seem to move backward, i.e., from
east to west.
e.g. retrograde motion of Mars (Zeilik)
Ptolemy’s model con’t.



To “save” the model
the Greeks proposed
Epicycle-deferent
system
here the planet was
assumed to describe,
with uniform motion,
a circle called an
epicycle. whose
center, in turn, moved
in a larger circle
concentric with the
earth and called
deferent. The path of
the planet is an
epicycloid.
 The geocentric model
survived for about
1000 years influenced
by Aristotelian ideas
of motion, religion, its
common-sense appeal
and Platonic doctrine
of philosophical truth.
COPERNICUS and the Heliocentric
Model of the Universe
*there is no one center of all celestial
spheres
*the center of the universe is the sun and
the planets move around it
*the earth is only the center of gravity and
the lunar sphere
*planets are arranged outward from the sun
*retrograde motion of planets consequence of relative motion of the earth
with respect to other planets
Copernicus con’t.
-faster moving planet soon “catches up”
with the outer planet and eventually
overtakes it. Outer planet appears to move
in the reverse direction relative to the stars
* The Copernican theory
a) conforms to Platonic view of circular
motion;
b) was against the prevailing religious
dogma; and
c) took about 100 years before its
acceptance.
Kepler’s laws of planetary motion
* Tycho Brahe - made precise and
accurate observations of apparent
planetary positions
I. Law of orbits - a planet moves in
an ellipse aaround the sun
II. Law of areas - planets sweep out
equal areas in equal times
III. Law of periods - the square of
the period of the planet divided by the
cubes of its average distance from
sun is constant
The law
of orbits
Law of
areas
Law of Periods
2 = k r3
k = 42/GMs
= 2.97 x 10-19 s2/m3
Ms = 1.99 x 1030 kg
-11
2
2
G = 6.67 x 10 N-m /kg
* Aristotle’s cosmological and motion theories
- the universe has four elements
- motion of an object depends on its most
predominant elemental component
- these elements are terrestrial in nature
-a body with heavier mass falls to the ground
first compared to that of a body with lighter
mass
* Galileo Galilei
- known as the father of experimental physics
-his contributions wer
* in astronomy
o spots on the sun and mountains on the
moon
o Venus and Mercury have phases like
the moon
Galileo con’t.
o four moons circled the universe
*popularized the Copernican system
*in mechanics
o concept of mass
o problems in pendulum motion
o uniform motion in straight line
o free falling bodies
o composite motion (projectile)
Galileo’s contributions con’t.
o foundations of the science of dynamics study of the laws of motion
o invention of the pendulum - precursor of
the “pulsometer”
* the period of the pendulum is independent
of the “amplitude” (the solution required
calculus which was later invented by
Newton)
* for a given length of the string, the period
of oscillation is the same & independent of
the mass of the bob attached at the end of
the string (the solution provided by
Einstein’s general theory of relativity)
Galileo’s contributions con’t.
* the motion of a pendulum is a special case of the
fall caused by the force of gravity.
* his observations were in conflict with the
generally accepted opinion of Aristotelian
philosophy according to which heavy objects fall
down faster than light objects.
o the laws of fall
- Galileo used a water clock in which time was
measured by the amount of water pouring out
through a little opening near the bottom of a large
container. The time it takes for a ball to roll down a
certain distance down an inclined plane was
measured using such a water clock.
* the steeper the plane, the corresponding
distances covered during the same time intervals
became longer but the ratios remained the same,
i.e. 1:3:5:7, etc.
* conclusion: in the limiting case of free fall, the
same law must hold
+ the total distance covered during a certain period
of time is proportional to the square of that time
(SQUARE LAW)
-----according to this law, the total distance covered
at the end of consecutive time intervals will be 12,
22, 32, 42, etc. or 1, 4, 9, 16, etc. The distance
covered during each of the consecutive time
intervals will be: 1, 4 - 1 = 3, 9 - 4 = 5, 16 - 9 = 7, etc.
conclusion: the observed dependence of distance
traveled on time, led Galileo to conclude that the
velocity of that motion must increase in simple
proportion to the time.
+ Nowadays, we call this: law of uniformly
accelerated motion; where
velocity = acceleration x time
and distance = (1/2) acceleration x time2
*For free fall, the acceleration is denoted by
g (for gravity) and has a value
g = 9.8 m/sec2 or 981 cm/sec2 or 32.2 ft/sec2
o idea of composite motion
- e.g. two-dimensional projectile motion
o construction of the first astronomical
telescope
Newton’s Principia
Definition I. The quantity of matter (mass) is
the measure of the same, arising from its
density and bulk (volume) conjointly.
*Nowadays we say that the mass of any given
object is the product of its density and its volume.
This defines the notion of mass.
Definition II. The quantity of motion is the
measure of the same, arising from the
velocity and quantity of matter conjointly.
* These days, the amount of motion (which is
simply momentum) is the product of velocity and
mass of the moving object. This defines the notion
of momentum.
Definition III. The innate force of matter, is a
power of resisting, by which every body, as
much as in it lies, continues in its present
state, whether it be of rest, or of moving
uniformly forwards in a straight line.
* force of inactivity is what we now call inertia. This
defines the notion of inertia.
Definition IV. An impressed force is an
action exerted upon a body, in order to
change its state, either of rest, or of uniform
motion in a straight line.
* such forces exist in the action only, e.g.
percussion, pressure.
Newton’s laws of motion
I. Every body continues in its state of rest,
or of uniform motion in a straight line,
unless it is compelled to change that state
by a force impressed upon it. (law of inertia)
II. The change of motion (i.e., of mechanical
momentum) is proportional to the motive
force impressed; and is made in the
direction of the right line in which that force
is impressed.(force law)
III. To every action there is always opposed
an equal reaction; or, the mutual actions of
two bodies upon each other are always
equal and directed to contrary parts. (law of
action-reaction)
Mass

…is measured in kilograms.

…is the measure of the inertia of an object.

Inertia is the natural tendency of a body to
resist changes in motion.
Force

…the agency of change.

…changes the velocity.

…is a vector quantity.

...measured in Newtons,
dynes, or foot-pounds
Newton’s First
Law

Law of Inertia

“A body remains at rest
or moves in a straight
line at a constant speed
unless acted upon by a
force.”
Newton’s First Law
 No
mention of chemical composition
 No mention of terrestrial or celestial
realms
 Force required when object changes
motion
 Acceleration is the observable
consequence of forces acting
Newton’s Second
Law
The Sum of the Forces acting
on a body is proportional to
the acceleration that the body
experiences
F  a
 F = (mass) a


F  ma
Net Force
 Fx  max
 Fy  may
 Fz  maz
Newton’s Third
Law

Action-Reaction

For every action force
there is an equal and
opposite reaction force
Weight

The weight of an object FW is the
gravitational force acting downward on the
object.

FW = m g
Tension
(Tensile Force)

Tension is the force in a string, chain or
tendon that is applied tending to stretch it.

FT
Normal Force

The normal force on an object that is being
supported by a surface is the component of
the supporting force that is perpendicular to
the surface.

FN
Coefficient of
Friction
 Kinetic
Friction
• Ff = mk FN

Static Friction
• Ff  ms FN

In most cases, mk < ms.
SOME EXERCISES
EQUATIONS OF KINEMATICS
0
1. <v> = [v1 + v2 ] / 2,
2. v2 = v1 + a t
3. 2ax = v22 - v12
4. x = v1t + (1/2)a t2
x
acceleration a = constant
EQUATIONS OF FREE FALL
y
1. <v> = [v1 + v2 ] / 2,
2. v2 = v1 + g t
3. 2gy = v22 - v12
0
4. y = v1t + (1/2)g t2
acceleration g =
constant
g = 9.8 m/s2
= 980 cm/ s2
= 32 ft/ s2
< 0, motion upward
> 0, motion downward
FOR ROTATIONAL MOTION
 change x for 
 change v for 
 change a for 
 linear distance s = r
 speed v = r 
 acceleration a = r 
s
r 
OTHER CONCEPTS
Work and Energy Equivalence
 kinetic energy T = (1/2)mv2
 potential energy U = mgh
 total work Wt = T + U
Power = work/time
Conservation of Energy
Momentum : linear and angular
OTHER CONCEPTS con’t.
Impulse
Torque
Kinetic Energy of Rotation
Moment of Inertia
Newton’s 2nd law for rotational motion
Characteristics of a physical law
simple
 mathematical in its expression
 not exact
 universal
 invariant

THE LAW OF GRAVITATION*: AN
EXAMPLE OF PHYSICAL LAW (see
Feynman’s treatise)
*considered as “the greatest generalization
achieved by the human mind”
* two bodies exert a force upon each other
which varies inversely as the square of the
distance between them and varies directly
as the product of their masses, or in
mathematical form:
F = G mm’/r2
, G= 6.67 x 10-11 Nt-m2/kg2
Gravity Questions

The constant G is a rather small number.
What kind of objects can exert strong
gravitational forces?

If the distance between two objects in space
is doubled, then what happens to the
gravitational force between them?
Historical development:
- Copernicus’ treatise on the motion of
the planets
- The recordings of Tycho Brahe on the
positions of the planets
- Kepler’s deductions from the
observations of Tycho leading to his
three laws of planetary motion
? What makes planets go around the sun
- Galileo’s discovery of the law of inertia
- Newton’s contribution, the concept of force
- Newton’s deductions
a) from the motion of Jupiter’s satellites: the
concept of gravitational force
b) on the relation of the period of the moon’s
orbit and its distance from the earth and the
length of time for an object to fall at the earth’s
surface
c) on the shape of the orbit if the law were
the inverse square
d) the phenomena of the tides
EXPERIMENTAL VERIFICATIONS OF THE
THEORY
I. Olaus Roemer’s (Danish astronomer)
verification that the moons of Jupiter moved
in accordance with Newton’s laws; as a
consequence he was able to determine the
velocity of light
II. Adams and Leverrier - the perturbations
in the motion of Jupiter, Saturn and Uranus
were due to the existence o f another planet,
later discovered as Neptune.
III. Einstein’s modification of Newton’s laws
to explain the motion of the planet Mercury
IV. The experiment of Cavendish to
determine G = 6.67x10-11Nt-m2/kg2
V. The measurements of Eotvos and Dicke
showing that the force is exactly
proportional to the mass
VI. The inverse square law in the electrical
laws
APPLICATIONS OF THE THEORY
1. geophysical prospecting
2. predicting the tides
3. working out the motion of satellites and
planet probes sent to space
4. predicting the planetary positions
precisely
5. formation of new stars
Examples
1. Centripetal acceleration of the moon
let m = mass of moon rotating about a frame of
reference attached to m’ (earth’s mass)
Force on m: F = mv2/r ; r = distance of moon to earth
v = speed of orbit = 2r/T
T = period of orbit
F = 42mr/T2
By Kepler’s 3rd law: T2 = cr3, hence F = 42m/kr2
F ~ 1/r2
a = v2/r = 42r/T2 ; r = 3.84 x 108m , T = 2.36 x 106 s
a = 2.72 x 10-3 ms-2 and g/a = 3602 ~ (60)2
Since RE = 6.37 x 106 m
(r/RE) = (384/6.37)2 ~ (60)2 = (g/a)
2. gravitational potential energy
m'
r
F
m
v
ur
m’ is at origin of coordinates, F is attractive
The gravitational potential energy UG = - Gmm’/r
The total energy of the system of two particles subject to their
gravitational interaction is
E = T + U = mv2/2 + m’v’2 – mm’/r
= mv2/2 –mm’/r if m’  m, m’ coincides with c.m., v’ = 0
a) Case when E < 0
If m rotates around m’, then
mv2/r = -Gmm’/r2 and mv2/2 = Gmm’/2r
E = - Gmm’/2r (negative energy characteristic of elliptical
or bound orbits; T is not enough to take the
particle a t infinity)
Other cases: E > 0, E =0
escape velocity = minimum velocity of body fired from
earth to reach infinity
mve2 – Gmme/Re = 0 or ve = (2Gme/Re)1/2 = 1.13 x 104 m/s
= 4.07 x 104 km/hr
b) If E > 0, T is sufficient to overcome U and bring object
to infinity; path is hyperbolic
c) If E = 0, T = U and the path is parabolic
Applications in placing artificial satellites in orbit
REFERENCE FRAMES (see “The Inertial
Reference Frame”)
* absolute reference frame - something
which had some fundamental advantage
over all other frames (are non-existent)
* inertial reference frame (sometimes called
Lorenz reference frame)- reference frames
moving with uniform velocity with respect to
each other and with respect to the fixed
stars. Such are unaccelerated, non-rotating
reference frames. Inertial frames are
necessarily always local ones, limited in
a certain region of space-time.
e.g. freely moving space ship. A free particle
at rest in this vehicle remains at rest in this
vehicle. When given a gentle push, it
moves across the vehicle in a straight line
with constant speed. Any reference frame
that moves with constant velocity relative to
an inertial frame is itself an inertial frame.
* all inertial reference frames are equivalent
for the measurement of physical
phenomena. Observers in different frames
may obtain different numerical values for
measured physical quantities, but the
relationships between the measured
quantities, that is, the laws of physics, will
be the same for all observers.
*due to its orbit around the Sun and about its own
axis, the Earth is not an inertial frame of reference.
ac = 4.4 x 10-3 m/s2 about the Sun
ac = 3.37 x 10-2 m/s2 toward Earth’s center
cf. gravity g = 9.8 m/s2 , these values are small
and can be neglected. So we can assume that
a set of points on the Earth’s surface constitutes
an inertial frame.
* all motion is relative to a frame of
reference. The choice of a frame of
reference for reckoning motion depends on
the situation. e.g. moving car, an electron in
an atom, a moving planet, etc.
THE GALILEAN TRANSFORMATIONS
* the positions and velocities as measured
by two observers in relative motion are
correlated; the time measurements are the
same.
EINSTEIN’S THEORY OF RELATIVITY
ENERGY AND CONSERVATION LAWS
1. conservation of charge
(electricity and magnetism)
2. conservation of energy
(kinetic , gravitational potential, elastic, heat,
chemical, electrical, magnetic)
3. conservation of linear momentum
(equilibrium conditions)
4. conservation of angular momentum
(torque, rotational motion)
5. conservation of baryons
(origin of universe)
6. conservation of leptons