Matter and Motion: Ancient View *world and human race had always existed and continue to exist indefinitely (Aristotle) *emphasis on natural philosophy,

Matter and Motion: Ancient ViewMatter 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

PtolemyPtolemy’s’s Geocentric Model 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

PtolemyPtolemy’s’s model model con’t. 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)

PtolemyPtolemy’s’s model model con’t. 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.

COPERNICUSCOPERNICUS and theand the Heliocentric Heliocentric Model of the UniverseModel 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

CopernicusCopernicus con’t. 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 motionKepler’s laws of planetary motion* * Tycho BraheTycho Brahe - - made precise and made precise and accurate observations of apparent accurate observations of apparent planetary positionsplanetary 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 = 42/GMs

= 2.97 x 10-19 s2/m3

Ms = 1.99 x 1030 kgG = 6.67 x 10-11 N-m2/kg2

* * AristotleAristotle’s’s cosmologicalcosmological andand motion theoriesmotion theories - the universe has four elements - the universe has four elements- motion of an object depends on its most - motion of an object depends on its most predominant elemental componentpredominant elemental component- these elements are terrestrial in nature- these elements are terrestrial in nature-a body with heavier mass falls to the ground -a body with heavier mass falls to the ground first compared to that of a body with lighter first compared to that of a body with lighter massmass* * Galileo GalileiGalileo Galilei- known as the father of experimental physics- known as the father of experimental physics-his contributions wer-his contributions wer** in astronomy in astronomy

oo spots on the sun and mountains on the spots on the sun and mountains on the moonmoon

oo Venus and Mercury have phases like Venus and Mercury have phases like the moonthe moon

GalileoGalileo con’t. 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)

GalileoGalileo’s contributions con’t’s contributions con’t..oo foundations of the science of dynamicsfoundations of the science of dynamics - - study of the laws of motionstudy of the laws of motionoo invention of the penduluminvention of the pendulum - - precursor of precursor of the “the “pulsometerpulsometer””** the period of the pendulum is independent the period of the pendulum is independent of the “amplitude” (the solution required of the “amplitude” (the solution required calculus which was later invented by calculus which was later invented by Newton)Newton)** for a given length of the string, the period for a given length of the string, the period of oscillation is the same & independent of of oscillation is the same & independent of

the mass ofthe mass of the bob attached at the end of the bob attached at the end of the string (the solution provided by the string (the solution provided by Einstein’s general theory of relativity)Einstein’s general theory of relativity)

GalileoGalileo’s contributions con’t.’s contributions con’t. ** the motion of a pendulum is a special case of the the motion of a pendulum is a special case of the

fall caused by the force of gravity.fall caused by the force of gravity.

** his observations were in conflict with the his observations were in conflict with the generally accepted opinion of Aristotelian generally accepted opinion of Aristotelian

philosophy according to which heavy objects fall philosophy according to which heavy objects fall down faster than light objects.down faster than light objects.

oo the laws of fallthe laws of fall- Galileo used a water clock in which time was - Galileo used a water clock in which time was measured by the amount of water pouring out measured by the amount of water pouring out

through a little opening near the bottom of a large through a little opening near the bottom of a large container. The time it takes for a ball to roll down a container. The time it takes for a ball to roll down a

certain distance down an inclined plane was certain distance down an inclined plane was measured using such a water clock.measured using such a water clock.

** the steeper the plane, the corresponding the steeper the plane, the corresponding distances covered during the same time intervals distances covered during the same time intervals became longer but the ratios remained the same, became longer but the ratios remained the same,

i.e. i.e. 1:3:5:7, etc.1:3:5:7, etc.* * conclusionconclusion:: in the limiting case of free fall, the in the limiting case of free fall, the

same law must holdsame law must hold+ + the total distance covered during a certain period the total distance covered during a certain period

of time is proportional to the square of that timeof time is proportional to the square of that time ((SQUARE LAW)SQUARE LAW)

-----according to this law-----according to this law, the total distance covered , the total distance covered at the end of consecutive time intervals will be 12, at the end of consecutive time intervals will be 12,

22, 32, 42, etc. or 1, 4, 9, 16, etc. The distance 22, 32, 42, etc. or 1, 4, 9, 16, etc. The distance covered during each of the consecutive time covered during each of the consecutive time

intervals will be: 1, 4 - 1 = 3, 9 - 4 = 5, 16 - 9 = 7, etcintervals will be: 1, 4 - 1 = 3, 9 - 4 = 5, 16 - 9 = 7, etc..conclusionconclusion:: the observed dependence of distance the observed dependence of distance traveled on time, led Galileo to conclude thattraveled on time, led Galileo to conclude that the the velocity of that motion must increase in simple velocity of that motion must increase in simple

proportion to the time.proportion to the time.

+ Nowadays, we call this: + Nowadays, we call this: law of uniformly law of uniformly accelerated motionaccelerated motion; where; where

velocity = acceleration x timevelocity = acceleration x timeandand distance = (1/2) acceleration x timedistance = (1/2) acceleration x time22

*For *For free fallfree fall, the acceleration is denoted by , the acceleration is denoted by g (for g (for gravity)gravity) and has a value and has a value

g = 9.8 m/secg = 9.8 m/sec22 or 981 cm/sec or 981 cm/sec2 2 or 32.2 ft/secor 32.2 ft/sec22

oo idea of composite motionidea of composite motion - e.g. two-dimensional projectile motion - e.g. two-dimensional projectile motion

oo construction of the first astronomical construction of the first astronomical telescopetelescope

----------------------------------------Reference : Galileo by George GamowReference : Galileo by George Gamow

Newton’s PrincipiaNewton’s Principia

Definition Definition I.I. The quantity of matterThe quantity of matter (mass) (mass) is is the measure of the samethe measure of the same, , arising from its arising from its

density and bulkdensity and bulk (volume) (volume) conjointlyconjointly..*Nowadays we say that the mass of any given *Nowadays we say that the mass of any given

object is the product of its density and its volume. object is the product of its density and its volume. This defines the notion of mass.This defines the notion of mass.

Definition Definition IIII. . The quantity of motion is the The quantity of motion is the measure of the same, arising from the measure of the same, arising from the

velocity and quantity of matter conjointly.velocity and quantity of matter conjointly.** These days, the These days, the amount of motionamount of motion (which is (which is

simply simply momentum)momentum) is the is the product of velocity and product of velocity and massmass of the moving object. of the moving object. This defines the notion This defines the notion

of momentum.of momentum.

Definition Definition IIIIII. . The innate force of matter, is a The innate force of matter, is a power of resisting, by which every body, as power of resisting, by which every body, as

much as in it lies, continues in its presentmuch as in it lies, continues in its present state, whether it be of rest, or of moving state, whether it be of rest, or of moving

uniformly forwards in a straight line.uniformly forwards in a straight line.

** force of inactivity is what we now call force of inactivity is what we now call inertiainertia. This . This defines the notion of inertia.defines the notion of inertia.

Definition Definition IVIV. . An impressed force is an An impressed force is an action exerted upon a body, in order to action exerted upon a body, in order to

change its state, either of rest, or of uniform change its state, either of rest, or of uniform motion in a straight line.motion in a straight line.

* such forces exist in the action only, e.g. * such forces exist in the action only, e.g. percussion, pressure. percussion, pressure.

This defines the notion ofThis defines the notion of forceforce..

Newton’s laws of motionNewton’s laws of motionI.I. Every body continues in its state of rest, Every body continues in its state of rest, or of uniform motion in a straight line, or of uniform motion in a straight line, unless it is compelled to change that state unless it is compelled to change that state by a force impressed upon it. by a force impressed upon it. ((law of inertialaw of inertia))II.II. The change of motion The change of motion (i.e., of mechanical (i.e., of mechanical momentum) momentum) is proportional to the motive is proportional to the motive force impressed; and is made in the force impressed; and is made in the direction of the right line in which that force direction of the right line in which that force is impressed.is impressed.((force lawforce law))III.III. To every action there is always opposed To every action there is always opposed an equal reaction; or, the mutual actions of an equal reaction; or, the mutual actions of two bodies upon each other are always two bodies upon each other are always equal and directed to contrary parts. equal and directed to contrary parts. ((law oflaw ofaction-reactionaction-reaction))

MassMass …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.

ForceForce …the agency of change.

…changes the velocity.

…is a vector quantity.

...measured in Newtons, dynes, or foot-pounds

Newton’s First Newton’s First LawLaw

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 LawNewton’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 Newton’s Second LawLaw

The Sum of the Forces acting on a body is proportional to the acceleration that the body experiences

F a

F = (mass) a

amF

xx maF

yy maF

Net Force

zz maF

Newton’s Third Newton’s Third LawLaw

Action-Reaction

For every action force there is an equal and opposite reaction force

WeightWeight The weight of an object FW is the

gravitational force acting downward on the object.

FW = m g

Tension Tension (Tensile Force)(Tensile Force)

Tension is the force in a string, chain or tendon that is applied tending to stretch it.

FT

Normal Force 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 Coefficient of Friction Friction

Kinetic Friction• Ff = k FN

Static Friction• Ff s FN

In most cases, k < s.

SOME EXERCISES

EQUATIONS OF KINEMATICS

0 x

1. <v> = [v1 + v2 ] / 2, acceleration a = constant

2. v2 = v1 + a t

3. 2ax = v22 - v1

2

4. x = v1t + (1/2)a t2

EQUATIONS OF FREE FALL

1. <v> = [v1 + v2 ] / 2, acceleration g = constant

2. v2 = v1 + g t

3. 2gy = v22 - v1

2

4. y = v1t + (1/2)g t2

0

y

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

r

s

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 lawCharacteristics 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 QuestionsGravity 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 massVI. 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

ExamplesExamples1. 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 = 2r/T

T = period of orbit F = 42mr/T2

By Kepler’s 3rd law: T2 = cr3, hence F = 42m/kr2

F ~ 1/r2

a = v2/r = 42r/T2 ; r = 3.84 x 108m , T = 2.36 x 106 sa = 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 energygravitational potential energy

m' Fr

m

v

ur

m’ is at origin of coordinates, F is attractive The gravitational potential energy UG = - Gmm’/rThe 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’ = 0a) Case when E < 0If 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 =0escape velocity = minimum velocity of body fired from earth to reach infinitymve

2 – 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 ownaxis, 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