Matter and Motion: Ancient View

1. 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, fireon earth *one element: etherin celestial world

2. Ptolemy’sGeocentric 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 universeshown in next slide

4. 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 planetsmove 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)

6. 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.

9. COPERNICUSand theHeliocentric 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

10. 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.

12. 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

13. The law of orbits Law of areas

14. Law of Periods 2 = k r3 k = 42/GMs =2.97 x 10-19 s2/m3 Ms = 1.99 x 1030 kg G = 6.67 x 10-11 N-m2/kg2

15. * Aristotle’scosmologicalandmotion 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 astronomyo spots on the sun and mountains on the moono Venus and Mercury have phases like the moon

16. 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)

17. Galileo’s contributions con’t.ofoundations of the science of dynamics - study of the laws of motionoinvention 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 ofthe bob attached at the end of the string (the solution provided by Einstein’s general theory of relativity)

18. 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.othe 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.

19. *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 thatthe velocity of that motion must increase in simple proportion to the time.

20. + Nowadays, we call this: law of uniformly accelerated motion; wherevelocity = acceleration x timeand distance = (1/2) acceleration x time2*For free fall, the acceleration is denoted by g (for gravity) and has a valueg = 9.8 m/sec2 or 981 cm/sec2 or 32.2 ft/sec2oidea of composite motion - e.g. two-dimensional projectile motionoconstruction of the first astronomical telescope--------------------Reference : Galileo by George Gamow

21. Newton’s PrincipiaDefinition 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.

22. 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 presentstate, 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. This defines the notion offorce.

23. Newton’s laws of motionI.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 ofaction-reaction)

24. 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.

25. Force • …the agency of change. • …changes the velocity. • …is a vector quantity. • ...measured in Newtons, dynes, or foot-pounds

26. 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.”

27. 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

28. Newton’s Second Law The Sum of theForcesacting on a body is proportional to the acceleration that the body experiences • F a • SF = (mass) a

29. Net Force

30. Newton’s Third Law • Action-Reaction • For every action force there is an equal and opposite reaction force

31. Weight • The weight of an object FW is the gravitational force acting downward on the object. • FW = m g

32. Tension (Tensile Force) • Tension is the force in a string, chain or tendon that is applied tending to stretch it. • FT

33. 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

34. Coefficient of Friction • Kinetic Friction • Ff = mk FN • Static Friction • Ff ms FN • In most cases, mk < ms.

35. SOME EXERCISES

36. EQUATIONS OF KINEMATICS x 0 • <v> = [v1 + v2 ] / 2, acceleration a = constant • v2 = v1 + a t • 2ax = v22 - v12 • x = v1t+ (1/2)a t2

37. EQUATIONS OF FREE FALL y • <v> = [v1 + v2 ] / 2, acceleration g = • constant • 2. v2 = v1 + g t • 3. 2gy = v22 - v12 • 4. y = v1t+ (1/2)g t2 g = 9.8 m/s2 = 980 cm/ s2 = 32 ft/ s2 < 0, motion upward > 0, motion downward 0

38. FOR ROTATIONAL MOTION • change x for  • change v for  • change a for  • linear distance s = r • speed v = r  • acceleration a = r  s r 

39. 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

40. OTHER CONCEPTS con’t. Impulse Torque Kinetic Energy of Rotation Moment of Inertia Newton’s 2nd law for rotational motion

41. Characteristics of a physical law • simple • mathematical in its expression • not exact • universal • invariant

42. 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

43. 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?

44. 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 ofplanetary motion

45. ? What makes planets go around the sun- Galileo’s discovery of the law of inertia- Newton’s contribution, the concept of force- Newton’s deductionsa) from the motion of Jupiter’s satellites: the concept of gravitational forceb) on the relation of the period of the moon’s orbit and its distance from theearth and the length of time for an object to fall at the earth’s surface

46. c) on the shape of the orbit if the law were the inverse squared) the phenomena of the tidesEXPERIMENTAL VERIFICATIONS OF THE THEORYI. 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

47. 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 MercuryIV. The experiment of Cavendish to determine G = 6.67x10-11Nt-m2/kg2V. The measurements of Eotvos and Dicke showing that the force is exactly proportional to the massVI. The inverse square law in the electrical laws

48. APPLICATIONS OF THE THEORY1. geophysical prospecting2. predicting the tides3. working out the motion of satellites and planet probes sent to space4. predicting the planetary positions precisely5. formation of new stars

49. Examples • 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)

50. 2. gravitational potential energy F m m' ur r v 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)