Chapter 11 Newton

1. When, in the year of Galileo's death, Newton, the mightiest of the sons of light, Was born to lift the splendor of this torch And carry it, as I heard that Tycho said Long since to Kepler, 'carry it out of sight, Into the great new age I must not know, Into the great new realm I must not tread." Chapter 11 Newton

2. Newton extended Galileo’s Law of Inertia "A body remains in its state of rest or uniform motion in a straight line, unless it is compelled to change that state by an outside force acting on the body." What is force? Force is the agency to realize a change in the state of motion, i.e. acceleration. Precise definition of Force was vague up to the arrival of Newton

3. a a F The more the mass, the more the resistance to any change in velocity, i.e., the more the resistance to acceleration. a = F/m Put together, the two concepts of force and mass spell: F = ma Key to the mechanical universe F = ma unifies all motion a a 1/m Force causes a change in the state of motion: acceleration. When the same force is applied to different objects, the acceleration is different. What makes acceleration different is a unique property of the body, Newton called it mass.

5. Force generated by fan motor on trolley (demo) Acceleration = Force/mass Increase mass Acceleration decreases as expected from a = F/m Force is a vector : direction and magnitude

6. Mass is an intrinsic property of all matter It unifies the motion properties of all matter Mass resists any change in the state of motion Mass is not weight - Weight is a force - the force with which the earth attracts a body = F = mg Unit of mass = kg Unit of Force = kg x m/sec2 = newton What is the force of "attraction of the earth" (gravity) on an object of mass = 1kg. m = 1kg a = g = 9.8 m/sec2 F = ma = 1 x 9.8 kg m/sec2 = 9.8 newtons

7. The meaning of mass can be understood from its relationship to • (a) how much force is needed to stop a moving object • (b) how much force is needed to accelerate an object • (c) how much force is needed to change the direction of an object moving at constant speed • (d) all of the above.

8. A body of the same mass can have different weights weight at equator is slightly different from weight at poles, why? Because the acceleration is different But mass is the same Later we will see how Your weight on the moon is less than your weight on earth Or Weight in orbiting satellite can be zero, but mass is the same F = mg F = (g – 0.0338) ---- Weigh Less at the Equator!!!

9. F = ma - The Key to the Mechanical Universe Newton used the same key in two ways First method: a = F/m If you know the force(s), you can deduce the motion - acceleration = a > velocity vs. time > distance vs. time, time taken to travel a particular distance, or the period of oscillation Vf = Vi + at Vf2 = Vi2t = 2ad d = vit + (1/2)at2 Example - Equilibrium You know that F = 0 (Also definition of equilibrium) => a = 0 => velocity is constant = v, => d = vt

10. Reaction = mg Weight = mg If you do not know the force, you can deduce the force from the motion. Example: We can deduce the existence of a Reaction Force a= 0  F net = 0 F1 F 1 = mg F net = F 1 + R = 0  R = -F 1

11. Look at gravity force, earth on mass Increase mass Acceleration stays the same! (remember Galileo at Pisa) What does this tell us about the gravity force? Earth’s gravity force also increases with mass Fgrava mass…… different from F= ma

12. Force generated by fan motor on trolley (demo again) Acceleration = Force/mass Increase mass Acceleration decreases as expected from a = F/m

13. As if the donkey pulls harder to provide the same acceleration knowing that the farmer doubles the load

14. Force of gravity on apple a (proportional) mass of apple F gravitya m F gravity = K m (K = constant) Apply F = ma (Newton’s Law of Motion) F = ma = F gravity = Km => ma = Km = > A = K Acceleration is constant. Constant acceleration motion revealed a fundamental property of gravity to Newton F gravitya m Different from F = ma!!!

15. Did Newton, dreaming in his orchard there Beside the dreaming Witham, see the moon Burn like a huge gold apple in the boughs And wonder why should moons not fall like fruit? Or did he see as those old tales declare... A ripe fruit fall from some immortal tree Of knowledge, while he wondered at what height Would this earth-magnet lose its darkling power? Would not the fruit fall earthward, though it grew High o'er the hills as yonder brightening cloud? Would not the selfsame power that plucked the fruit Draw the white moon, then, sailing in the blue? Then, in one flash, as light and song are born, And the soul wakes, he saw it-this dark earth Holding the moon that else would fly through space To her sure orbit, as a stone is held In a whirled sling; and, by the selfsame power, Her sister planets guiding all their moons; While exquisitely balanced and controlled In one vast system, moons and planets wheeled Around one sovran majesty, the sun.

17. The earth exerts the force of gravity on a falling apple. Any object on the surface of the earth experiences a downward force of magnitude mg. How far does this force of gravity extend? Could it possibly stretch all the way up to the moon? Newton was sitting in his garden at home contemplating such questions when he had a brilliant insight - a pivotal event for physics If the moon goes around the earth in circular motion, there must be a centripetal force acting on the moon, a force directed toward the center of the earth. Could that centripetal force be the very same attraction of the earth? Could it be gravity which attracts the apple to the center of earth? Was the centripetal acceleration = g= 9.8 m/sec2 ? He was eager to try out the numbers.

18. = 1021 m/sec am = m/sec2 Earth rotation speed = 464 m/s With this data Newton had the numbers to calculate the centripetal acceleration of the moon Much smaller than the value of g = 9.8 m/sec2

19. At this point a lesser mind would have given up the idea! But Newton saw an interesting pattern in the numbers.. Recognizing mathematical patterns in nature…again

20. 9.81/0.00271 = 3609 ≈ 3600 There is a nice relationship between 60 and 3600 => The moon’s acceleration is 602 times smaller than the apple’s acceleration. a m = g/3600 Why? Because the earth’s attractive force on the moon must be 602 times weaker, since the moon is 60 times farther away from earth-center than the apple is = F moon = Mmoon g/3600 The force of gravity decreases with the square of the distance

21. If the moon were at the surface of the earth, the earth would attract the moon by a much larger force, larger by a factor of 602

22. The reason why the moon orbits the earth is because the earth attracts the moon with the force of gravity…but the force is weaker It is the same reason why the apple from the tree falls to the ground - to the center of the earth. The moon also falls to the earth! Why does it not fall all the way to the earth? Newton's answer: As the moon moves horizontally, it falls just enough toward the earth to follow the curvature of the earth. The moon is forever falling, but its horizontal motion makes it move in a curved path, which happens to be a circle (nearly).

23. Calculate how far an apple will fall in one second. Use d = (1/2)gt2 (g = 9.8m/sec2). Then calculate how far the moon will “fall” in one hour, keeping in mind that the acceleration of the moon is much less than g. Compare the two answers. A) The apple falls much farther, since it is close to earth. B) The moon falls much farther because of the longer time C) They both fall about the same distance. D) There is not enough information given. In one second the apple will fall (1/2)gt2 = (1/2)9.8 (1) 2 = 4. 9m = 5m In one second the moon will fall (5/(60) 2) = 5/3600 = 1.4mm In 3600 seconds the moon will fall 5m Gravity force of earth on moon is diluted by 602 = 3600

24. No bouts of ecstasy! Newton’s explanation that the moon is “falling” to the earth in just the same way as the apple falls to the earth rested on an important assumption: The earth and the moon are far from each other so it is reasonable to treat both the earth and the moon as single points when considering the force that the earth exerts on the moon. For the apple the earth behaves as if it exerts its attractive force from a single point at the center of the earth.

25. Spherical Symmetry

26. Newton’s Two Crucial Insights in Understanding the Motion of the Moon 1) The attractive force of gravity which is responsible for the apple falling to the earth is also responsible for the moon’s orbit around the earth. • The same force applies to motion of earthly bodies as to motion of a heavenly body. • It was not the first time that someone had perceived unity between heaven and earth. • Galileo: the moon has mountains  the moon is made of the same stuff as the earth 2) The force of gravity decreases as 1/R2 - A mathematical description for the force of gravity Now Could the force of gravity be also responsible for the motion of the planets around the sun?

27. Newton could prove from Kepler’s 3rd Law : The inverse square law of gravitational attraction. Here is his proof for a circular orbit: If each planet moves in a circular orbit The acceleration for this circular motion is produced by a force between the sun and the planet. The force is directed toward the center of the circle, i.e. the sun.

28. Ah! so the earth and everything on it does fall toward the center of the sun. Just as the moon’s circular motion is equivalent to falling into the earth. The earth is also forever falling into the sun, but its horizontal motion makes it move in a curved path, which happens to be a circle (nearly).

29. If the sun attracts the planets (like earth) with “a force of gravity”: Does that force of gravity also decrease as 1/R2 Yes! Newton could prove this from Kepler’s 3rd Law and much more. Again, we see how a new theory embraces what is previously understood, and guides us to new understanding.

30. Newtonian Synthesisof Celestial and Terrestrial Motion

32. Kepler's harmonic law [R3/T2 = constant] seems mysterious, but Newton established that it is the same as the inverse square law of gravitational attraction. The force which the sun exerts on each planet is the same force of gravity that Makes an apple fall to the ground from a tree Makes the moon go round the earth Makes planets go around the sun Byron: When Newton saw an apple fall, he found In that slight startle from his contemplation A mode of proving that the earth turn’d round [the sun] In a most natural whirl, called “gravitation”; And this is the sole mortal who could grapple, Since Adam, with a fall, or with an Apple. The force of gravity Varies as 1/R2. Varies as mass of the object it acts on (apple, moon, planet)

33. “And God said let there be a firmament in the midst of the waters, and let it divide the waters from the waters. And God made the firmament, and divided the waters which were under the firmament from the waters which were above the firmament.” -Genesis Von Carolsfeld

34. Artists always separated heaven from earth

35. Corregio, Heaven and Earth1526-30

36. St. Ignazius Church(1770)

37. Pietro da Cortona. Glorification of the Reign of Urban VII

38. If the force of gravity between the sun and an orbiting object varies with distance as 1/R2, then the orbit of the object could be • (a) a circle • (b) ellipse • (c) parabola • (d) any of the above

39. + Newton’s Law of Motion: a = F/m (Newton’s Second Law) Together: Reveal the mechanics of the universe Grander Synthesis Newton’s Law of Gravitation M1 M2 R 12 1 2

40. Deeper Insight…Mutual Attraction We have discussed the -1st Law of Motion (principle of inertia…Galileo) -And the 2nd Law of Motion: F = m a Now we come to Newton’s 3rd Law of Motion, the third pillar of mechanics. -A single particle, or a single body by itself, can neither exert nor experience any force at all. -Forces emerge only as a result of interactions between two or more entities. Whenever there is an interaction between bodies (A and B) The force exerted by body A and on body B is equal and opposite to the force exerted by body B on body A.

41. Demo

42. Symmetry Sun attracts planets with gravity force : Planets attract sun with SAME gravity force, and opposite in direction Asymmetry

43. If the earth attracts the apple with a certain force, then the apple attracts the earth with the same force. So, why does the earth not accelerate toward the apple? It does, but the acceleration of earth is much smaller, because the mass of earth >> mass of apple. Demo If the earth attracts the moon with a certain force then the moon attracts the earth with the same force.... (One of the causes of Tides ! - later)

44. F grav-on-applea m apple F grav-on-eartha m earth F grav- on-suna m sun Symmetry Commands: F grav-on-earth = - F grav-on-sun How can that be?? masssun >> massearth F on-eartha m earthm sun F on-sunam sunm earth

45. Put it all together F grava m planet m sun (1/ Rsun-planet2) Is this true for earth-moon – Yes Is this true for earth-apple – Yes Generalize to mass1 (m) and mass2 (M). Replace proportionality by constant:

46. The sublime principle of universal gravitation: “All matter moves as if every particle attracts every other particle with a force proportional to the product of their masses, and inversely proportional to the distance between them. That force is universal gravitation.” Poet Francis Thompson captures the universality and reciprocal symmetry of gravity: “All things by immortal power Near or far Hiddenly To each other linked are That thou canst not stir a flower Without the troubling of a star.” This was the first synthesis.