Newton s Law of Universal Gravitation

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Sec. 4.1 The Law of Universal Gravitation Sec. 4.2 The Universal Gravitational Constant, G. What's all this universal" stuff?Prior to Isaac Newton, it was generally believed that gravity (the falling of objects) was something that only occurred on Earth.Newton proved that gravity extends indef

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Newton s Law of Universal Gravitation

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1. Chapter 4 Newton’s Law of Universal Gravitation

2. Sec. 4.1 The Law of Universal Gravitation Sec. 4.2 The Universal Gravitational Constant, G What’s all this “universal” stuff? Prior to Isaac Newton, it was generally believed that gravity (the falling of objects) was something that only occurred on Earth. Newton proved that gravity extends indefinitely out into the universe.

3. To the Moon Newton realized that the same force that causes an apple to fall could explain how the Moon orbits the Earth. Yeah, I’ve always wondered about the Moon. If gravity pulls on the Moon, what holds the Moon up?

4. Gravity and the Moon This is the wrong question to ask. The Moon is moving and wants to travel in a straight line (Newton’s 1st Law). What prevents the Moon from flying away? The Moon is moving and falling.

5. The Moon “falls into an orbit”. Like a baseball curving downwards as it flies over the field, the combination of its forward velocity and the downward acceleration due to gravity give a curved path.

6. The Moon is Falling! Objects near the surface of the Earth are all pulled downward with an acceleration of 10 m/s2. Given the size and period of the Moon’s orbit, Newton worked out that the acceleration due to gravity where the Moon was was about 4000 times less. This and other information allowed Newton to work out an exact formula for gravity.

7. Newton’s Law of Gravity All objects attract each other with a force given by F = G m1 m2 / d2 where F = force G = gravitational constant m1, m2 = masses of the two objects d = distance between centers of objects

8. G Newton proved that the formula worked but he didn’t know the value of G. That was later determined by Cavendish. G = .000 000 000 066 7 N m2 / kg2 = 6.67 x 10-11 N m2 / kg2 Don’t confuse G with g (g = 9.8 m/s2), let’s clarify how they are related.

9. The gravitational force exerted by the Earth on an object of mass m near its surface is (using Newton’s law of gravity): F = G ME m / RE2 (ME, RE are mass and radius of the Earth). The combination of factors (G ME / RE2) is equal to 9.8 m/s2 = g. So a shortcut for calculating the gravitational force (weight) acting on a body is F = m g.

10. The story of g Suppose the only force acting on an object is the gravitational force from the Earth (so F = mg). That object will accelerate according to Newton’s second law (F = ma), giving mg = ma or a = g All objects at the surface of the Earth will accelerate at g = 9.8 m/s2 if acted on solely by gravity.

11. Sec. 4.3 Gravity and Distance: The Inverse-Square Law I may have already done this and it is common usage so be warned: ‘particles’ or ‘objects’ are often called “masses”. The strength of the gravitational pull between two objects depends on their masses and their separation distance.

12. Gravity: masses and distances The bigger the masses are, the greater the gravitational force; if both masses were doubled, the gravitational force would increase by a factor of four. The greater the distance between masses, the smaller the force; if the distance between two masses is doubled, their gravitational attraction is four times less.

13. Dependence on Distance Why four times less? Because in the gravitational force law (F = Gm1m2/r2), we divide by the distance squared. 6 is twice what 3 is, 62 = 36 is four times more than 32 = 9. So doubling r will make r2 four times larger.

14. Inverse-Square Law Because we multiply by the inverse of the distance squared, we call this type of dependence on distance an “inverse-square” law. Okay, but why does gravity obey an inverse square law?

15. 1/r2 is very common Electrical forces, light intensity, sound intensity, spray paint, and many other things all obey inverse-square laws. Inverse square behavior is actually the normal and natural behavior for anything that spreads out into space.

16. The spray paint spreads out into space. After a distance of 1 the paint will all hit one square. At a distance of 3 it will land on 9 squares, and after a distance of 4, 16 squares.

17. Spray Paint Density The further away we go, the less dense the paint will be - same amount of paint spread over a greater area. Moving the screen twice as far away (1 to 2 or 2 to 4) meant four times more area (1 to 4 or 4 to 16). Three times further, 9 times more area.

18. Inverse Paint Law The darkness or strength of the paint obeys an inverse square law, the further the target is from the can, the less dense the paint will be on the target, following an inverse square law. Light, like from a light bulb, spreads out into space and also obeys an inverse square law (1/4th the illumination when twice as far from the source of light).

19. 3-D Space Causes Inv-Square Gravity, sounds, electric forces all spread out into space and weaken. They all obey inverse-square laws. Not everything spreads out into space (e.g. laser beams, water waves), those things do not obey inverse square laws.

20. Sec. 4.4 Weight and Weightlessness You feel weightless when in free fall. Like momentarily on a roller coaster. Or an astronaut in an orbiting shuttle.

21. What do we mean by weight? In chapter 1, we defined weight to be the force of gravity from a planet (usually the Earth) acting on a body. Fine, but then shouldn’t “weightless” mean a situation when the weight is zero? That’s a problem, because in none of the “weightless” examples we just had was the weight equal to zero.

22. No gravity? The gravitational force decreases with distance but does not go to zero anywhere. The astronaut in orbit is pulled with nearly as much gravitational force (about 97%) as when he or she is standing on the Earth. Gravity is acting there! It is causing the astronaut to fall and move in an orbit.

23. Free Fall So, is nothing “weightless”? The better term is free fall. Free fall is when an object is acted on by gravity (its weight) only. Objects in free fall are those that have the sensation we commonly call weightless.

24. What to do about weight? I think we should continue to use “weight” as the force of gravity from the Earth. And try to banish the improper use of “weightless” (and the equally bad “micro-gravity”). But our text takes an alternate route, one that I haven’t seen used elsewhere.

25. Redefinition of Weight Our text chooses to re-define weight as follows: Your weight is the support force acting on you. (Specifically, the force something under your feet – like a scale – would exert, or what the scale would read.) So “weightless” would mean a situation in which no support forces act on you, only gravity is acting on you. Now weightless does mean the same as free fall.

26. Very Confusing, I Know So now, according to the text, the force of gravity is not the same as weight. When you let go of a ball, its weight drops to zero and it falls down. That’s not what I’m used to and it is not the usage you’ll see elsewhere. This is very tricky and confusing so we need to spend extra time talking about this.

27. You Don’t Feel Gravity Because gravity affects every molecule in your body, you don’t feel it. Really, you don’t! When you are in free fall, every molecule is accelerating the same, you feel weightless. Yes, you feel different, but you are not feeling any direct consequence of gravity.

28. You Do Feel Support Forces Support forces are contact forces with your surroundings. They push against only one part of your body. When standing, you feel the support force on your feet. When sitting, on your butt.

29. Weight = support force So, the text’s equating weight with the support forces acting on you does make it agree with what you “feel” your weight is. But I still don’t like it. What I want is for you to understand gravity and the like so well that you can understand what anyone else might mean no matter how they use (or misuse) gravity and weight.

30. Example: Standing Normally Me: Have weight, not in free fall. Weight and support force at feet balance. Text: Gravity acts and is balanced by support force. Have weight because of support force.

31. Example: Skydiver Me: Always has weight (force due to gravity). Never weightless. In free fall during fall (but not once air-resistance forces build up). Text: Gravity always acts. Weightless and in free fall during early moments of the fall but not once air-resistance forces build up.

32. Example: Astronaut in Orbit Me: Has weight (same as force due to gravity), not weightless. Is in free fall. Text: No weight, does feel force due to gravity, weightless (and in free fall).

33. Example: Elevator accelerating upwards Me: Person in elevator has normal weight, contact (support) force with floor is more upward than weight is downward. Text: Greater than normal weight, upward weight/support force exceeds gravitational force downward explaining upward acceleration.

34. Example: Accelerating Spaceship Spaceship accelerating “upwards” but located far from any planets. Me: No weight, weightless, no force due to gravity, only support force acting. Text: Has weight (same as support force of floor), not weightless, no force due to gravity.

35. Artificial Gravity The accelerating spaceship can cause a support force due to the floor. Rotating spacecraft can generate the same effect. This is called artificial gravity or simulated weight. The text would consider this as real weight.

37. Sec. 4.5 Tides If we are running behind, section 4.5 will be skipped. The Moon pulls on the Earth with gravity. But the Earth isn’t one single thing, it’s rocks, dirt, lakes, etc.

38. Gravity weakens with distance The Moon pulls on these things differently because the distances vary. [And because the masses vary? No, that’s not important because while heavier objects feel more gravitational force, they also have more inertia, the accelerations would be the same regardless of mass as long as the distances are the same.]

39. The Moon pulls more strongly (causes greater accelerations) on the parts of the Earth that are closer to the Moon. What happens if you pull more strongly on one side of an object than the other? It stretches, like this:

40. The Moon Stretches the Earth It is mostly just the water (oceans) on the Earth that stretch. This causes high tides on both the side towards and the side away from the Moon.

41. The Sun also stretches the Earth For the same reason, the force (acceleration) from the Sun on the near side is more than that on the far side of the Earth. The Moon causes bigger tides than the Sun because the Moon makes up for its smaller mass by being much closer.

42. When the Sun, Earth, and Moon are in a line, the tides on Earth are larger (Sun and Moon work together). Called “spring tides”, because the water springs up higher, not anything to do with the spring season. Spring Tides

43. Neap Tides When the Sun-Earth and Moon-Earth lines are perpendicular, the stretchings from the Sun and Moon work against each other. Tides are smaller. Called “neap tides” (not sure why).

44. Tidal Frequency The side of the Earth towards the Moon (and the side away) have high tides. The Earth rotates, so different parts of the Earth experience high and low tides. The rotation of the Earth (and Moon’s orbital motion) cause each location to have a high tide about once every 12 1/2 hours.

45. Question We are made mostly of water (about 60%). So, during high tide, are we stretched taller? And, during low tide are we fatter? Yes, measurably so. Yes, but negligibly small. No, because the water in our bodies is not free to flow.

46. Answer: (b) If our heads (or feet) are closer to the Moon than our feet (heads), there will be a stretching effect. The text does some calculations to estimate this and finds that it is incredibly small, in fact tides induced by nearby buildings, people, apples, etc. will cause much more stretching than the Moon.

47. Sec. 4.6 Universal Gravitation The text tells the interesting story about the discovery of Neptune. Actually, the story is even more interesting than what the text tells. Let’s start with the standard story.

48. What’s Uranus Doing? The planet Uranus was discovered in 1781, but subsequent observations showed that it was not following exactly the path expected of it. Even accounting for the various pulls on it by other planets. Maybe due to an unknown 8th planet?

49. Explaining Uranus English Astronomer: J. C. Adams French Astronomer: Urbain Leverrier Both attempted to calculate the position and mass of an 8th planet that could explain the observed motion of Uranus using Newton’s Laws. Leverrier succeeded, Adams didn’t.

50. Calculator beats Telescope Neptune has been described as the planet that was discovered by mathematics. Shortly after receiving Leverrier’s calculated location for Neptune, an astronomer colleague found it. It really shows the power of Newton’s Laws.

51. English Claims The English claimed that Adams had made the same correct calculation but that others had been slow to actually search for the planet. Adams and Leverrier shared credit for the discovery.

52. The Truth The truth has only recently been revealed. Adams never did the correct calculations. English scientists hid this fact to protect Adam’s reputation (and their own pride). Leverrier should be given sole credit for the discovery (maybe he already has but most textbooks, like ours, don’t yet reflect this).

53. Another one bites the dust That’s the end of chapter 4. Uh oh, that means it’s time for another exam.

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