Planetary Astronomy Lecture 04b Gravity and Orbits Part 1 — Transcript

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1. 0:01Now we're going to see how the laws of motion and  energy apply to gravity and orbits. A good way to
2. 0:07think about what we'll be doing here is to look  at this picture: this is the international space
3. 0:13station in orbit around the Earth. The space  station travels at eight kilometers per second
4. 0:19going around the earth every hour and a half.  It's been doing this for many years now,
5. 0:25but look closely. The space station is a spindly  contraption with lots of pieces sticking out every
6. 0:31which way. There is no way you could put a rocket  on this and give it any significant push: it would
7. 0:37simply fall apart. So how does something like  this go racing around the Earth at this tremendous
8. 0:43speed for years on end? That's one of the big  questions we'll be looking at in this section.
9. 0:52We'll start by looking at the force  of gravity and learn how it behaves.
10. 0:57Next, we'll see how we can use force to  understand what an orbit is. Finally,
11. 1:03we'll see how gravity can produce tides,  and why this is important for astronomy.
12. 1:11Let's try it by dropping a soccer  ball and a truck and see what happens.
13. 1:18As you can see, they hit  the ground at the same time.
14. 1:21In fact, all bodies on Earth are pulled  towards the ground with the same acceleration:
15. 1:279.8 meters per second squared, no matter how  heavy they are. However, objects that are very
16. 1:34lightweight or have a large surface area like a  piece of paper or a feather will be slowed down by
17. 1:40friction with the air. If you take away  the air they fall just as fast as a rock.
18. 1:46This is a general rule about gravity: everything  on the surface of a given world will be pulled
19. 1:51down with the same acceleration. That acceleration  will be different for each world though.
20. 1:57For example, on the Moon the acceleration is less  so things fall more slowly there. You can click on
21. 2:04a link to see this being demonstrated by the folks  on the Apollo 15 mission to the moon back in 1971.
22. 2:13In the 1600s, Isaac Newton figured out the law  of universal gravity. I could just give you the
23. 2:20formula here, but i think you should get a chance  to see how Newton came up with this equation.
24. 2:25It's not that difficult to see  where most of the parts come from.
25. 2:29The first step is to start  with a basic observation:
26. 2:33everything feels the pull of gravity. In other  words the gravitational force acts on all matter.
27. 2:41However, think for a minute about  Newton's third law of motion:
28. 2:46for every force there is an equal and opposite  reaction force. So if every bit of matter feels
29. 2:52gravity, then every bit of matter must also have  its own gravitational force. That's why we call
30. 2:59this the universal law of gravity. It's universal  because all matter exerts a gravitational pull:
31. 3:07planets, moons, stars, buildings, people,  guinea pigs, and so on. We all have our own
32. 3:15gravity. So now let's figure out how strong this  force is. We'll call the force of gravity f sub g:
33. 3:25f for force and g for gravity. So we can start  by saying that the gravitational force is equal
34. 3:32to something: we're just not sure what. If  everything experiences a gravitational pull,
35. 3:38try holding a ball close to your head  and then let it go and see what happens.
36. 3:43Of course, the ball will be pulled to the ground  not to your head. So why is that? Right the
37. 3:50Earth is a lot bigger than your head is. This  means that the gravitational pull must depend
38. 3:56on how big something is, or more accurately  how much mass it has. So we can say that the
39. 4:02gravitational force the ball feels depends  on the mass of the thing pulling on it.
40. 4:08Let's write this out as gravitational  force equals something times m sub 1.
41. 4:15Next, think about what we just said about  falling bodies: everything falls with the same
42. 4:21acceleration. Imagine that instead of dropping  a truck and a soccer ball you are trying to push
43. 4:28them instead. You know from Newton's second law  that if you want to push the truck it takes a
44. 4:33lot more force than it does to push the soccer  ball, because the truck has a lot more mass.
45. 4:40That must be what's happening with gravity: if a  giant truck gets the same acceleration as a ball,
46. 4:46then the truck must feel a much larger  gravitational force than the ball.
47. 4:50If it were the same force, the  ball would race toward the ground
48. 4:54while the truck would just sit there floating  in midair hardly moving downward at all. So
49. 4:59the force depends on the mass of the thing being  pulled as well as the mass of the thing pulling.
50. 5:06We can write this down by saying that the force  is equal to something times mass 1 times mass 2.
51. 5:15Let's think about that ball again.  We know that when we drop a ball,
52. 5:19it falls to the ground. Now the planet  Jupiter is 300 times as big as the Earth,
53. 5:26but somehow even when Jupiter is high in  the sky we never have to worry about the
54. 5:31ball falling towards Jupiter instead  of down to the ground. Why is that?
55. 5:37Of course, Jupiter is far away so gravity  must get weaker as objects get farther apart.
56. 5:45We can write this by saying that  the force is equal to something
57. 5:49times the two masses and divided by the distance.  Note that this is distance, not diameter.
58. 5:58But what distance? Is it the distance between the  surface of the ball and the surface of the Earth?
59. 6:04Try this experiment. Set the ball on the ground.  At this point the distance between the surface
60. 6:10of the ball and the ground is zero. If you put  a distance of zero into the formula we've got,
61. 6:16you'll find that whatever masses are there  no longer matter because the force blows up
62. 6:21to infinity and the ball should be stuck to the  ground forever. Since you can pick the ball up
63. 6:27off the ground, the distance in the formula can't  be between the surfaces. In fact, the distance
64. 6:35we need is between the centers of the objects.  If you want to know the force between the ball
65. 6:40and the Earth, you use the distance between the  center of the ball and the center of the Earth.
66. 6:47Since the Earth is much bigger than the ball, this  will usually be simply the radius of the Earth.
67. 6:53We've almost got the law of universal gravity:  there are just two more things we need to do.
68. 6:59First, when we actually calculate how the force  behaves, it turns out that it gets weaker with
69. 7:05distance even faster than what we have here.  In fact it drops off with the distance squared
70. 7:11instead of just the distance. Secondly, we need  to put something in there to get the units right.
71. 7:18We've got force units on one side of the  equation, and we've got mass squared divided
72. 7:23by distance squared on the other side. They  need to match up, so we put in something called
73. 7:28the gravitational constant, capital G. We'll  look at this more closely on the next slide.
74. 7:35That's Newton's universal law of gravity:  the force of gravity between any two bodies
75. 7:41depends on both of their masses and  the distance between them squared.
76. 7:47Let's look a little more closely  at the gravitational constant.
77. 7:52First off it's a universal constant. The  gravitational constant is the same on the Earth
78. 7:59as on the Moon and anywhere else in the universe.  Its value is 6.67 times 10 to the minus 11
79. 8:07meters cubed per second squared kilograms. So  anytime you see this in a formula you know what
80. 8:15you'll have to put in. You don't have to memorize  the value though: I'll give it to you on exams.
81. 8:22As I said on the last page, this constant makes  the units match up. This is the part that tells
82. 8:28you how much force you get from one kilogram  masses at one meter distance. At the same time,
83. 8:37this forces you to use certain units. Anytime  you see the gravitational constant in a formula,
84. 8:43you know that all the rest of the units in the  formula need to match up with this. So masses
85. 8:50have to be in kilograms, distances in meters  and times in seconds. If any of the values you
86. 8:56are given are in different units, you'll have to  convert them before you can calculate or you will
87. 9:02get the wrong answer. Finally let's think about  what the gravitational constant actually means.
88. 9:09That negative exponent tells you that the constant  is small, meaning that the force of gravity must
89. 9:16be incredibly weak. In fact, it's the weakest  force we know of in the universe, much weaker
90. 9:23than the other force we experience regularly,  the electromagnetic force. To appreciate this
91. 9:29think about standing up. The gravity of the entire  Earth, 6 times 10 to the 24 kilograms of it,
92. 9:37is trying to drag you into the ground.  However, just a few kilos of bone are able
93. 9:43to resist the pull of the entire Earth and  hold you up. Those bones get their strength
94. 9:50from the electromagnetic forces in the atoms. So  gravity is weak - the only reason we notice it
95. 9:58is simply that it adds up. There isn't any  negative gravity, so the gravity of the
96. 10:04entire Earth can combine to pull on you all  at once. On the other hand, electric forces
97. 10:10come in positive and negative charges, so  most of the time they cancel each other out.
98. 10:19There's one last thing about the law of gravity  that we want to point out before we go on, and
99. 10:25that's the philosophical point about scientific  method. Notice how Newton worked. He based his
100. 10:31three laws of motion on observation of the world  around him. He developed his law of gravity by
101. 10:39assuming that the laws of motion applied to parts  of the universe that were beyond his ability to
102. 10:45observe or measure. Above all, he was trying to  develop a law of gravity with logical consistency.
103. 10:52The force of gravity that an object feels  should be the force that free fall acceleration
104. 10:58indicates. This logically consistent thinking is  characteristic of the way scientific theories are
105. 11:05developed and this is one of the times in  the course where we discuss all the steps
106. 11:10in the process. So what is the next step in the  scientific method? You should know that by now,
107. 11:16but if you don't go back and take a  quick look at the science section.
108. 11:21Or your instructor may assign a discussion  of this question as a Your Turn. If so,
109. 11:27you may also be asked to consider whether  this is a step that Newton himself could take.
110. 11:33Do you think it's a step that could have  been taken in Newton's own lifetime?
111. 11:40Now let's have a look at an example of how to  calculate the force of gravity. We'll figure out
112. 11:45how much gravitational force we would feel if we  were standing on the surface of the Moon - this is
113. 11:51our weight on the Moon. Here's a picture showing  what's going on. You're an astronaut standing on
114. 11:57the Moon. The things you'll need to know here  are your mass, the mass of the Moon, and the
115. 12:04distance from the center of you to the center of  the Moon - this is effectively the moon's radius.
116. 12:12So let's remind ourselves what needs to go into  the gravity formula. We need the gravitational
117. 12:18constant, which is always the same. We need  the mass of the Moon in kilograms. Since it's
118. 12:25currently in Earth masses we'll have to convert to  kilograms. We need to have our mass in kilograms,
119. 12:32and we need the distance from the center of us  to the center of the Moon. If we're standing on
120. 12:38the Moon this will be the Moon's radius in meters.  Once again, we'll need to convert from Earth radii
121. 12:45to meters. So let's take care of those unit  conversions before we go on to calculate the
122. 12:51force. For the mass of the Moon, remember that  one Earth mass is 6 times 10 to the 24 kilograms,
123. 13:01so we multiply the mass of the moon, 0.012 Earth  masses, by 6 times 10 to 24 kilograms divided by 1
124. 13:11Earth mass, and that gives us the mass of the Moon  in kilograms: 7.2 times 10 to the 22 kilograms.
125. 13:21If you don't know your own mass in kilograms, you  can always calculate this by taking your weight in
126. 13:26pounds and dividing by 2.2. For example, someone  who weighs 170 pounds has a mass of 73 kilograms.
127. 13:38The radius of the Moon works similarly to  the way that the mass did. We know that
128. 13:43one Earth radius is 6400 kilometers; however,  we want our answer in meters not kilometers.
129. 13:51So we remember that one kilometer is 1000 meters.  This gives us one Earth radius is 6400 times
130. 14:021000 meters, which equals 6.4 times 10 to the  sixth meters. So we take the Moon's radius,
131. 14:100.27 Earth radii, and multiply that by the  conversion factor 6.4 times 10 to the 6
132. 14:17meters divided by 1 Earth radius. This  gives us 1.7 times 10 to the sixth meters.
133. 14:26This is a good place to do a sanity check. The  Moon may be smaller than the Earth, but it's
134. 14:32still a world. You would expect a large number of  meters and the fact that we're getting millions
135. 14:38of meters fits well here. One common mistake  is to just assume that the mass and the radius
136. 14:44are already in units of kilograms and meters.  If you do that you'll see that the mass is much
137. 14:50less than a kilogram and the radius is much less  than a meter: in other words you'd be saying the
138. 14:56Moon is about the size and weight of a volleyball:  that's not going to give you a realistic answer.
139. 15:04Now we can go ahead and do the calculation. As  usual we write down the formula, then we put in
140. 15:11all the values with their units,  including the gravitational constant.
141. 15:16This is a good place to check that we've got  all the units right here. Make sure that the
142. 15:21distances are all in meters, the masses are all  in kilograms, and the times are all in seconds,
143. 15:28and wherever you can try to make sure that the  values look realistic. Now we can go ahead and
144. 15:35calculate the force. Just multiply through  the numbers on the top and divide the result
145. 15:40by the radius squared on the bottom. This  gives 1.21 times 10 squared or 121 Newtons.
146. 15:50Remember earlier we saw that we could calculate  weight on Earth using Newton's second law of
147. 15:54motion: w equals ma. If we do this calculation for  someone with a mass of 73 kilograms, we find their
148. 16:01weight is 715 Newtons. So we'd weigh a lot less  on the Moon than on the Earth. This makes sense
149. 16:10since the Moon is a lot smaller than the Earth,  its gravity is weaker and you weigh less on it.
150. 16:18Now you can try working with this gravitational  force law. Calculate your weight on this planet.
151. 16:24There are several steps here so you need to  take it slow and make sure you write down
152. 16:29what you need to do each step of the way. Remember  to show your work and be careful of the units.
153. 16:38Remember this table? You should be using  it to help you keep track of the equations
154. 16:43we use in this course. Fill in the f equals  g m m over d squared section of this table
155. 16:51now. Keep it and update it every  time we use a new equation.
156. 16:55You'll find this page useful doing  homework and reviewing for exams.
157. 17:00Depending on your section you may be required to  submit a copy of the completed version for credit.