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Gravitation
Outline
1. Forces in the Universe
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Gravity
Electromagnetic Forces
Weak Nuclear Force
Strong Nuclear Force
2. Gravitation
* Dictates large-scale structure in the universe
3. Falling Bodies
* Galileo: objects fall to Earth with constant acceleration (9.8 m/s/s)
* Acceleration same for all masses  F  m
* Objects attracted to Earth pull on Earth! (3rd Law of Newton)
4. Universal Gravitation
* All bits of matter attract all other bits of matter:
> 1. means force increases when masses increase; decreases when masses decrease
> 2: "Inverse-Square Law"
F  M1M 2
F
1
d2
* Universal gravitation:
F 
GM 1M 2
d2
(G  ‘gravitational constant’)
> Acts through empty space
> Explains how gravity works - not why
5. Weight
* Measure of gravitational force of Earth (or other planets) on you:
W 
GmM
R2
* Weight is different on different planets
* Weight can be made to apparently increase/decrease
> Accelerating elevator
> Weightlessness
6. How Do The Planets Go?
* Ptolemy
> Geocentric solar system
> Circular orbits & epicycles
* Copernicus
> Heliocentric solar system
> Circular orbits
* Galileo
> Telecopic observation of Jupiter’s moons & Venus’ phases
7. Orbits
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Motion controlled by gravity is orbit
Circle: object continuously falling
Circles, Ellipses: closed orbits
Parabolas, Hyperbolas: escape orbits
Earth: circle: 5 mi/s; escape: > 7 mi/s
8. Kepler's Laws
* Kepler's use of Brahe's data
* Kepler’s Laws
> Planets' orbits are ellipses with Sun at one focus
~ Semi-major axis (a), perihelion, aphelion
~ Eccentricity: 0 < e = c/a < 1
> Planet-sun line sweeps out equal areas in equal times
~ Planet's speed varies around orb
3
2
* a = P (P = orbit period)
* Newton's form of Kepler's third law:
a3
M1  M 2  2
P
9. Sun's Mass
* ~ 330,000 Earth masses
10. Center-of-Mass Orbits
* True story of orbits
> Sun pulls on planet; planet pulls on sun: sun moves, too
> Sun/Jupiter & Earth/Moon center of mass
Questions
1. Ques. #5, pg. 95.
2. Ques. #6, pg. 95.
.
3. Ques. #7, pg. 95.
4. Ques. #15, pg. 95.
5. Prob. #4, pg. 96.
6. In part c) of the last question, what would happen to the force between Earth and Sun if you
simultaneously moved Earth to twice its present distance from the Sun?
7. If you fall unimpeded toward Earth's surface, your acceleration will be (a constant) 9.8 m/s/s.
What will be your speed after 1 sec of fall? 2 sec? 10 sec?
8. If Galileo had dropped a lead ball and a wooden ball, both the same diameter, from the top of
the Leaning Tower of Pisa, he should have observed (in principle) that both hit the ground at the
same time. Explain in words why both balls experience the same acceleration.
9. On a planet with Earth's radius, but twice Earth's mass, would you weigh more or less than
you do on Earth? Why?
10. On a planet with twice Earth's radius, but Earth's mass, would you weigh more or less than
you do on Earth? Why?
11. The above three orbits have identical semi-major axes, but very different eccentricities. For
which will the orbit period be longest? shortest? Explain your answer.
12. A line joining the Sun and an asteroid was found to sweep out 5.2 square AU of space in
1994. How much area was swept out in 1995? In a span of 5 years?
13. Consider the orbit of a planet about the sun. (See diagram above.) At perihelion, the planet
lies 4 AU from the sun.
a)
b)
c)
d)
e)
What is the planet's average distance from the sun?
What is the planet's aphelion distance?
What is the eccentricity of this orbit?
What is the planet's orbit period, in years?
At what point in its orbit is this planet's speed largest? Smallest?
14. What would be the average distance from the sun of a planet with orbit period 8 years?
15. Suppose the Sun were nine times as massive as it now is, and Earth's average distance from
the Sun were unchanged. Would the year be longer or shorter than it now is? Justify your
answer. [Hint: Use Newton's form of Kepler's 3rd law.]
16. The planet Xenon is orbited in a circle by a moon (see above diagram). Xenon's mass is 5
times the mass of its moon.
a) Xenon exerts a gravitational force on the moon. What path would the moon follow if the
force (F) suddenly became zero? Which of Newton's laws is relevant to this situation?
b) In terms of the force applied to Xenon's moon, how large is the force applied to Xenon by its
moon? Which of Newton's laws is relevant in answering this question?
c) Compare the acceleration experienced by Xenon due to its moon with the acceleration
experienced by the moon due to Xenon. Which of Newton's laws is most relevant here?
d) Consider the common point orbited by Xenon and its moon: Is it closer to Xenon, or to its
moon? Can you justify your answer?
e) Try modeling this system using the orbit animation applet (click here) and use the following
settings:
m_red/m_blue = 5
t = 10
p=5
e=0
Click on the "center-of-mass" and "show orbit" buttons; then click on "Stop/Start."
f) In part e), which body follows the smaller orbit? Which body has the greater orbit speed?
g) In the orbit animation applet, change m_red/m_blue to 100 (so that Xenon is now 100 times
more massive than its moon, which is close to the situation for Earth and
its moon). Leave all other settings unchanged and run the animation. Why does Xenon move so
little now?
Answers
1. Mass is a measure of the quantity of matter in a body; weight is a measure of the force of
gravity acting on that same body. Mass does not depend on the location of the body; weight
does. For example, you weigh less on the Moon than on Earth, though your mass is the same in
both places.
2. An object in free fall is falling at a rate equal to the local acceleration due to gravity; e.g., near
the surface of Earth, anything falling at 9.8 m/s/s is in a state of free fall. To experience weight
requires that your acceleration be less than the acceleration due to gravity. Think of yourself
standing motionless on a floor (with zero acceleration, of course). You aren't accelerating, so the
floor must be pushing upward on you just hard enough to balance the downward pull of Earth's
gravity, which is always present. It's actually this upward push by the floor that leads to our
ordinary experience of 'weight.' Now imagine the floor suddenly removed - you immediately
start falling downward at the acceleration due to gravity - you're in free fall. Jump from one step
to another down a staircase - you're in free fall. So it's not difficult to find yourself in free fall.
Astronauts in orbit about Earth are in an extended state of free fall as they are continually falling
toward Earth, but never reach it (until they land, of course). The astronauts' free fall simply takes
them from the straight-line path Newton's first law tells them they should take, to the circular (or
elliptical path) enforced by the pull of gravity. They are continuously falling toward that orbital
path.
3. The space shuttle in orbit is effectively falling toward Earth. Imagine launching a cannonball
from a cannon pointed sideways on a high mountain. The cannonball goes sideways, but
eventually falls to Earth. Giving the cannonball more speed out of the muzzle of the gun causes
it to go farther before landing (i.e., before Earth 'gets in the way.') As the cannonball travels,
Earth below is falling away (because Earth is curved). Eventually, you might imagine making
the cannonball go so fast that it's falling toward Earth just as rapidly as Earth's surface is falling
away from it - now we have an orbit - the cannonball never lands. Indeed, in principle it
eventually returns to its launch point on the mountaintop. Just as the cannonball requires high
speed to go into orbit (so that it never collides with Earth), so, too, the space shuttle requires high
speed to attain orbit. If space shuttle were launched with the escape velocity it would never
return to Earth.
4. Bound orbits: circles and ellipses. A spacecraft in a bound orbit follows one of these closed
curves. Unbound orbits: parabolas and hyperbolas. A spacecraft following one of these open
curves escapes from the place (planet) where it was launched.
5. a) Triple the distance and the force is reduced to 1/9 its original value. (1/9 = (1/3)2)
b) The gravitational force acting between Jupiter and the Sun will be 318 times the force acting
between Earth and the Sun: Force is proportional to the product of the masses. For the sake of
argument, suppose the Sun's mass is 10,000 times Earth's mass. Then for Earth and the Sun: F =
1*10,000 = 10,000. For Jupiter and the Sun: F = 318*10,000 = 3,180,000. But this is just 318
times the force between Earth and the Sun.
c) The gravitational force between Earth and the Sun would double. For Earth and present Sun:
F = 1*10,000 = 10,000. For Earth and twice-massive Sun: F = 2*10,000 = 20,000. So the force
doubles.
6. Well, we have the effect of changing the mass and the distance simultaneously. We can write
the force equation like this: F = M1M2/d2. So, F = 2*10,000/22 = 20,000/4 = 5000. 5000 is
one-half the orginal force value (10,000), so we conclude that doubling the Sun's mass while at
the same time moving Earth to twice its present distance from the Sun would result in a
gravitational force between Earth and Sun that is one-half its present value.
7. 1*9.8 = 9.8 m/sec. 2*9.8 = 19.6 m/sec. 10*9.8 = 98 m/sec.
8. We assume air resistance is zero. We then observe that both objects have the same
acceleration. The only way this can occur is if the forces on the two objects are different, as they
evidently have different masses: Newton's 2nd law tells us: a = F/m. So, if different masses
have the same acceleration, then the forces must be different. Now, the gravitational force law
tells us: F = GMm/R2. For all objects at Earth's surface, R (Earth's radius) is the same, as is the
mass of Earth (M). So, the gravitational force law tells us that the force which accelerates an
object toward Earth is porportional to that object's mass: when the mass doubles, the force
doubles; when the mass is halved, the force is halved; etc. Thus, the gravitational force acts in
just the way required to give all masses (no matter how big) the same acceleration - namely, the
acceleration due to gravity.
9. Were Earth's mass doubled without changing its radius, your weight would double. The
reasoning here goes just as it did in #6 b) above. Double the mass of one body interacting with
another gravitationally, and the force of gravity between them doubles.
10. Were Earth's radius doubled, without changing its mass, your weight would change to 1/4 its
present value. The reasoning here follows the reasoning used in #6 a) above. Think of Earth
reduced to a small object (a baseball, say), so that you and Earth are effectively separated by one
Earth radius. Now, double that radius. How the does the force change? According to the
gravitational force law, F = 1/(22) = 1/4. So, the force is 1/4 its original size. Note that the '2' in
the denominator of the last equation is the factor by which the distance changes (it's doubled),
not the actual distance.
11. Orbit periods are the same for all because all have the same semi-major axis (a). According
to Kepler's 3rd law, a and period (P) are related like this: P2 = a3. This equation includes no
terms that depend on eccentricity. So eccentricity doesn't matter. Only a matters in determining
P.
12. 5.2 sq AU was swept out in 1995, because, according to Kepler's 2nd law, equal areas are
swept out in equal times. So, if in 1994 5.2 sq AU was swept, we must find the same area swept
out in any one-year interval. 5*5.2 = 26 sq AU. So, 26 sq AU will be swept out in a 5 year
period.
13. a) The average distance from the Sun is 5 AU, half the distance across the long (major) axis
of the orbit.
b) At aphelion, the planet is 6 AU from the Sun: 6 AU + 4 AU = 10 AU.
c) e = 1/5 = 0.20. 1 AU is the distance between the center of the orbit and the Sun (the yellow
disk). We know this because half the distance across the orbit is 5
AU, and at aphelion the planet is 4 AU from the Sun: 5 AU = 4 AU + 1 AU.
d) P2 = a3: P2 = 53 = 125, so P = (125) = 11.2 yrs.
e) Its orbit speed is largest at perihelion (when 4 AU from the Sun), and smallest at aphelion
(when 6 AU from the Sun).
14. a3 = P2 = 82 = 64, so a3 = 64. To find a, ask the question: What number multiplied by itself
twice gives you 64? The answer is 4: 4*4*4 = 64. So the average distance from the Sun must be
4 AU. Like, awesome, man.
15. Here's Newton's form of Kepler's 3rd law:
M  m 
a3
P2
According to the problem, we keep a unchanged (the orbit diameter doesn't change), and make
M (the Sun's mass) nine times bigger. So what happens to P? The equation tells us that in these
circumstances, P2 must decrease; so P must decrease, as well. Thus, increasing the Sun's mass
nine times would result in a (much) shorter orbit period for Earth.
17. a) If the force became zero, the moon would go off on a straight line (specifically, a line
tangent to the orbit at the point where the force disappears). Newton's first law is relevant to this
situation.
b) Xenon's moon pulls on Xenon just as hard as Xenon pulls on its moon; the forces are equal.
This result is required by Newton's 3rd law of motion.
c) The forces are equal, but the moon has a much smaller mass, so its acceleration is much
greater than Xenon's acceleration. This result is required by Newton's
2nd law of motion.
d) Xenon is closer to the common point (the center of mass) because it is the more massive of
the two bodies.
f) The red object (the one closer to the center) follows the smaller orbit. The blue object has the
greater speed (it follows a big circle in the same amount of time the red object follows a small
circle).
g) Xenon moves so little because its mass is so much larger than the moon's mass.