Gravitation

1.  Forces in the Universe
*  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
*  Falling objects pull on Earth (3rd Law)
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"
*  Universal gravitation:

>  Acts through empty space
>  Explains how gravity works - not why
5.  Weight
*  Measure of gravitational force of Earth (or other planets) on you:

*  Weight changes on different planets
*  Weight can be made to apparently increase/decrease
>  Acclerating 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
7.  Orbits
*  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
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
*  a3 = P  (P = orbit period)
*  Newton's form of Kepler's third law:

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

Animations

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)  What is the planet's average distance from the sun?
b)  What is the planet's aphelion distance?
c)  What is the eccentricity of this orbit?
d)  What is the planet's orbit period, in years?
e)  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.  Consider Newton's form of Kepler's 3rd law:

Here, the Sun's mass is M, and a planet's mass is m.
Divide both sides of this equation by a3.  Now look at what you have on the right side of the new equation:  In the solar system, we know that M >> m, even for the most massive planet.  a)  What does this fact suggest to you about the value of P2/a3 in the solar system, for all planets?  b)  Based on your answer to part a), do you think that Kepler's 3rd law (as Kepler wrote it) would work in the solar system if planets were about as massive as the Sun?  Explain.

16.  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.]

17.  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                                        p = 5

t = 10                                      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*10,000/2 =  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 at 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 = sq root (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.

15.  a)  P2/a3 is about the same for all planets.  Think of it this way:  the only terms on the right side of the (new) equation not constant are the planet's mass (m) and the Sun's mass (M).  Let M = 1:  then for the largest planet, m = 0.001.  Is M + m = 1.001 very much different than 1?  The answer, of course, is no:  adding a very small value of m to the very large value of M changes the right side of the equation hardly at all, so it's about the same for all planets; thus P2/a3 is about the same for all planets.
b)  Kepler's 3rd law (P2 = a3) would not work in the solar system if planet masses were close to the Sun's mass.  Let's go back to part a), and write an equation with
P2/a3 on the left, and everything else in Newton's form of the 3rd law on the right.  Suppose that M = 1.  Now suppose that two planets have masses m = 0.3 and
m = 0..6.  Then, for the first planet: m + M = 1.3, and for the second planet, m + M = 1.6.  1.3 is substantially different from 1.6, so P2/a3 would be much different for these two planets.  And we would observe a solar system in which P2/a3 is not the same for all planets - which is what Kepler's (original) 3rd law requires!

16.  Here's Newton's form of Kepler's 3rd law:

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.