Physics 116a02: Extra Hints for Assignment 12
Department of Physics and Astronomy, Vanderbilt University
This hints page was completed on March 23.
Things to Think About for Assignment 12
- Exercise 11.9
- No extra hints needed. After getting the answer to part A you might
wonder what the problem statement means by "small weight" compared to the
sizes of all the other forces in the problem.
- Exercise 11.10
- No extra hints needed.
This problem is very similar to the Lancelot-on-ladder
problem discussed at length in class, with even the same 3-4-5 right triangle.
- Exercise 11.21: A couple. (The forces in a couple exert toques in
opposite directions. Hence, these forces act to produce rotational accelerations in
opposite directions.)
- For part A: The problem should have stated "produce a net torque whose magnitude
is 6.40 N about the left end of the rod".
- For part B: What is the answer in part B inconsistent with the way the
part A problem is being stated by the MasteringPhysics program?
- For parts C and D: Is the net torque (magnitude and direction) the same
for all pivot points along the rod for a couple? This would certainly not be
the case for a single force acting on the rod.
- Exercise 11.2
- Test Your Understanding 11.1: Conditions for Equilibrium
- For part A: The book prescribes two conditions for equilibrium, and there
is no such thing as the "zero condition of equilibrium" in case you were
wondering. (There is a "zeroth law of thermodynamics", however, in addition
to the other three and better known laws of thermodynamics.)
- Exercise 12.48
- For part A: Reminder that to enter a number in scientific notation, for example 12345,
you would use 1.2345*10^4. Don't use 1.2345 x 10^4, or
1.2345e+04, which are marked as wrong answers.
- For part B: The ratio to the Sun's mass is to be given in part C.
- For part D: The event horizon is the essentially same as the Schwarzchild
Radius, named after the German astronomer Karl Schwarzchild. The equation
for this quantity is given on page 406. Coincidentally, the German word for the
color black is schwarz, and "black hole" translates as
schwarze Bohrung, but the name "black hole" was coined 130 years before
Schwarzchild. How does your answer to part D compare with the orbital radius
of the Earth about the Sun? Do you have any idea of how astronomers can
measure directly the speeds of stars in far away galaxies? We have not covered
this piece of physics yet, but it is based on the Doppler Effect for light.
- Exercise 12.10
- Exercise 12.33
- No extra hints needed. Be careful that the Part A answer is requested in
units of km/sec.
- Exercise 12.54
- Orbiting satellite
- No extra hints needed. After you give the correct answer for part C, the
program will respond with information about the Virial Theorem, which
you should find interesting.
- Problem 12.73: Binary Star-Equal Masses
- For part A: No extra hints needed. All the answers are in terms of
the three parameters G, R, M.
- For part B: You should notice that the answer which you get is not
the same expression (i.e. sqrt(GM/R)) that you would get for the
speed of a planet in a circular orbit of radius R about the Sun in our solar system. Do you
know why you are getting a different answer with the Binary Star system?
- Problem 12.78: A Moon of Uranus
- No extra hints needed. Notice that the gravitational acceleration at the
surface of Uranus is only 13% larger than that of the Earth. If you were walking on Uranus, you would
hardly notice the effect, given that you would be wearing a heavy space-suit. A
nice Web site for getting planetary information is
http://www.nineplanets.org, which was
originally named when there were nine planets recognized. Now Pluto is called
a dwarf planet. You should realize that the mass calculated for Uranus in this
problem differs significantly from the actual mass in solar system data tables.
- For part B: Notice that the altitude of Miranda above the surface of
Uranus is being directly provided, and not the orbital radius of Miranda about Uranus.
- A Satellite in Orbit
- For part B: Notice that the correct answer is a high fraction of the
satellite's weight. As the extra comments say, objects in orbit about the
Earth are not weightless, but are in a continuous state of free-fall.
The next time that you see astronauts floating inside the space shuttle
while it is in orbit, you can think that they are continuously falling and cannot
get closer to the surface of the Earth.
- Problem 12.55: Geosynchronous Satellites
- No extra hints needed. Similar to problem 12.78, it is the altitude above the
surface of the Earth that is being requested.
- Problem 12.68: Energy (Work Done) to Move a Satellite
- For part A: The problem is ambiguously worded. Is the question how much work (energy) is
required to move a mass from the Earth's surface to a point somewhat
above the Earth's surface, or is it how much energy is required to put the satellite
into near-Earth circular orbit, meaning an orbit whose radius is approximately,
but just larger than R_E? It is the second question which is being asked,
effectively what is the approximate kinetic energy of a mass m which is in circular
orbit just slightly above the Earth's surface. You can't answer the first
question since that depends on how high the final position is above the Earth's
surface. Similarly the exact answer to the second question also depends
on how high the orbit is above the Earth's surface. Neither answer depends on
the zero position of potential energy since what is being asked is a
change in energy.
- For part B: This problem is asking how much energy is required to move the
mass almost infinitely far away from the Earth, such that it has zero total
energy at the end. From a practical point of view, you can think of how
much kinetic energy would have to be added to the satellite's motion, firing
an on-board rocket engine for a short period of time, such that the satellite
could coast to an infinitely distanct point from the Earth. Such a "burn"
would put the satellite into an unclosed and unbounded parabolic orbit, compared
to the closed elliptical orbit which is characteristic of a gravitationally
bounded mass.
- Exercise 12.18
- No extra hints needed. How does the answer to part B compare with the
density of water, which is 1 gram/cc?
- Understanding Newton's Law of Universal Gravitation
- For part B: You should look at the answer options in Figure B.2 for
the Earth-Sun relative positions shown in Figure B.1 .
- For part F: After all the work you have done so far
you should be able to give this numerical
answer almost instantaneously. Astronomers typically express
masses in terms of solar mass units, meaning the ratio
of mass to the Sun's mass. Similarly, for solar systems,
the length unit is an Astronomical Unit (AU), meaning the
average radius of the Earth's orbit. Finally, times are expressed
in units of years. If one uses such an astronomy-based system
of units for length, mass, and time (instead of the MKS system),
then the product GM_sun is simply equal to 2pi in these
astronomy units.
See if you can derive that numerical result from Kepler's Third Law.
- Exercise 12.6
- Escape Velocity
- No extra hints necessary. The inclusion of an angular momentum
discussion in the concept of escape velocity is somewhat of an unnecessary
detour. Kepler's Second Law (equal areas in equal times) is equivalent
to angular momentum conservation. In turn angular momentum conservation
is true for any central force meaning a force which is directed
along a line between the centers of the masses, such as Universal Gravity.
Interestingly, Kepler's Second Law is the only one of the three Kepler
Laws which does not depend on Univeral Gravity being an inverse square
law. Any kind of radial dependence would work to prove Kepler's Second Law.
An inverse square law is needed to prove Kepler's First Law and
Kepler's Third Law, however.
- Understanding Mass and Weight
- For parts A and B: The abbreviation MN is for Mega-Newtons.
- A Satellite in Circular Orbit
- For part E: This is a strange question, especially in view of the correct
answer to part C. Actually, in advanced mechanics courses and in modern
physics, the laws of motion are derived from knowing the functional form
of the potential energy and the kinetic energy. One constructs what is
called the Hamiltonian or the Lagrangian for a system,
and there are differential equations which are valid for the Hamiltonian
or the Lagrangian function. In some physical systems, with complicated
contraints on the motion, the Hamiltonian or Lagrangian equations are
easier to solve than are the equations from Newton's Second law.
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This page was last updated on March 23, 2008