1.  How can we represent oscillations graphically?
    First you will try your hand at sketching qualitatively how the position of an oscillator varies in time,
    then you will make measurements to quantify your graph.   This will prepare you for interpreting the orbit plots in part 2.

    (a) Qualitatively (without numbers):  Consider a swinging pendulum. At rest, it hangs at equilibrium. Call this position zero. When it's displaced from equilibrium, it oscillates about this zero point.  In your own words, how does the angular displacement (or height) change with time?
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    Everyone in the group should sketch your own representation of the pendulum's displacement versus time. Don't worry about assigning scales to your graph's axes yet.
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    Then compare and discuss your graphs. What are the main features shared by the graphs which your group agrees on?
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    Sketch your team's graph on the board to compare with other groups.
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    (b) Quantitatively: For the demonstration pendulum in the classroom, measure the period (in seconds) and amplitude (in cm or degrees).

    Use those data to label the axes on your sketch above: amplitude vs period.

    (2):  Investigate relationships between the size and period of a gravitationally bound circular orbit.

    (a) Obtain a graph of the motions of Jupiter's moons. Analyze the graph qualitatively first, to get a general overview and to practice your basic graph-interpreting skills. First answer each question on your own.

    Which moon has the fastest orbit? The slowest?  How can you tell?
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    Which moon orbits furthest from Jupiter? which orbits closest?  How can you tell?
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    Discuss your results with teammates until you reach a consensus.  Predict your moon's rank, in orbit speed and distance from Jupiter.
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    (b) Choose one of Jupiter's four moons to analyze carefully. Talk with all the tables around you to make sure they are analyzing different moons, because you will need their results on those other moons.  Which is your moon?
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    (c) Before you do any calculating, first qualitatively predict how your moon's orbit will compare with the others. Second shortest period? Third largest orbit radius?  Explain your reasoning.
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    (d) Quantitative analysis of your moon's orbit: Check the time and space axes on your graph to make sure you understand their units.

    What changes as you go across the page, left to right?  What is the circle in the middle?
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    What changes as you go down the page?  What are the units?
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    Measure the period of your moon as carefully as possible, with a clear plastic ruler. Your result will be more accurate if you measure over several periods and divide by the number of periods. How far off could your result be? You probably will have no more than three significant figures in your answer ( 1.53 seconds, +/- 0.02 sec) Don't waste time on meaningless over-precision (such as 1.5286149 seconds).
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    Measure the orbit radius of your moon as carefully as possible, with a clear plastic ruler. What is the uncertainty in your measurement? (r +/- Dr)
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    Post your results on the chart on the front board, and tabulate the class results in your notebook.  INCLUDE YOUR UNITS!
    Group initials
     
     
     
     
     
     
     
     
    Moon
    Io
    Io
    Europa
    Europa
    Ganymede
    Ganymede
    Callisto
    Callisto
    Orbit Radius
     
     
     
     
     
     
     
     
    Period







    Speed (pt.4)







    (3) : Use Newton's and Kepler's laws to find Jupiter's M from your moon's T and r.

    (a) Using the class data on the board, plot (period T) versus (orbit radius r) for Jupiter's four moons. Is this a straight line? Do you expect it to be?
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    (b) Recall the derivation of Kepler's 3d law from class.  Start with F=ma (Newton's 2d law) and derive an algebraic relationship between T and r for a circular orbit.
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    Where is the central mass represented, e.g. Jupiter?  Where is the mass of the moon, or the body orbiting at a distance r?
    How does the moon's mass affect its orbit?
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    (c) What power of T should you plot against what power of r to get a straight line graph?

    Do it with the class data on Jupiter's moons. Compare to your graph in (a) above.  Explain the curve in your own words.
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    (d) You can also use your measurements of your moon's period and orbit radius to find the mass of Jupiter!
    First, solve your equation for M, the central mass about which the moons are orbiting.
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    Now plug in your moon's data to get a number for Jupiter's mass, M.  Do your units work out?
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    How uncertain is your answer?  If your radius could be off 4% and your period could be off 5%, then since M is proportional to r^3 and T^2, M could be off 3*4% + 2*5% = 22% (this is probably an overestimate - errors could either cancel or add).
     
    % uncertainty in r your estimate of M
    % uncertainty in T times % uncertainty in M
    % uncertainty in M DM = uncertainty in M

    Check your Jupiter mass against the results of other groups, with each group using only their own moon's r and T. Don't look it up in the textbook!
    If any values are terribly different, discuss with that group to try to resolve the difference.
    When every group is confident of their calculations, then average the results.  As a class, what do you find for Jupiter's mass M +/- DM?
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    (4): Extend your understanding to stars orbiting in the galaxy,
    and find the distribution of dark matter in the Milky Way!

    (a) The "velocity profile" or "rotation curve" of a system of orbiting masses tells you how the system's mass is distributed, whether you the mass emits light or not. Graph speed v versus R for Jupiter's moons, where speed v=distance/time = 2*Pi*R/T and R is the moon's distance to Jupiter and T is the orbit period, as usual.
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    (b) Where is most of the mass in the system consisting of Jupiter and its moons? Your graph is typical of a system dominated by a central "point" mass.
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    (c) Vera Rubin found that galaxies have a different rotation profile than our solar system: v is nearly constant with respect to R. Sketch that relation on a v versus R graph. Do you expect that most of a galaxy's mass is located at a central point? Galactic light is concentrated in the center.
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    (d) "The Sun orbits the center of the Milky Way galaxy with an orbital speed of about 250 km/sec. The distance to the center of the galaxy is ... about 9.1 kpc. ... [For stars nearer the edge, at] a distance of 15kpc from the center of the Milky way, the rotational speed is still 250 km/sec. ... What percentage of the mass of the Milky Way lies between 9.1 and 15 kpc? ... What can one say about the mass gravitationally revealed in the outer regions of the Milky Way," where there is little mass visible in the form of stars? (from Ferguson's Introductory Astronomy Exercises, p.160)
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    (5): Reflect on your collaborative work. Each team, work together to fill out a workshop feedback form. Turn it in before you leave. Keep your workshop in your portfolio - you will use it again later this quarter.