This is a simulated lab in which the periods and radii of several planets are provided for analysis. 
Suppose you're an astronomer studying a newly discovered planet and its moons.
By observing the motion of the moons, you'll determine the mass of the
planet they are orbiting.
In Solving Gravitation Problems using
Proportional Reasoning, you saw that the acceleration due to gravity at
the surface of the Earth is given by a_{g} = GM_{e}/R_{e}².
In the case of an arbitrary planet of mass, M_{p}, the acceleration
of a planetary moon in a circular orbit at distance, R_{m}, from the
center of the planet would be a_{m} = GM_{p}/R_{m}².
For all moons of the planet, the value of GM_{p} would be a
constant. Thus, a_{m} would be proportional to R_{m}^{2}.
Suppose you took data on acceleration and orbital radius for several moons
of a particular planet and plotted a graph of acceleration vs. orbital
radius. If you did a power law fit to the equation y = Ax^{n} with n = 2, y = acceleration, and x = orbital radius, then you would expect
the fit coefficient A to represent GM_{p}. Knowing the value
of G, you could then calculate the mass of the planet. That's the plan
of this exercise. You'll use fits for two other relationships
to verify the value obtained for the mass of the planet.
Submit the prelab questions in WebAssign L144PL.
 Orbital acceleration, speed, and period
can all be expressed as a coefficient multiplied by a function of R. That is, each dependent variable has a powerlaw relationship with R of the form y = AR^{n}, where the coefficient, A, is a
function of G and M. This is shown in the table below for the acceleration. Complete the rows for Speed and Period. For Units of Coefficient, reduce the units to a combination of meters and seconds only. (Use m and s.)
Fit 
Dependent variable 
Relationship 
Coefficient, A 
Units of
Coefficient 
Power of R 
1 
Acceleration, a 
a = (GM)/R² 
GM 
<_> 
2 
2 
Speed, v 
v = <_> 
<_> 
<_> 
<_> 
3 
Period, T 
T = <_> 
<_> 
<_> 
<_> 
 Now you'll collect data from this animation. You should see five small circles around the center of the screen – those are the planet and 4 moons.
The planet is the stationary blue circle at the origin. Coordinates of the
moons are displayed in the Outputs panel. The grid spacing is 4 x 10^{7} m. For each moon, determine the period and radius of the orbit to 3 significant figures.
Enter the data into the data table. The radius and period of the Orange moon's orbit are given to you.
Moon 
Radius (m) 
Period (s) 
Red 
<_> 
<_> 
Green 
<_> 
<_> 
Black 
<_> 
<_> 
Orange 
3.54 x 10^{8} 
5.07 x 10^{5} 
You'll submit the analysis in WebAssign L144.
A review of curvefitting methods: In L131, you used the method of reexpression to linearize the data. This is the default method that we use in this course for curve fitting, and it's also the method that you would be expected to use on the AP exam for a problem requiring data analysis. (There is often one such freeresponse problem on each AP exam.) In L113, you used a different method. In that lab, you fit the position vs. time data for dropped and projected marbles to a quadratic. That choice was made because of the expectation that the position of a freefalling object was quadratice in time. In the present exercise, you'll use a similar method to fit the satellite data. Rather than reexpressing a variable to obtain a linear fit for each of the dependent variables, you'll select an appropriate function for fitting the data.

Do the following.

Examine your prelab results. Make any necessary corrections before continuing with the analysis.

Start
Logger Pro. Double click on the
title box of the first column and rename it Orbital Radius (Radius for short) with units of m.
Change the title of the second column to Period with units of s. Enter the data for the 4 moons.

Create a calculated column for the Orbital Speed (Speed for short). Click on Data, New
Calculated Column. Type in appropriate labels for a speed column.
Use 2πR/T for the equation. Select the variables R and T from the dropdown box.
Select
the constant π from a dropdown box as well.

Create another
calculated column for Acceleration. Use v²/R for the equation.

Now prepare your graphs.
Begin with a graph of Acceleration vs. Radius. Title and format it
appropriately.

Click on Insert, Graph.
Plot Speed vs Radius for this graph.

Repeat the last step for a graph
of Period vs. Radius.

Click on Page, Auto Arrange so that all your graphs will show without overlap.

For each graph in turn, select
Analyze, Curve Fit. Select the AR^n (variable power) fit, and type
in the appropriate power. Apply the fit.
 Enter the numerical values for the fit coefficients into the table below. Don't include units. (This, together with the table in item 2 of the prelab, will serve as your matching table.)
Fit 
Dependent variable 
Value of Coefficient 
Mass of Planet
(kg) 
Percentage
Difference 
1 
Acceleration, a 
<_> 
<_> 
0 
2 
Speed, v 
<_> 
<_> 
<_> 
3 
Period, T 
<_> 
<_> 
<_> 
 Knowing the values for the fit
coefficients and the expressions for the coefficients from the prelab, calculate
the mass of the planet for each relationship and enter the result in the
table above. For the acceleration
relationship, for example,
Fit Coefficient, A = GM;
therefore, M = A/G.

You calculated
the mass of the planet using three different relationships. You would
expect nearly the same result for each calculation. Calculate the
absolute value of the percentage difference between the value of mass obtained for each of
Fits 2 and 3 and that obtained for Fit 1. Agreement to within a few percent can be
expected. If your values for mass vary significantly from one
another, then you may have a mistake in your data or calculations.

Check that your Logger Pro data table and graphs are formatted and labeled properly. Then upload your file.
