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Stars and Slopes |
Day 1 focuses on log-log plotting and determining the slopes of such plots.
Materials
| Teacher |
Students |
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| overhead to emphasize points in intro and throughout lesson |
copies of all plots and charts of data |
| copies of all plots and charts of data |
few sheets of log-log and Cartesian graph paper |
(pdf version available)
NOTE: Reading pdf files requires the Adobe Acrobat Reader, which is available for free download from http://www.adobe.com/products/acrobat/readstep.html.
First, let us remember what a base 10 logarithm is: when you calculate
the logarithm of a number (log), you are calculating the power to which 10 is
raised to obtain that number. For example, log1=0 (i.e., 100 = 1),
log10=1 (i.e., 101 = 10), log100=2 (i.e., 102 = 100),
etc. A sample of logarithmic graph paper is shown below. Notice that is has
logarithmic scales on both axes (as opposed to the linear scales you may be
used to) and the tick marks, or grids, are made on the axes according to the
logarithms of numbers. The divisions from 1-10, 10-100, 100-1000 along the
axes are called "cycles".
Let us now turn to a simple physics lab for an example. For many
relationships in the real-world, if you plot the data on a rectangular
coordinate system, you do not get a straight line. Instead, you get a curve
such as a hyperbola, a parabola, or some form of power or exponential curve. In
this lesson, we concentrate on power laws, i.e. variables which
have a relationship which can be expressed as Y= k Xn, where n
can have any value.
The following data have been gathered from an experiment meant to
determine the relationship which exists between the diameter of a ring and
its period as a pendulum. Five steel rings with varying diameters were
individually suspended from a knife edge mounted on the wall, and caused to
swing back and forth about this axis. Each diameter was measured, and each
period was determined by measuring the number of cycles per unit of time.
| Ring Diameter (cm) | 3.51 |
7.26 |
13.7 |
28.5 |
38.7 |
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| Time for Completing 25 Swings (sec) |
9.35 |
13.3 |
19.2 |
26.98 |
32.88 |
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We expect these data to follow a relationship of the form T = Adn
where T is the period of oscillation, A is the constant of
proportionality, d is the diameter of the ring, and n is a constant. Given
this, and our desire to end up with a straight line on our graph, we consider
the following:
if T = Adn, then log T = log A + n log d.
This equation should have a very familiar form to you - it is the equation
for a straight line if you plot log T vs. log d, regardless of the value of
n.
Now, plot the data in the following ways: (1) as log T versus log d on
Cartesian graph paper; and (2) as T versus d on log-log graph paper.
Compare the two plots and answer the following questions:
- What do you see?
- What are the values of A and n?
- How do these values compare with the equation for the period of
oscillation of a simple pendulum,
where g is the acceleration due to gravity, l is the length of the pendulum,
and pi = 3.14?
- For a pendulum and a ring to have equal periods of oscillation, what
must be true?
- Given any set of measurements for T and d, what value for the
acceleration due to gravity here on Earth do you calculate?
Shown below are the appropriate plots of data from the previous experiment.
(pdf version available)
Students will plot the following data in log-log format.
| X | 1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
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| Y | 4 |
32 |
108 |
256 |
500 |
864 |
1372 |
2048 |
2916 |
4000 |
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After completing the log-log plot, do you see a straight line?
What does this tell you about the equation?
What is the slope of the line?
What is the equation of the line?
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