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The
Simple Pendulum: An Exercise in Measurement and Graphical Analysis
A phenomena which repeats some action over and over again in a regular
way is said to be periodic. The
time for such a system to exactly complete the repetitive action once
is called the period. For
example, the period of the Earth in its orbit is one year and the period of
your heartbeat is about one second. Periodic
phenomena are very common in nature and when they are sufficiently reliable
they are sometimes used to measure time.
One of the simplest devices which exhibits periodic motion is the simple
pendulum. A simple pendulum consists of a heavy object suspended by a light
weight string. If you displace
the bob to one side and release it, then the period is the time for the bob to
return to the point of release. This
lab exercise uses a simple pendulum to illustrate how one can learn something
about natural phenomena by a combination of measurement and graphical
analysis.
This simple experiment illustrates a technique used by scientist in
many fields. The scientist wishes
to understand some physical system which in this case is the simple pendulum.
He identifies some important property of the system like the period of
the pendulum. Then he tries to
determine how this quantity depends on other parameters of the system.
In this case the other parameters might be the weight of bob, the
length of the pendulum, the size of the arc through which the pendulum swings,
and possibly other things. He varies a parameter (say the weight of the bob or length of
the string) and measures the value of the important quantity (the period) for
each value. The scientist is
careful to keep all other properties the same.
Then he varies another parameter and so on.
Using the methods of a scientist, you will attempt to learn how the
period of pendulum depends on its length and the mass of the bob.
You should always keep the arc through which the pendulum swings small
(15 º or less.) In order to
measure the period of the pendulum, measure the amount of time for 20 complete
swings.
To eliminate one source of error, do not use the first swing. Start
timing and counting when the pendulum bob returns to its starting point.
Use any available timers.
Repeat this two to three times and average your total times.
The period of the pendulum is then the total time measured divided by
20 (i.e., the amount of time for one swing.)
The length of the pendulum is the distance from the point of support to
the center of mass of the bob. Using the
same mass you should measure the period for at least 4 to 6 different lengths,
ranging from 0.50 m to 1.50 m. Rather
than actually measuring the mass of the bob, you may use the mass of one
sample (a metal sinker) as your unit of mass.
A good modification of the experiment would be to change the amount of mass of the bob
and measure the
period again for two to three of the lengths already used.
All data should be recorded in a table.
Column headings might be total time measured, length, mass of bob,
period and period squared (a calculated value.)
Always record your measured quantities as you make the measurement.
Calculations should be performed later.
In order to see more clearly what the data shows, you need to make some
graphs. Plot
the period as a function of length of the pendulum.
Also plot the period squared as
a function of the length of the pendulum.
The length will be on the x-axis and period (or period squared) on the
vertical or y-axis.
Discussion Questions:
1. Did you obtain a
straight line for any of your plots?
2. Can you write an
equation for the line?
3. How does the period, T,
depend on length?
4. By interpolating from
your graph determine the period of a simple
5. By extrapolating
determine the length of a pendulum which has a
6. Using your data and the formulas given below (formula #1) solve for
g. We know that the actual
value is 9.807 m/s2.
Additional bonus credit: Use the computer program Graphical Analysis to
plot the two graphs using your table of data. How do they
compare with your hand drawn graphs? Look at the statistical information
the computer program can provide and make an attempt to correlate that
information with your collected data.
For
further information: The motion of this type pendulum is simple harmonic in
character with a period proportional to the square root of the pendulum's length and
independent on the mass of the bob.
The following are general
equations for simple pendulums: T = 2 p
Ö
(L/g)
T2 = 4 p2 (L/g) Graphical
Analysis for Windows Section Double click on the "X" in the data table. Here you will name the variable and its units. Do the same for the "Y" variable Input data for each variable. Click on the F (x) button on the task bar. Type the in the name of the new column, "Period Squared" and the units "sec^2". You now need to give a formula for this data. Click on the "columns" button and pick "period" Then press the "^2" button on the keypad. Press "OK". On the graph window, double click on "Period (sec)". Click the box beside "Period Squared" then press "OK". Select all data on the graph by dragging a dotted box around every point. Click on Analyze on the menu bar. Then Regression. Click on "Period", then on "OK". Repeat for "Period Squared". This data box gives the Correlation (Corr), the slope (M), and the y-intersept (B). Use the data in the boxes to finish answering the question from the experiment. Teacher Initial for Graph _______________ Correlation of Period ______________ Slope of Period vs. Length _______________ Correlation of Period2 _____________ Slope of Period2 vs. Length _______________ |
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Page Last Updated: Friday March 02, 2007 Webmaster: Larry Jones Pickens County School District |