### Pendulum Laboratory: Hanging Around

#### Purposes:

• To determine the acceleration of gravity.
• To learn to plot points on a graph.
• To make a best line fit to data.
• To determine the slope of a line.

#### Set Up

Hang a fishing weight or any other compact and dense object by fishing line from a height such that the maximum length of the pendulum can be on the order of two meters. Hang them anywhere.

Use any chronograph (timer) that can count seconds: many digital watches have timers built in. If these are not available just use any watch that has a smallest division of seconds and take measurements to the nearest second.

#### Experimental Procedure

Possible assignments for students in multi-student groups:

• Recorder of the data
• Keeper of the timer
• Wrangler of the equipment
• Measurer of lengths
1. Start with a pendulum of the maximum possible length, probably around 2.4 meters.
2. Time ten complete swings (ten periods) of the pendulum in seconds. One pendulum period is one swing from right to left and back to the right.  Record the length L in meters for the pendulum and the time for ten cycles t in a table.
3. Shorten the pendulum by roughly 30 centimeters.  Return to step 2, measuring another ten swings.
4. When the pendulum length is roughly 30 centimeters long or shorter, stop the experiment.

#### Table

10t/s t/s L/m t²/s² *** 10t/s t/s L/m t²/s²
______ ______ ______ ______ *** ______ ______ ______ ______
______ ______ ______ ______ *** ______ ______ ______ ______
______ ______ ______ ______ *** ______ ______ ______ ______
______ ______ ______ ______ *** ______ ______ ______ ______
______ ______ ______ ______ *** ______ ______ ______ ______

#### Graphing the Data

1. Calculate t and t² for each row of the table.
2. Graph L on the x-axis and t² on the y-axis.

#### Calculations

1. Fit a best line to the points.
2. Determine the slope.
1. Pick two points, one near each end of the best fit line.
2. Determine the coordinates of the two points, (x1 , y1) and (x2, y2)
3. Use the formula for determining the slope from two points to obtain the slope: (y2- y1) / (x2- x1)
3. The constant g can be calculated by dividing 39.5 (4p²) by the slope: g =  39.5/slope.
##### g = 39.5/_______ = ____________

The value of g on planet Earth is roughly 9.8 m/s².  Use this value to calculate the percentage error

##### percentage error = (your value for g - 9.8)/9.8 = _____________

The equation of the line on the graph is t² = (4p²/g)L.    This equation can be used, along with the g determined above, to calculate the period for any length of pendulum.

##### Question:  What will the period be for a pendulum with a length of 4 meters.   Use your value of g.

Values for g from 07 March 2000: 12.7 m/s², 9.4 m/s², 7.5 m/s², 9.3 m/s², 8.3 m/s². Lab section average: 9.44 m/s².

Here is a sample table and sample data and graph, your data and graph will have different numbers!

10t/s t/s L/m t²/s²
27.24 2.724 1.890 7.42
24.92 2.492 1.565 6.21
22.80 2.280 1.305 5.20
20.47 2.047 1.045 4.19
17.34 1.734 0.740 3.00
14.98 1.498 0.540 2.24

Sample run: g =  39.5/3.8517 = 10.2 m/s².

The result should be on the order of 9.8 m/s². The sample data above led to a result that is 5% high.  This result is not unreasonable given that the data was gathered by swinging a Kooshkin from a yarn line tied to a ceiling fan.

Computer extension: Use Excel to create the table.  Make an XY (Scatter) graph of L versus t².  Use the Chart: Add Trendline feature to add a best fit line.  In the options one can select to have the equation displayed.

Fun extension: Find a really high place from which to hang a pendulum. A cliff, a palm tree, or a tall building. Do not violate private property laws or endanger yourself or your students. Measure the height. If it is a cliff you can drop a rock and time the fall and reverse engineer the height of the cliff if need be. Or if at the waterfall in Kepar, Kitti, simply toss off the least well behaved student and time their fall (check to make sure they can swim first). Have the students predict the period by calculation based on their lab data. Then time the period.

Keep it simple features:

• Single runs of ten swings at a length avoid having to calculate an average for a series of 10t runs.
• Single 10t consumes less time than the traditional format where multiple runs are averaged.
• Any mass and line can be used hung from any suspension mechanism.
• Dividing by ten t involves moving only a decimal.
• Calculation is of the square not the square root as is usually done in a pendulum lab.

#### Images:

Setup for sample data.  A Kooshkin suspended by a yarn line from a ceiling fan.

One period:  Successive shots from left to right.

#### Credits:

Equipment used in the production of this page was paid for in part by a Title III grant from the United States Department of Education.  Development of these laboratories was made possible in part through the cooperation and assistance of the following personnel:

Science PrePilot Investigative Research Team
Brenna Lorenz, University of Guam
Casiano Soram, Pohnpei Islands Central School
Dana Lee Ling, College of Micronesia-FSM

Chuuk Star Support Team
Aier Willyander, FSM Dept. of Education
Joe Habuchmai, College of Micronesia-FSM
Solpisio Salvador, Pohnpei Islands Central School

Note: no kooshkin was harmed in the performance of this lab.

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