This experiment doesn't require a major time commitment.  If you mess something up you will get feedback from the instructor, and will have an opportunity to fix it.  However, as always, do your best to get it right the first time.

In this experiment you will make a simple pendulum and observe how its frequency of oscillation varies with its length.

The goals of this experiment include the following:

• Practice sketching graphs.
• Practice interpreting graphs.
• Obtain a concrete example which will be useful in understanding the essential concept of rate of change.
• Enhance your ability to make accurate observations.
• Quantify limits on the accuracy of data and the effect of these limits.
• Obtain information necessary to use a pendulum as an accurate timing device.

To make a pendulum tie a light string or thread, about as long as you are tall, around a relatively small dense object.  In the absence of anything more convenient you could use a couple of CDs or DVDs with a string or thread tied through the middle.  A ring might also be a good choice (but avoid using a ring with a lot of value).  A smaller and denser object would be preferable, but don't take a lot of time trying to locate the smallest densest object you can find.  Just about anything will do if it is smaller than an average fist and dense enough to sink if it is placed in water (you won't actually be placing the object in water).  The string or thread should be about 4 feet long.

If you have a ruler or a measuring tape marked in centimeters, you may use it.  If you don't, open and print out the file at the link ruler, which can be used to obtain sufficiently accurate measurements. 

You will count and time oscillations of a series of simple pendulum, of 9 different lengths.

Determine the lengths of the pendulums to be used in your experiment:

Take your height in inches, subtract 30 and divide your result by 5.  Round your result off to the next whole number.  This will be the length in centimeters of your first pendulum.  (For example if you are 91 inches tall (unlikely but not impossible), you would subtract 30 to get 61, then divide 61 by 5 to get 12.2.  This rounds off to 12, so your pendulum would be 12 cm long).

Double the length of your first pendulum.  This is the length of your third pendulum.

Double the length of your third pendulum.  This is the length of your fifth pendulum.

Double the length of your fifth pendulum.  This is the length of your seventh pendulum.

Double the length of your seventh pendulum.  This is the length of your ninth pendulum.

Write down the numbers 1 through 9 in the first column of a table, and the lengths you have obtained in the second column, each length opposite the appropriate number.  For example if your first pendulum was 12 cm long your table would look something like the following:

Sketch a rough graph of your lengths

You are going to make a graph of this information. 

Sketch the graph by hand (perhaps using the template below).  Don't use the computer to construct your graphs in this experiment.

If you want to print and use the template below you may do so, or you may make a rough sketch something like the figure below.  Don't bother to use a ruler and make a meticulous graph.  Any reasonable freehand sketch is fine (or again, if you wish, you can print the figure below and use it).

Now make a graph of pendulum length vs. number, based on your table.  This graph will have length on the vertical axis and the pendulum number on the horizontal.  The graph will look something like the one below, which however is based on the unlikely 12 cm initial length for the 91-inch-tall student.

Sketch a smooth curve through the points on your graph.  This will give you a graph much like the one below:

Estimate the pendulum lengths which would correspond to numbers 2, 4, 6, and 8.

Complete the table by filling in the lengths corresponding to pendulums number 2, 4, 6 and 8, as estimated from your graph.  Try to make reasonable estimates, but don't take a lot of time to make your estimates exact.  If you are 91 inches tall your table might be as indicated below:

Time and count oscillations for the nine different pendulums

Hold the string so that the length from the point at which you are holding it to the center of the object is equal to the first distance on your table (e.g., for the table given above that distance would be 12 cm).

• Start the pendulum oscillating, but don't make it swing too far--keep the distance from one end of the swing to the other less than half the length of the pendulum.  To keep it swinging you might need to use the fingers in which you are holding it to slightly nudge the pendulum.
• Using a clock with a second hand, determine how many times this pendulum oscillates in 60 seconds.  A complete oscillation, or a complete cycle, is from one extreme point to the other and back.  If you count every time your pendulum changes direction, you are counting half-cycles (it's fine to count half-cycles; if you do just be sure to divide by 2 to get the number of complete oscillations).  Repeat your count at least a few times, and continue until you are sure you know to the nearest whole cycle (or if you prefer to the nearest half-cycle) how many times it oscillates back and forth in a minute.  It shouldn't take more than a few trials to get in the habit of counting accurately, but if you don't count carefully it might.  A surprising number of students have difficulty making an accurate 60-second count, and some need the practice.

Repeat this for each of the nine lengths on your table.

Construct a table of the number of oscillations observed vs. the length of the pendulum.  For the example given here, corresponding to a person 91 inches tall, the table might look something like the following (the numbers in this example shouldn't be regarded as particularly accurate or useful; your counts should be more accurate than these):

Graph your data

You will graph the data in your table.

You may if you wish use the template below to help you construct your graph; you may if you wish print out the figure, or simply make a hand sketch--again be reasonably accurate but don't take the time to be be overly meticulous.

A graph of the sample data for the very tall student would look like the following:

Figure out the time required per oscillation at each length

You know how many times each pendulum oscillated in 60 seconds.  From this information you can figure out for each length how many seconds, and/or what fraction of a second, was required for a single oscillation.  You should be able to figure out how to do this.  Make a table showing the number of seconds required per oscillation vs. the length of the pendulum in cm.  The figure below corresponds to the example of the 91-inch-tall individual, and these results are not to be regarded as particularly accurate.

Construct a graph of the time required for a single oscillation vs. the length of the pendulum.  You may use the graph below as a template:

For the present example the graph might look like this:

Sketch a smooth curve to represent the data.  A good smooth curve for the sample data represented above might look like this:

The curve should come as close as possible, on the average, to the data points, but it should not 'wobble around' in an attempt to actually go through any of the data points.  The figure below represents a smooth curve that does wobble around a lot, and this isn't what you want here:

In the box below report your data, using the precise format described below.  All reported data will be collected and reported back to the rest of the class (your data will be reported anonymously--only I will know whose data is which), and if anyone's data is not in the prescribed format, it might be necessary to ask everyone to report their data again.

You should always compose your responses in a text editor or word processor, not in the form.  If you lose your connection or if your computer malfunctions, anything that hasn't been submitted will disappear.  Compose your responses elsewhere, save them so you don't lose them, then copy and paste them into the box.  Keep your copies; if the data is required later you will then be easily able to access it.

On the first line report your vccs email address, the length of the shortest pendulum in centimeters, the number of cycles counted for this pendulum in 60 seconds, and the time required for one complete cycle.  Put a comma between each pair of entries.  So for the example of the 91-inch-tall student, the first line would read

abc123@email.vccs.edu, 12, 85, .7

Each subsequent line will appear in the same format.  So the next two lines for this example student would be

abc123@email.vccs.edu, 15, 75, .8

abc123@email.vccs.edu, 24, 60, 1

It should take you only a couple of minutes to enter this information.  You should as usual use copy-and-paste to insert your email address (this will save you time and will ensure that you have given the correct address).

A typical report might appear as follows:

abc123@email.vccs.edu, 8, 100, .59
abc123@email.vccs.edu, 12, 80, .70
abc123@email.vccs.edu, 16, 70, .85
abc123@email.vccs.edu, 24, 60, 1.00
abc123@email.vccs.edu, 32, 55, 1.15
abc123@email.vccs.edu, 48, 42, 1.40
abc123@email.vccs.edu, 64, 36, 1.50
abc123@email.vccs.edu, 96, 30, 2.00
abc123@email.vccs.edu, 128, 25, 2.30

(the calculations on the above report are not necessarily accurate and you shouldn't use them as an example)

Answer questions:

Copy the questions below into a text editor or word processor and insert your answers.

1.  According to your graphs, complete the following tables

length in cm	number of cycles	time for one cycle
10
30
50
70
90
110
130

 

 

length in cm	number of cycles	time for one cycle
	10
	30
	50
	70
	90

 

 

length in cm	number of cycles	time for one cycle
		0.5
		0.9
		1.3
		1.7
		2.1
		2.5

 

2.  Is the graph of # of cycles vs. length in cm constant, increasing or decreasing?  Is it doing so at an increasing, constant or decreasing rate? 

A typical graph shows the number of cycles decreasing with length.  The graph gets less and less steep, so is said to be decreasing at a decreasing rate. 

The graph shown below contains fewer data points than you observed, but you should recognize that the general shape of the graph is similar to yours.

This wasn't requested, but for future reference a smooth curve depicting the trend of the graph might look like the following:

3.  Is the graph of time required for one cycle vs. length in cm constant, increasing or decreasing?  Is it doing so at an increasing, constant or decreasing rate?

The time required for a cycle increases with increasing length.  The steepness of the graph is decreasing, so we would say that the graph increases at a decreasing rate.

For possible future reference, a smooth curve depicting the trend of the graph might look like the following:

4.  How much difference is there between your first two lengths, and how much difference between the number of cycles counted in 60 seconds?

5.  How much difference is there between your first two lengths, and how much difference between the corresponding times required to complete a cycle?

6.  How much difference is there between your last two lengths, and how much difference between the number of cycles counted in 60 seconds?

7.  How much difference is there between your last two lengths, and how much difference between the corresponding times required to complete a cycle?

8.  Is your graph of number of cycles counted vs. length in cm steeper, on the average, between the first two lengths or between the last two lengths?

The slope of the graph is negative, but the steepness of this graph (steepness refers to the magnitude or absolute value of the slope) is decreasing.  The graph will be much steeper between the first two lengths than between the last two.

9.  Is your graph of time required to complete a cycle vs. length in cm steeper, on the average, between the first two lengths or between the last two lengths?

The steepness of this graph is decreasing.  The graph will be much steeper between the first two lengths than between the last two.

10.  The curve you sketched for your graph of (time required to complete a cycle) vs. (length) cannot possibly pass through the center of each of your points. 

• What is the greatest vertical distance between a point of your graph and the curve? 
• What do you think is the least vertical distance? 

(For example, in the figure below a curve has been constructed based on three data points.  The first and third data point lie slightly above the curve, the second point slightly below.  The second point is probably the one which lies furthest from the curve, at a distance of approximately .03 vertical units below.  This distance is roughly estimated based on the scale of the graph.  The first point is perhaps .01 vertical units above the curve, and the third is perhaps .02 units above.)

Once you have inserted your answers, submit them by copying and pasting them into the box below:

You should have several points on your graph, and a smooth curve approximating the trend of your points.  If you had trouble answering this question consider the following series of questions: 

• Which of your graph points lie above your smooth curve (e.g., in the above sketch the first and third graph points lie above the curve). 

• Which lie below the curve (e.g., in the above sketch the second point lies below the curve). 

• If none of your points lie either above or below the curve, then the curve goes exactly through the center of each of your points.  If this is the case then one of the following things is likely: 

You drew a very thick line for your curve. 

You measured the lengths of your pendulums accurate to within a fraction of a centimeter (very possible) and you counted the swings accurately to within a small fraction of a cycle (unlikely but not out of the question) and your data is therefore sufficiently precise that your curve 'pegs' every one of the points (again, not impossible but not likely). 

You let your curve weave around in such a way as to go through all the points (not the way to sketch the trend of the data).   

• Which point lies the furthest from the curve, and how many units above or below the curve is it?  What are the distances of some of the other points above or below the curve?

In the box below

• Describe how you constructed your pendulum and out of what (what you used for the mass, its approximate dimensions, what it is made of, what sort of string or thread you used--be as specific as possible). 
• Describe its motion, including an estimate (you don't have to measure this, just give a ballpark estimate) of how far it swung from side to side and how this distance varied over the time you counted.
• Describe what you mean by a 'cycle'.  Different people might mean different things, but there are only a couple of reasonable meanings.  As long as you describe what you mean we will all understand what you measured.
• 'Frequency' means the number of cycles in a unit of time.  Your counts are frequencies, in cycles/minute.  'Period' means time required for a cycle.  Explain how you used your observed frequencies to obtain the periods of the nine pendulums in this experiment.

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