GRAPHING PENDULUM MEASUREMENTS

OBJECTIVE:

An object swinging at the end of a string forms a pendulum. In this

investigation you will let a metal weight swing from different lengths of string.

Make sure you use the same weight for all of the trials in the experiment.

For each length, you will count the number of swings the pendulum

makes in one minute. This value represents the frequency of the pendulum's

motion. You will then graph your data and use the graph to predict frequency

values for pendulum lengths you did not actually measure.

MATERIALS:

String, 1 m long

metal weight with a hook on the end

meter stick

Masking tape

Graph paper

Clock or watch that indicates seconds

HYPOTHESIS:Make a guess as to how you think the length of the pendulum will affect

the frequency at which it swings.

PROCEDURE:

1. Attach one end of a 1-meter-long string to a metal weight. Find the point on

the string that is 80 cm from the center of the weight (center of mass). Place

the bottom edge of a piece of masking tape across the string at that point.

Use the masking tape to suspend the pendulum from the side of a table or

other support.

2. Hold the pendulum ball about 10 cm to the side and release it. Make sure

the ball swings freely. Count the number of complete swings in one minute. A

complete swing is from the point of release to the other side and back to

the same point. Record your observation in the table provided. Record the

total number of complete swings your pendulum had for one minute, in DATA

TABLE 1.

3. Change the length of the pendulum to 70 cm. Again count the number of

complete swings in one minute. Record your observation in DATA TABLE 1.

4. Repeat the measurement of the number of complete swings with the pendulum

length at 60 cm, 50 cm, 40 cm, and 30 cm. Again record your observations in

DATA TABLE 1.

DATA TABLE 1: Number of complete swings observed per minute for

different lengths of a pendulum

Pendulum Length in centimeters Number of complete swings in one minute

_______80_________________________________________________________

_______70________________________________________________________

_______60________________________________________________________

_______50_________________________________________________________

_______40_________________________________________________________

_______30_________________________________________________________

CALCULATIONS:

There are many ways to deal with raw data in order to put it into a form that we

can both understand and predict results for other events. A graph

is a wonderful tool for interpreting data, and in this investigation,

the best method of dealing with our data.

5. Make a graph of your data manually. Remember there are some basic "rules" for

graph construction. Every graph needs a title and both axes of the graph

need numerical values and labels telling what they represent. Also the axes need the

units of measurement the numbers are in. In our case cemntimeters and number of swings.

By convention, we graph a variable (the thing we observe) and a

constant (that which we have set). We put the variable on the vertical axis,

and the constant on the horizontal axis. Our variable here is the number of

complete swings in one minute, and the constant is the length of the

pendulum.

After you plot the points from the pairs of data in your DATA TABLE, the

next step is to fit a line to the data points. When we take data in an

investigation, it often contains some error from our observations. This is

normal. We could have miscounted, or our timing was not exact. We fix this

in the graph, by fitting the line to average our data. First, look at the

data points and decide if it is a straight line, or a curved line. Now we

draw a line through our data points that best fits all the points. If it is

a straight line we always use a ruler or straight edge to draw the line, and

if its is a curved line we have to draw it free-hand. We have to be careful

to draw a smooth curve. All of our data points will not necessarily be on

the line, but the line best "averages" our points.

GRAPHING WITH THE TI-83

Enter your data by going to Stat button, Edit Mode and typing the length of the

pendulum under the List One column(L1) and the number of swings under List Two(L2).

After entering your data go back to the Stat button, Calc Mode and go to linear

regression (#4) to calculate the slope-intercept equation. After this you may want to

VARS, specifically Y-VARS, Functions, Y1 and you will be able to graph the equation

automatically on the Calculator and see the line of best fit on the

TI-83 as well.

Now you know how to properly fit a line to experimental data for future

investigations.

PREDICTING FROM YOUR GRAPH:

6. Describe the line that passes through the points plotted on your graph.

_________________________________________________________________

7. Now use your graph to predict the behavior of a pendulum in conditions

you did not measure, but are represented on your graph.

How many swings per minute would you predict for a pendulum 55 cm long?

_________ swings / minute

How long a pendulum would you predict will make 60 swings in one minute

(this is one per second)?

_________ cm

CONCLUSION:

How does the number of swings change as the pendulum is shortened?

_________________________________________________________________

FURTHER INVESTIGATION:

Now let us change things a bit. Instead of the length of string being changed we will

change the mass of the metal weight that is on the end of a 50 centimeter string.We

will determine the effect of changing the weight on the end of the string and the

frequency of the pendulum. Make sure that you keep the string length the same for

all of the trials and only change the mass of the weight on the end of the string.

Put the answers in DATA TABLE 2.

DATA TABLE 2: Number of Complete Swings observed per minute for different

masses of metal weights

Masses of Metal Weights in grams Number of Complete Swings per minute

_____________1000__________________________________________________________________ 1000

______________500__________________________________________________________________

______________250__________________________________________________________________

______________100_________________________________________________________________

_______________50_________________________________________________________________

_______________25_________________________________________________________________

Make a graph that shows how the frequency of the pendulum changes as the mass of

the metal weight changes on the end of a 50 centimeter string.

Don't forget what you learned about graphing when you did the first

graph. Do the points on your graph lie on a straight line?Determine the equation of

your line. Determine what the frequency of the pendulum would be if you used a 800

gram weight on the end of the string. How about a 165 gram weight on the end of the

string, how would that affect the frequency?

________

There are other investigations you could do with a pendulum. Write a

hypothesis stating how the variables of frequency of complete swings relates to

the variable of how far back from center you start the

pendulum swinging.

Now design an investigation for your hypothesis. There may be other aspects

of a pendulum that you can think of to investigate. Think about it.