The Pendulum

A pendulum consists of an object suspended from a fixed point so that it can freely swing back and forth. The period of a pendulum (i.e. the length of time it takes to swing from a certain point and return to that same point) is dependent only on the length of the string that is holding the mass, a fact first discovered by Galileo. Although this is absolutely true only for an ideal pendulum (i.e. one in which the mass is concentrated at a point, the string has no mass, and there is no wind resistance on the mass as it swings), properties of ideal pendulums can still be used to mathematically determine the period of less than ideal pendulums. This is the reason that pendulums have been used in clocks for hundreds of years.

More than three hundred years ago Isaac Newton used a pendulum to estimate the speed of sound. Tour guides at Cambridge University in England demonstrate Newton’s technique by stopping at a colonnade in Neville's Court and clapping their hands so that a nice echo comes back with a slight delay. The guides explain that this was the place where Newton determined the speed of sound using an echo. He measured the length of the hallway and doubled it. This gave him the distance that the sound traveled. He then had to find the length of time between clapping his hands and hearing the echo. By dividing d the total distance the sound traveled, by t, the time needed for the sound to travel there and back, he would be able to compute the speed of sound.

This all sounds fine except for one thing. The time difference between the clap and the echo returning was less than a second. While there were clocks in Newton's day, there were no stopwatches that would measure to the accuracy needed. How did Newton measure the time? He used a pendulum. Newton knew the relationship that existed between the length of the string and the period of a pendulum. To measure the time, he varied the length of a pendulum until the period matched up with the time between the clap and its return. Through this experiment he calculated the speed of sound to be between 920 and 1085 feet per second. Not too bad with such simple instruments!

A formula relating the period (time) and the length of the pendulum would be useful in understanding the relationship (and in using a pendulum for a timepiece).
 

1. Construct a slightly less than ideal pendulum using thread and the weight provided. Adjust the length of the pendulum as accurately as possible and count the number of periods (swings) of the pendulum in a one minute interval. Record your information in the second column of the table below.

• Complete the table by computing the time (in seconds) that it takes for one period of the pendulum. Round your answer to two decimal places.

• Graph the data from the table with length of string on the horizontal axis and length of the period on the vertical axis (put length in L1 and time in L2). Is your graph linear (i.e. y = ax+b would fit the graph for some constant values of a and b)?  If not, some other possible models might be exponential (y=a*b^x for some constant a and b), logarithmic (y=a*log(x) +b), or a power function (y=a*x^b).  Do any of these seem possible?  Which one(s)?

It may be useful for the following to enter the log of values in columns L1 and L2 into columns L3 and L4, respectively. To do this, go to the STAT menu and hit EDIT. Move the curse to the "L3" at the top of column L3. Enter log(L1). Each data value in this column is now the log of the string length.

• If an exponential model for the time as a function of length is appropriate, explain why the graph of log(time) in list L4 versus length in L1 should be approximately a straight line. (Hint:  take the log of both sides of the equation y=a*b^x and "simplify").

• If an logarithmic model is appropriate, explain why a scatterplot of time versus log(length) should be a straight line.

• If a power function is appropriate, explain why a scatterplot of log(time) versus log(length) should be a straight line.

• When you have decided which you think might be the best model , use the calculator to construct such an appropriate  scatterplot. You will need to adjust the window size and make a rough copy of the plot.  Is your prediction confirmed?  Use the regression feature on you calculator to determine the suitable constants. (Ideally:  Use linear regression to fit a straight line to the converted data, and use this to determine the a and b values for your model.  Note that because of the "logs," the "a" and "b" values from your calculator using linear regressions are not the actual a and b values you are seeking.  Less ideal:  try the power function, exponential, or log function regression key on your calculator.)  Graph the regression curve with the original data to see how well the curve "fits."