PhysicsLAB: Simple Pendulums: Class Data

Purpose: To let students become familiar with measuring with meter sticks and triple beam balances, identifying vibrations and to develop critical thinking skills. Secondly, to introduce student to a technique of data analysis using linear regression.

Each group needs the following equipment:

one prepared pendulum in classroom
computer station
meter stick
EXCEL spreadsheet

Procedure: Students will work in teams of two. On each team, one member will work with one of the suspended pendulums provided along the crossbeams at either the front or the back of the classroom while the second member will record the data provided in an EXCEL spreadsheet.

The team member working with the pendulum must initially measure the length of his assigned pendulum. The length of a pendulum is from its point of suspension (in this case, the top knot on the beam from which is pivots) to the center of mass of its bob (the mass hanging from its end).

The team member working with EXCEL is to open this worksheet and get ready to record data. When all teams are ready, the first data to be entered in column B of the spreadsheet will be the length of each pendulum in centimeters.
Next we will count the number of vibrations each pendulum completes in three 30-second trials. A vibration is one complete back-and-forth motion. In the diagram shown below, if the pendulum is released at point A, then a complete vibration will have occurred once the pendulum bob has traveled from A to B and back to A. When releasing the pendulum, make sure that its amplitude is very small, less than 15º, as larger angles may result in inaccurate results.

Counting vibrations:

One member of the class is now to serve as a class timer. All teams are going to count the number of vibrations at the same time. When the class timer announces start, all pendulums will be released. When the class timer announces stop, all teams are to stop counting the number of vibrations of their pendulums.

For the purposes of this experiment, we can estimate quarter cycles - that is, if the pendulum is on its way towards B and the class timer tells you to stop while the pendulum is passing through its equilibrium position, then a ¼ cycle has passed. If the pendulum is at B then a ½ cycle has passed. If the pendulum is returning from B towards A and is passing through equilibrium, then 3(¼) of a cycle has passed.

Once all teams have reported and all teams have recorded the number of vibrations for every team for Trial 1 in column C of the spreadsheet, the timing, counting, and recording process will be repeated for Trial 2 (column D) and Trial 3 (column E).

Calculating frequency and period:

Once all teams know the data for all of the suspended pendulums, the focus will switch from all teams working together, to each team completing its own spreadsheet.

To do this, each team must first calculate the average number of vibrations for each pendulum in column F, the average frequency in column G, the average period in column H, and the square of the average period in column I.

In EXCEL, the following formula should be typed in to cell F4, =average(c4:e4). Once entered, you should see the average of the number of vibrations for the first pendulum's three trials. To copy that formula for the remaining pendulums, highlight column F from cell F4 to F18 and use the key strokes (Alt Edit Fill Down) to fill down the formula through the remainder of the column.

In cell G4, type =F4/30 to calculate the average frequency. Once entered, you should see the average frequency calculated for the first pendulum. Frequency is defined as the number of vibrations in one second.

frequency = number vibrations / total time
it is measured in vibrations per second (vib/sec or hertz)

So our formula in G4 took the average number of vibrations in F4 and divided it by 30, the total number of seconds allowed for each trial. To copy that formula for the remaining pendulums, highlight column G from cell G4 to G18 and use the key strokes (Alt Edit Fill Down) to fill down the formula through the remainder of the column.

In cell H4, type =1/G4 to calculate the average period. Once entered, you should see the average period calculated for the first pendulum. Period is defined as the number of seconds required for each vibration. It is the reciprocal of the frequency.

period = total time / number vibrations
it is measured in seconds per vibration (sec/vib or sec)

So our formula in H4 took the average frequency in G4 and calculated its reciprocal by dividing it into 1. To copy that formula for the remaining pendulums, highlight column H from cell H4 to H18 and use the key strokes (Alt Edit Fill Down) to fill down the formula through the remainder of the column.

In cell I4, type =H4^2 to calculate the square of the average period. Once entered, you should see the square of the value in cell H4. To copy that formula for the remaining pendulums, highlight column I from cell I4 to I18 and use the key strokes (Alt Edit Fill Down) to fill down the formula through the remainder of the column.

Once you have complete column I, columns K and L will automatically be filled out for you from the data you programmed respectively in columns C and I. In addition, you will see a graph of your data with the equation of its trend line, or line of best fit.

After you fill in the requested information in column M, save your file in your period folder as

LastnameLastnamePendulum.xls

where each member's last name is included in the file's name.

After your EXCEL file has been completed and saved, come to the print station and print a copy of each file for your lab report and for any group member who would like to keep a copy.

Conclusions and Error Analysis:

What is your group's EXCEL spreadsheet's filename?

1(a) Using the form, y = mx + b, write the specific equation for the regression line shown on your printout. Do not use x and y, use the appropriate variables for each axis.

What is the equation of your line?

What are the units on your line's slope?

1(b) Under your equation on your printout show your calculations to extrapolate the period of a pendulum whose length equals the height of an average flag pole, 10 meters.

What would be the period of this pendulum?

2. We will now solve for the gravitational field strength (g) in our lab room by using the slope of your regression line. To do this, set the numerical value of your line's slope equal to the expression 4π²/g and solve for the value of g.

Since the expression 4π²/g represents the slope of your graphs, the units for measuring g would be the reciprocal of the slope's units: sec²/m. Hence, g is measured in m/sec². Be sure to include these units on your value for g. Show your work on your graph's printout.

What is the value of gravity according to your experiment?

3. If the accepted value for the gravitational field strength at sea level is 9.8 m/sec², calculate your experiment's percent error. Show your work on each printout.

The formula to calculate % error is

What is the percent error for your experiment?

4. State an experimental error, procedural or system, and a correction which would minimize its affects on the outcome of future experiments. Perhaps this physlet animation entitled "The Pendulum" can give you some guidance.

source of error:

correction:

5. In our original data, there were two pendulums that had more massive bobs. We now want to consider whether the mass of the pendulum's bob affected the frequency of the pendulum. Support your answer by comparing or contrasting the data for the more massive bobs.

Here are some things you might want to think about when discussing your answer: were the results for these masses outliers or did they merge systematically with the results of the smaller masses? were there similar lengths with different masses that had the same or closely related frequencies?

Lab Report. When finished, each lab member is to keep their own copy of the EXCEL graph for their notebooks. As a group, you are to turn in one graph in to the one-way box with your calculations for conclusions #1, #2 and #3.