Physics 40 Lab 6: The Simple Pendulum
In this experiment, your job is to test equations that describe the motion of a simple pendulum.
The pendulum shown has bob mass m, string length l (= R), and
initial displacement angle given by .
Applying Newton’s second law to the tangential and radial
components of force we can derive equations of motion as a
function of the displacement angle, where at is the tangential
acceleration, v is the tangential velocity, and T is the tension in
the string:
 F  mg sin  ma
t
at  g sin 
F
t
r
 T  mg cos 
 v2

T  m   g cos 
R

(1)
mv 2
R
(2)
An analysis of the oscillatory motion gives an equation for the period of oscillation, P (s):
P  2
l
g
(3)
Notice that the equation for the period of oscillation for a simple pendulum is independent of the
mass of the bob and the angle of release. It only depends on the length of the string and the
acceleration due to gravity. We will test to see if indeed this is validated by experimental data.
Definitions
Period of the Pendulum: the time measured in seconds for the pendulum bob to go from one
extreme position to the other and back again: the time it takes to complete one cycle.
Amplitude: The size or angle of the arc the pendulum is displaced from equilibrium, for this lab
measured in degrees.
Equipment
Lab Jack, Table clamp, Long Rod, Pendulum clamp, 3 pendulum balls – (wood, aluminum,&
brass), Single pendulum string, Protractor, Lab Pro, Motion detector, force probe & cables
Procedure Set up the pendulum and probes as shown by your instructor. Open an Excel data
sheet and the LoggerPro file in the Physics 40 Lab 6 Pendulum folder. You will need to open a
word document for the analysis part of your group lab report.
Part I: Testing Equations (1), (2) and (3)
The first part is mostly a qualitative – we re not taking a lot of data or doing error analysis. You
will explore the motion of a simple pendulum and test the equations to see if they are believable.
Set up the pendulum with the brass ball but first measure the mass of the brass bob and the length
of the string and record these in your data sheet. Use the protractor to set the angle of release to
something less than 15 degrees and collect data so that you get a smooth sinusoidal curve for the
data plots. If there is ‘clipping’ (the bob and data curve disappears) reduce your release angle.
Once you find a good angle to use, note it in your data sheet.
Set the bob in motion and collect both F vs t and x vs t data plots in Logger Pro. Copy and paste
these into your report and do/answer the following (you can copy and paste these too!):
1. Describe the type of motion given by the graphs. Do equation (1) and (2) agree with the
data plots? Find the values for the acceleration, velocity and the tension from the data
plots at the release angle and at the bottom of the path? Note these in the data sheet.
Calculate those values given by (1) and (2) (show the calculations – you can leave space
and do them by hand) for the release angle and at the bottom of the path (you will have to
use the velocity found from the data plot to use equation (2)). Do the values agree? What
is the percent discrepancy?
2. Determine the period of oscillation from your graph by measuring the total time for 10
cycles and dividing by 10. Note this in your data sheet. Copy and paste the plot into your
lab and show how you found the time for 10 cycles. Now calculate the value for the period
using equation (3). Show the calculation and put the value in your data sheet. Do they
agree? What is your percent discrepancy?
3. Briefly summarize your results.
Part II: The Period of a Simple Pendlum: Dependence on mass, length and amplitude
Effect of Mass
You will set up a pendulum with a 3 different bobs of different masses while keeping the string
length and amplitude constant to find if mass affects the period of oscillation.
Effect of Mass
1. Weigh the bobs & string to get the total mass. Measure the length of the string. Set up your
pendulum with the heaviest bob and determine a fixed angle (use the one from above.) Enter
these values into your data table.
2. Use the motion detector to graph the periodic motion for as many cycles for which the data is
good (about 10) and find the period for one cycle. Calculate the period using formula (3).
Enter into these values into your spreadsheet. Repeat for the two other bobs, keeping the
string length and amplitude constant.
3. Determine the % difference between the theory and experiment. Does the period change with
changing mass?
Effect of Amplitude
1. Set up a pendulum with the heaviest bob keeping the string length and mass constant. Measure
the period as above for at least 3 different angles, ranging between 10-40 degrees. Calculate the
period for the pendulum using equation (3) for the three different angles. (opps – it doesn’t
depend on the three angels!! ;-) Find the percent difference. Does the period change with
changing mass?
Effect of Length
1. Set up a pendulum with the heaviest bob. Keeping the mass and amplitude constant, measure
the period as above for at least 5 different string lengths. Be sure to take data three times and get
an average period and an average deviation for each string length.
2. Using equation (1) and the values of your length, calculate the period for the pendulum and
enter the value into the data table. Find the percent difference.
3. Graph your experimental data, Period (s) vs Length (degrees). Label your graph
You may have to change the maximum values on the axes.
4. Do a power-fit for the graph. What is the value of the exponent? What is the value of the
coefficient? What is the R squared value? Does your graphical data and analysis support the
theoretical equation (3)?
5. Determine a value of g from this coefficient and give a percent error from the accepted value of
9.80m/s2. How does this method compare to the other ones you have tried in this lab class?
Conclusion: What can you conclude about the effects of mass, length and amplitude on the
period of a pendulum? Write a summary of your results.