EXPERIMENT
Pendulum and the
Calculation of g
Peter Jeschofnig, Ph.D.
Version 42-0269-00-01
Review the safety materials and wear goggles when
working with chemicals. Read the entire exercise
before you begin. Take time to organize the materials
you will need and set aside a safe work space in
which to complete the exercise.
Experiment Summary:
The students will learn about pendulums and how
to calculate the period of a pendulum which is
dependent on the length of the pendulum. They will
construct a pendulum and vary the mass and the
length of the pendulum to see how these differences
affect the period. The students will use the data
collected in these experiments to calculate the
acceleration of gravity (g).
www.HOLscience.com
1
© Hands-On Labs, Inc.
Experiment
Pendulum and the Calculation of g
Objectives
●●
To calculate the acceleration due to gravity by observing the motion of a pendulum
●●
To investigate the effect of varying mass on the period of a pendulum
●●
To investigate the effect of varying the length of a pendulum on the period
Estimated time required to complete this experiment: 3 hours
www.HOLscience.com
2
©Hands-On Labs, Inc.
Experiment
Pendulum and the Calculation of g
Materials
MATERIALS FROM:
Student Provides
LABEL OR BOX/
BAG:
QTY
From LabPaq
String & Weight Bag
ITEM DESCRIPTION:
1
1
Support for the pendulum
Tape
1
1
1
1
1
Set of washers
Protractor
Scale-Spring-500-g
Stopwatch-digital
Tape measure, 3-m
1
String - Qty-4.0 Meters
Note: The packaging and/or materials in this LabPaq may differ slightly from that which is listed
above. For an exact listing of materials, refer to the Contents List form included in the LabPaq.
www.HOLscience.com
3
©Hands-On Labs, Inc.
Experiment
Pendulum and the Calculation of g
Discussion and Review
A pendulum is a weight hanging from a fixed point so that it swings freely under the combined
forces of gravity and momentum. A simple pendulum consists of a heavy pendulum bob (of mass
M) suspended from a light string. It is generally assumed that the mass of the string is negligible.
If the bob moves away from the vertical to some angle θ, and is released so that the pendulum
swings within a vertical plane, the period of the pendulum is given as:
Equation 1:
Table 1: Contents of Formula
Symbol
T
L
g
θ
Description
Period of a pendulum to complete one cycle
Length of string
Acceleration due to gravity: 9.81 m/s2
Angle of pendulum in relation to point of attachment
The period is the time required for the pendulum to complete one cycle of movement. That is, if
the pendulum is released at point P, the period is defined as the time required for the pendulum
to swing along its path and return to point P.
Jean Bernard Léon Foucault was a French physicist who
invented the Foucault pendulum in 1851 to demonstrate that
Earth rotates on its axis. This pendulum typically moves back and
forth, but as Earth rotates, the direction of the pendulum appears
to move to different locations in the path located below the bob.
Some pendulums trace lines in sand whereas others, such as the
one shown in Figure 1, have numbers that align with the changing
direction of the pendulum. Earth’s rotation causes this apparent
change in direction.
www.HOLscience.com
4
©Hands-On Labs, Inc.
Experiment
Pendulum and the Calculation of g
Figure 1. Foucault pendulum in the Pantheon in Paris ©Ellas Design
If the angle of the pendulum is 30o or less, Equation 1 for the period of the pendulum can be
greatly simplified, as shown in Equation 2.
Equation 2:
T = 2π
www.HOLscience.com
5
L
g
©Hands-On Labs, Inc.
Experiment
Pendulum and the Calculation of g
Figure 2. Pendulum diagram
Table 2: Items for pendulum Figure 2
Number Explanation
1
Bob with a mass: location of highest potential energy and
lowest kinetic energy
2
Pendulum at equilibrium: location of highest kinetic
energy and lowest potential energy
3
Bob’s trajectory
4
Angle θ
5
String or rod (in equations for this lab, this is assumed to
be massless)
6
Pivot point (in equations for this lab, this is assumed to be
frictionless)
7
Amplitude: distance between points 1 and 2
www.HOLscience.com
6
©Hands-On Labs, Inc.
Experiment
Pendulum and the Calculation of g
Note that the period in this equation is independent of the pendulum’s mass at initial angle θ. Note
also that the Equation 2 is most valid for small angles. There is only 1.7% error in measurements if
the angle is kept at 30° or less. This error rises to 7.3% if the angle is increased to 60°, and to 18%
if the angle is increased to 90°.
During the cyclic swinging motion of a pendulum, there is a constant yet gradual exchange
between kinetic energy and potential energy. In order to describe this phenomenon, some terms
should be defined.
●●
Bob – The mass on the end of the pendulum
●●
Cycle – One swing of the bob back and forth
●●
Displacement – The distance from the pivot point straight down to the bottom of the bob.
See the dotted line between #6 and #2 in Figure 2
●●
Period (T) – The length of time the bob requires to swing back and forth
●●
Periodic motion – This is a motion in which the object returns to the point of origin repeatedly
●●
Frequency – The number of complete cycles per unit of time. In Figure 2, this is illustrated as
the path the bob takes starting at position 1 and returning to position 1 over a period of time
●●
Amplitude – The distance the pendulum travels from the center point out to the point of
maximum displacement. See #7 in Figure 2
In the last year of his life, while he was completely blind, Galileo
Galilei designed a clock based on the use of a pendulum. The pendulum
clock was later refined and built by Christiaan Huygens in 1657. Variations
of this kind of clock have since been produced the world over and are still
in use today.
www.HOLscience.com
7
©Hands-On Labs, Inc.
Experiment
Pendulum and the Calculation of g
Procedure: Experimenting with the pendulum
In this lab, you will vary three components of the pendulum apparatus to see if these changes
affect the period.
Part 1: Changing the amplitude
Before beginning, find a solid support from which to hang the pendulum. Ideally, there should be
a wall close to the support so the protractor and tape measure can be attached for recording the
pendulum’s movements. A bathroom or kitchen towel bar is ideal for this purpose.
A support similar to that shown in Figure 3 can be constructed and placed on a narrow shelf
or tabletop. It is important not only that the support allows the pendulum to hang freely, but
also that you are able to read and record measurements from the protractor and tape measure.
Do not allow the pendulum string to touch anything or be obstructed from any direction. The
pendulum apparatus must also be sturdy enough so that it does not bend, flex, or move in any
manner as this will introduce error into the experiment. See Figure 4 for an example setup with
the pendulum bob hanging from an over-the-door hanger.
Figure 3. Pendulum apparatus
1. Attach a small plastic bag to the spring scale.
2. Add washers to the plastic bag until the scale measures approximately 25 g total. The filled
bag will hereafter be referred to as the bob. Record this value as “Mass of bob” in the place
provided in Data Table 1.
www.HOLscience.com
8
©Hands-On Labs, Inc.
Experiment
Pendulum and the Calculation of g
Data Table 1: Trial values at varying degrees
Length of string: _____ cm = _____ m
Placement
Amplitude
of Bob
(bob horizontal
Degrees displacement) cm
Trial 1 (s)
5 cycles
Mass of bob: _____ g = _____kg
Trial 2 (s)
5 cycles
Trial 3
(s)
5 cycles
Avg. Time
(s)
5 cycles
Period
1 cycle
5o
10 o
15 o
20 o
25 o
30 o
3. Measure a piece of string that is approximately 120 cm in length. Tie the string around the
top of the bag so that the washers cannot fall out. Suspend the bob from this string so that it
measures exactly 1 m (100 cm) between where it attaches to the support and the bottom of
the bob.
4. Use tape to affix the protractor behind where the string is attached to the support so you
can measure the pendulum’s amplitude in degrees. The center hole in the protractor should
be located directly behind the pivot point. The string should hang straight down so that the
string lines up with the 90o mark on the protractor. See Figure 4 as an example of the correct
placement of the protractor.
5. Stretch the measuring tape horizontally and use tape to affix it to the wall or door so that its
50-cm mark is directly behind the bob at rest.
6. Displace the bob out to the 5o mark and hold it there. Then observe the bob’s location during
its first cycle as it swings relative to the tape measure and record the distance in centimeters
as “Amplitude (bob horizontal displacement)” in Data Table 1.
www.HOLscience.com
9
©Hands-On Labs, Inc.
Experiment
Pendulum and the Calculation of g
Figure 4. Example setup of pendulum
IMPORTANT: The pendulum must swing without obstruction and should not strike the background
as it swings.
7. With a stopwatch ready to begin timing, release (do not push) the bob and begin timing how
long it takes the bob to move through five complete cycles. Record this first trial time in Data
Table 1 for Trial 1. Repeat the procedure for the second and third trials. Then average the
three trial times to calculate the average period for one cycle, and record this value in Data
Table 1.
8. Repeat this procedure, releasing the bobs at 10°, 15°, 20°, 25°, and 30°, and recording the
results for each of the angles in Data Table 1.
Part 2: Changing the mass
9. Add more weights to the bag until the mass has doubled to approximately 50 g. Record this
value as “mass of bob” in grams into the line provided next to Data Table 2.
10.Repeat the procedure used in Part 1 using only a 10o amplitude for the starting point of the
bob. Record the data in Data Table 2.
Bob
weight
(g)
Data Table 2: Trial values for bob masses
Length of string: ________ cm = _______ m
Amplitude: 10°
Bob
weight
(kg)
Trial 1 (s)
www.HOLscience.com
Trial 2 (s)
10
Trial 3 (s)
Avg Time (s)
©Hands-On Labs, Inc.
Period
Experiment
Pendulum and the Calculation of g
Part 3: Changing the length of string
11.Remove the weights until the original mass used in Part 1 (approximately 25 g) is inside the
bag. Record this “mass of bob” in grams into the line provided next to Data Table 3.
12.Put the original bob containing the washers back onto the pendulum. Use a 10o amplitude and
perform three trials each with successively shorter lengths of string. For example, 1 m, 0.75
m, etc. Record the time in seconds into the columns labeled “Trial #1, 2, or 3 s” in Data Table 3.
Data Table 3: Trial values for string length
Mass of bob: ________ g = _______ kg Amplitude: 10o
Length (m)
.25
.50
.75
1.0
Trial 1 (s)
Trial 2 (s)
Trial 3 (s)
Avg Time (s)
Period
Part 4: Calculations
13.Solve the pendulum formula for g using the values derived from this experiment. Equation
3 will be used in calculating “g.” Substitute the average data for time and the length of
the pendulum into the formula. Calculate to three significant figures. Then calculate your
percentage error as compared to the accepted value for g, which is 9.81 m/s2. See Equation 4.
Equation 3:
Where:
●●
g = acceleration due to gravity
●●
t = time in seconds
●●
L = length of pendulum string in meters
Note: If you get very large errors, such as 20% or more, in this lab, double-check your calculations.
Equation 4:
% error = experimental value – theoretical value × 100
theoretical value
www.HOLscience.com
11
©Hands-On Labs, Inc.
Experiment
Pendulum and the Calculation of g
Questions
A. How did the change in the mass of the bob affect the resulting period and frequency?
B. How did the change in amplitude affect the resulting period and frequency?
C. How did the change in the length of the pendulum affect the period and frequency?
D. What would happen if you used very large amplitudes with the same length of string? Check
your hypothesis by experiment. What amplitude(s) did you use? What were the results?
E. Hypothesize about how a magnet placed directly under the center point would affect an iron
bob. As an optional activity, design an experiment to see if a magnetic will affect the period
of a pendulum.
F. What was the percent error in conducting this experiment? What might be a few sources for
error in your experimental data and calculations?
G. What would you expect of a pendulum at a high altitude, for example on a high mountaintop?
What would your pendulum do under weightless conditions?
www.HOLscience.com
12
©Hands-On Labs, Inc.
Experiment
Pendulum and the Calculation of g
Pendulum and the Calculation of g
Peter Jeschofnig, Ph.D.
Version 42-0269-00-01
Lab Report Assistant
This document is not meant to be a substitute for a formal laboratory report. The Lab Report
Assistant is simply a summary of the experiment’s questions, diagrams if needed, and data tables
that should be addressed in a formal lab report. The intent is to facilitate students’ writing of lab
reports by providing this information in an editable file which can be sent to an instructor.
Data Table 1: Trial values at varying degrees
Length of string: _____ cm = _____ m
Placement
Amplitude
of Bob
(bob horizontal
Degrees displacement) cm
Mass of bob: _____ g = _____kg
Trial 1 (s)
5 cycles
Trial 2 (s)
5 cycles
Trial 3
(s)
5 cycles
Avg. Time
(s)
5 cycles
Period
1 cycle
5o
10 o
15 o
20 o
25 o
30 o
Bob
weight
(g)
Data Table 2: Trial values for bob masses
Length of string: ________ cm = _______ m
Amplitude: 10°
Bob
weight
(kg)
Trial 1 (s)
Trial 2 (s)
Trial 3 (s)
Avg Time (s)
Period
Data Table 3: Trial values for string length
Mass of bob: ________ g = _______ kg Amplitude: 10o
Length (m)
.25
.50
.75
1.0
Trial 1 (s)
Trial 2 (s)
www.HOLscience.com
Trial 3 (s)
13
Avg Time (s)
©Hands-On Labs, Inc.
Period
Experiment
Pendulum and the Calculation of g
Questions
A. How did the change in the weight of the bob affect the resulting period and frequency?
B. How did the change in amplitude affect the resulting period and frequency?
C. How did the change in length of the pendulum affect the period and frequency?
D. What would happen if you used very large amplitudes? Check your hypothesis by trial. What
amplitude did you use? What is the result?
E. Hypothesize about how a magnet placed directly under the center point would affect an iron
bob? Try it and find out. Did your trial verify your hypothesis?
F. How close was your calculation of the value of g at your location? What might be a few sources
for error in your experimental data and calculations?
G. What would you expect of a pendulum at a high altitude, for example, on a high mountain
top? What would your pendulum do under weightless conditions?
www.HOLscience.com
14
©Hands-On Labs, Inc.