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 Formulasymbol / Description
T / Period of a pendulum to complete one cycle
L / Length of string
g / Acceleration due to gravity: 9.81 m/s2
θ / 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.
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 = 2p L
g
Figure 2. Pendulum diagram
Table 2: Items for pendulum Figure 2Number / 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
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.
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.
Data Table 1: Trial values at varying degrees
Length of string: 125 cm = 1.25 m Mass of bob: 30 g = 0.03 kg
Placement of Bob Degrees / Amplitude (bob horizontal displacement) cm / Trial 1 (s)
5 cycles / Trial 2 (s)
5 cycles / Trial 3 (s)
5 cycles / Avg. Time
(s)
5 cycles / Period
1 cycle
5 o / 8 / 10.38 / 10.44 / 10.39 / 10.40 / 5.51
10 o / 14.5 / 10.94 / 10.87 / 10.89 / 10.90 / 7.24
15 o / 20.8 / 10.85 / 10.81 / 10.85 / 10.84 / 8.70
20 o / 29 / 10.85 / 10.84 / 10.85 / 10.84 / 10.27
25 o / 37 / 10.65 / 10.69 / 10.63 / 10.66 / 11.70
30 o / 45 / 10.65 / 10.66 / 10.66 / 10.66 / 12.90
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.
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.
Data Table 2: Trial values for bob massesLength of string: 125 cm = 1.25 m Amplitude: 10° (15.5)
Bob
weight
(g) / Bob
weight
(kg) / Trial 1 (s) / Trial 2 (s) / Trial 3 (s) / Avg Time (s) / Period
50 / 0.05 / 10.47 / 10.56 / 10.53 / 10.52 / 2.24
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 lengthMass of bob: 30 g = 0.03 kg Amplitude: 10o
Length (m) / Trial 1 (s) / Trial 2 (s) / Trial 3 (s) / Avg Time (s) / Period
.25 / 5.91 / 5.91 / 5.94 / 5.92 / 0.28
.50 / 8.39 / 8.41 / 8.37 / 8.39 / 0.28
.75 / 9.21 / 9.22 / 9.28 / 9.24 / 0.35
1.0 / 10.43 / 10.44 / 10.44 / 10.44 / 0.36
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
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?