A Simple Pendulum’s Oscillation Period
Abstract: An experiment was conducted to derive a mathematical relationship to predict the period
of a simple pendulum as a function of mass, length, and angle of oscillation. The period was found
to be independent of angle for oscillation angles less than 30.0 degrees. The period was found be
independent of mass ranging from 5.90 grams to 88.2 grams. The period was found to be dependent
on string length ranging between 10.0 cm and 100.cm. For oscillation angles less than 30.0 degrees
and masses between 5.90 grams and 88.2 grams the relationship between period and length was
found to be: T = 2.01 L 0.51 . The oscillation period for a 1.771 meter length was predicted to be
2.86 seconds and measured to be 2.64 seconds, a 1.83% difference.
Introduction: The purpose of this lab is to experimentally determine and verify a mathematical
relationship that predicts the period of oscillation for a simple pendulum as a function of oscillation
angle, mass and length.
Equipment and Materials:
1. Ring Stand
2. Cross bar for ring stand
3. Clamp for ring stand
4. String
5. Stopwatch
6. Drilled ball set- 7 different masses
7. Protractor
8. Meter stick
9. Masking tape
10. Scale-500. gram capacity
11. Sharpie marker
12. Scissors
Experimental Procedure for Changing Angle, Constant Mass and Length
1. Set up ring stand, cross bar and clamp as shown below. Clamp the ring stand onto the lab
table. The edge of the ring stand base should be even with the edge of the lab table.
clamp
clamp
Lab Table
2. Measure out approximately 1.2 meters of string. Tie a knot in one end of the string, and tape
the other end of the string with masking tape, so it can be easily threaded through the
pendulum balls.
3. Choose one pendulum ball to begin with. Mass the ball and record its mass in grams in the
data table. Thread the taped end of the string through the hole in the ball. Pull the string all
the way through the ball, so the knot is in the center of the ball.
4. Measure 1.0 meter from the center of the ball on the string. Mark this spot with a Sharpie
marker. Measure 90 cm from the center of the ball and mark this location with the Sharpie.
Continue marking the string in 10 cm increments until the 10 cm position is reached.
5. Align the 1.0 meter mark with the end of the cross-bar on the ring stand. Tape it in place with
masking tape on the top of the cross-bar.
1 m mark on string
6. Be sure that the masking tape does not get in the way of the string moving freely. It should
only be covering the string on the top of the bar, not on the end.
7. Confirm that the length of the string is exactly 1.00 meter to the center of the ball by using
the meter stick after the tape is secured.
8. One person will position the string at an angle of 5.0 degrees, while holding onto the ball,
while another person confirms the angle using a protractor. See below.
5.0 degrees
9. Release the ball without putting any external force on it. Allow the ball to swing freely for 2
cycles, then start the stopwatch and begin counting at “zero”. Time ten oscillations of the
pendulum and record the time for those ten oscillations in the data table.
10. Repeat steps 5 – 9, while increasing the oscillation angle for each trial.
Experimental Procedure for Changing Mass, Constant Length and Angle
1. Keep the ring stand and cross-bar set-up the same as in the previous experiment. For all
masses: String length = 1.00 m and Oscillation angle = 20.0 degrees.
2. Begin with the same pendulum ball as used in the varying length experiment. Measure and
record it’s mass.
3. Using 20.0 degrees as the angle of oscillation, position and release the ball. Time 10 cycles.
Record the time.
4. Now, remove the pendulum string from the bar, and remove the ball from the string. Choose
another pendulum ball, mass it, and record it’s mass.
5. Slide the new ball onto the string and re-attach the string to bar. Check that the length from
the top of the bar to the center of the ball is still 1.0 m.
6. Repeat steps 3-5 for the remaining 5 balls.
Experimental Procedure for Changing Length, Constant Mass and Angle
1. Keep the ring stand and cross-bar set-up the same as in the previous experiment. Choose one
pendulum ball to work with, mass it and record the mass.
2. Mark the length of the string with a Sharpie every 10.0 cm.
3. Thread the ball onto the string, and attach it the bar so that the length of the string from the
bar to the center of the ball is 1.00 m.
4. Using 20.0 degrees as the angle of oscillation, release the ball and time 10 cycles. Record the
time.
5. Now, remove the pendulum string from the bar, and move it up to the 90.0 cm mark. Re-tape
it in place. Repeat step 4, and record the time for 10 cycles. Change the string to 80 cm in
length and repeat step 4. Continue to do this for all lengths until 10.0 cm is reached.
Raw Data – Table 1:
Experimental Measured Data for Changing Angle, Constant Mass and Length
Trail #
1
2
3
4
5
6
7
8
9
Mass (g)
+/- 0.1 g
62.1 g
62.1 g
62.1 g
62.1 g
62.1 g
62.1 g
62.1 g
62.1 g
62.1 g
time (s)
+/- 0.1 s
20.4
20.4
20.3
20.3
20.7
20.8
21.1
21.8
22.4
Cycles (#)
10
10
10
10
10
10
10
10
10
Angle (degree)
+/- 0.5
5.0
10.0
20.0
30.0
40.0
50.0
60.0
80.0
90.0
Length (cm)
+/- 0.1 cm
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
Raw Data – Table 2:
Experimental Measured Data for Changing Mass, Constant Length and Angle
Trail #
1
2
3
4
5
6
7
8
9
Mass (g)
+/- 0.1 g
62.1
62.1
62.1
62.1
62.1
62.1
62.1
62.1
62.1
time (s)
+/- 0.1
19.3
18.3
17.1
15.7
14.1
11.4
11.1
9.0
6.4
Cycles (#)
10
10
10
10
10
10
10
10
10
Angle (degree)
+/- 0.5
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
Length (cm)
+/- 0.1 cm
90
80
70
60
50
40
30
20
10
Raw Data – Table 3:
Experimental Measured Data for Changing Length, Constant Mass and Angle
Trail #
1
2
3
4
5
6
7
*Prediction
Mass (g)
+/- 0.1 g
71.1
88.2
65.6
51.2
5.9
21
62.1
62.1
time (s)
+/- 0.1
20.4
20.3
20.6
20.4
20.2
20.4
20.5
26.4
Cycles (#)
10
10
10
10
10
10
10
10
Angle (degree)
+/- 0.5
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
Length (cm)
+/- 0.1 cm
100
100
100
100
100
100
100
177.1
Results:
For all trials, oscillation period was calculated from the raw data by dividing the time for ten
oscillations by 10 using the formula T = time for ten oscillations/10. The resulting calculations
for Table 1 Raw Data are shown below.
Table 4: Results for Changing Angle, Constant Mass and Length
Trial #
1
2
3
4
5
6
7
8
9
Angle (degree)
5.0
10.0
20.0
30.0
40.0
50.0
60.0
80.0
90.0
T (s)
2.04
2.04
2.03
2.03
2.07
2.08
2.11
2.18
2.24
Data from Table 1 was analyzed and the period of oscillation was determined to be independent
of angle, if the angle was held to 30.0 degrees or less. 20.0 degrees was chosen as the angle of
oscillation in order to eliminate angle as a variable. The graph is shown in figure 1 below.
Figure 1: The period of oscillation changes as the oscillation angle gets
large. For angles 30 degrees or less, the period is constant.
2.30
2.25
Period (s)
2.20
2.15
2.10
2.05
2.00
0.0
10.0
20.0
30.0
40.0
50.0
60.0
Oscillation Angle (degrees)
70.0
80.0
90.0
100.0
Table 5: Results for Changing Length, Constant Mass and Angle
Trial #
1
2
3
4
5
6
7
8
9
Length (m)
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
Period (s)
1.93
1.83
1.71
1.57
1.41
1.14
1.11
0.901
0.636
Period was plotted as a function of length and the data analyzed using Excel’s curve-fitting
algorithm. The graph is shown in Figure 2 below.
Figure 2: The oscillation period is proportional to the squareroot
of the string's length for a simple pendulum.
2.50
Period (s)
2.00
1.50
1.00
T = 2.01 L 0.506
R2 = 0.989
0.50
0.00
0
0.2
0.4
0.6
0.8
1
Length (m)
A prediction was made using the equation of the best fit curve shown above for a string length of
1.77 meters. The predicted oscillation period was 2.68 seconds. The measured oscillation period was
2.64 seconds which is 1.83% different from the predicted value.
Table 6: Experimental Results for Changing Mass, Constant Length and Angle
Trail #
1
2
3
4
5
6
7
Mass (kg)
0.071
0.088
0.066
0.051
0.006
0.021
0.062
Period (s)
2.04
2.03
2.06
2.04
2.02
2.04
2.05
Period was plotted as a function of mass and graphed using Excel. See figure 3 below:
Figure 3: There is a constant relationship between oscillation
period and Mass.
2.5
Period (s)
2
1.5
1
0.5
0
0
0.02
0.04
0.06
0.08
0.1
Mass (kg)
Conclusions:
For angle of oscillation 30.0 degrees or below, period was found to be independent of angle. For
masses between 5.90 grams to 88.2 grams, period was found to be independent of mass. For lengths
between 10.0 cm and 177.1 cm, period was found to have a square root relationship to length and the
equation for that relationship was found via curve-fitting to be T = 2.01 L0.51. A prediction was made
for a 1.771 m string length and was verified to within 1.83%.