Determining the Empirical Equation for the Period of a Simple Pendulum

Tyler Harvey

Partners: Gabriel Moore and Christopher Hoffman

Dr. Clyde Smith, Physics 101 Laboratory

16 September 2010

On my honor, I have neither given nor received unauthorized aid on this report.________________________________

Introduction

An ideal simple pendulum is nothing more than a massive weight, called a bob,

suspended from a massless rod or string, and attached at fixed, frictionless pivot point. Such a

pendulum will oscillate back and forth around a central point of equilibrium that is exactly in line

with the pivot point in the direction of the acceleration due to gravity. The amplitude of the

pendulum describes the angle between the equilibrium line and the rod when the pendulum is

farthest from equilibrium. This angle is the same as the initial angle of a pendulum swing. The

period of the pendulum is measured in seconds and describes one complete swing of the

pendulum.

The properties of the simple pendulum were first studied by Galileo in 1602. Through his

research, he discovered isochronism, or the independence of the period and amplitude of a

pendulum. It is this fact that makes pendulums useful for clockmakers, since they are able to

keep a constant time as long as they continue moving. Galileo also observed that the period is

independent of mass of the bob, and directly proportional to the square root of the length of the

string or rod. From these facts, the equation for the period of a pendulum has been accepted as

Τ=2 π ( lg )

12, where l the length of the string or rod is and g is the acceleration due to gravity.

The goal of this experiment is to prove Galileo’s observations by calculating the

empirical equation through experimentation. The specific objectives are to 1) determine the

correlation between mass of the pendulum bob and period, 2) determine the correlation of the

amplitude or initial angle of the pendulum and period, 3) determine the correlation between

string length and period, and 4) to develop an empirical equation based on the findings and

compare it to the accepted equation.

Page 2

Methods

In order to determine the correlation between mass of the pendulum bob and the period of

the pendulum, the apparatus was set up as in Diagram 4.1, with an initial angle (amplitude) of

20° and a string length of .55 meters. Seven bobs of varying masses were attached to the

pendulum and the period was recorded using a photogate timer in pendulum mode. The results

were recorded in Table 4.1 and then graphed in Figure 4.1 using Microsoft Office Excel 2010.

Diagram 4.1 – The pendulum apparatus used for this experiment.

Page 3

The same apparatus was used to determine the correlation between the amplitude and the

period. The string length was kept at a constant .55 meters and the mass of the bob was also

constant at 87.4 grams. Seven trials were performed with a different value each time. Again, the

period of the pendulum was recorded using a photogate timer in pendulum mode. The results

were recorded in Table 4.2 and then graphed in Figure 4.2 using Microsoft Office Excel 2010.

For the third objective of determining the correlation between the string length and the

period, the apparatus was set up as in Diagram 4.1 with a constant amplitude of 20° and a

constant bob mass of 87.4 grams. Eight trials were performed, with the string length being varied

by a constant increment in each. The period was recorded using a photogate timer in pendulum

mode and the results were recorded in Table 4.3 and then graphed in Figure 4.3 using Microsoft

Office Excel 2010. The square root of the string length was then taken and the results were

graphed in Figure 4.4, again using Microsoft Office Excel 2010.

Results

Table 4.1 – Mass of Pendulum Bob vs Period of PendulumMass (grams) Period (s)

1.5 1.47701.6 1.47832.1 1.48192.5 1.484656.0 1.498787.1 1.487787.4 1.482493.6 1.4992

Page 4

Table 4.2 – Initial Angle of Pendulum vs Period of PendulumAngle (°) Period (s)

10 1.486020 1.496330 1.514540 1.535950 1.564160 1.507170 1.6533

Table 4.3 – String Length vs Period of PendulumLength (m) Period (s)

.85 1.8767

.75 1.7541

.65 1.6345

.55 1.5061

.45 1.3582

.35 1.2052

.25 1.0339

.15 0.8146

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

R² = 0.407598870269321

Period vs Mass

Mass (grams)

Perio

d (s

)

Figure 4.1 – Period of Pendulum vs Mass of Pendulum Bob

Page 5

0 10 20 30 40 50 60 70 800

0.5

1

1.5

2

R² = 0.588903367262383

Period vs Angle

Angle (°)

Perio

d (s

)

Figure 4.2 – Period of Pendulum vs Initial Angle of Pendulum

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.5

1

1.5

2

R² = 0.999499535380902

Period vs String Length

String Length (m)

Perio

d (s

)

Figure 4.3 – Period of Pendulum vs String Length

Page 6

0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

f(x) = 1.97951665707721 x + 0.0405697609925684R² = 0.999627209923444

Period vs √String Length

√String Length (√m)

Perio

d (s

)

Figure 4.4 – Period of Pendulum vs Square Root of String Length

Discussion

Based on the findings of this experiment, it was determined that there is no correlation

between the mass of the pendulum bob and the period of the pendulum. This was determined

from the graph of the data, upon which a linear regression was performed. The r2 of the

regression was .4076.

It was also determined that there is only a small positive correlation between amplitude

and period. The r2 of the linear regression performed on the graph was .5889, indicating that the

two variables are approximately independent.

The last finding of this experiment is that there is a strong positive correlation between

string length and period. The r2 of the quadratic regression is .9995. In order to create a linear

relationship, the square root of the string length was plotted against period. The r2 of the linear

regression performed on the graph was .9996 indicating that the two variables are approximately

Page 7

directly proportional. The slope of this trendline was found to be 1.9795 s/√m. The empirical

equation for the period of a simple pendulum was found to be T = 1.9795 s/√m (ℓ) + 0.0406s.

Error Analysis

The accepted slope for the equation of the period of a simple pendulum is 2π/√g or

2.007089923 s/√m. The experimental slope calculated from this experiment was 1.9795 s/√m.

% Error = |(experimental – accepted)/accepted| * 100

% Error = |(1.9795 s/√m - 2.007089923 s/√m)/ 2.007089923 s/√m| * 100

% Error = 1.37%

This small degree of error was most likely caused by human error when measuring the

string length. This could be minimized or eliminating by using a more precise measuring tool to

measure the length.

Conclusion

This experiment was successful in all four of its objectives. Galileo’s observations that 1)

period is independent of bob mass, 2) period is independent of amplitude, and 3) period is

directly proportional to the square root of string length were all verified. Additionally, an

empirical formula for the period of a simple pendulum was determined and found to be within

2% of the accepted value, verifying that accepted equation is correct.

Page 8