Section 1: Experiment and Observation
A. Objective
The object of this experiment is to demonstrate the properties and calculations of
simple harmonic motion. In the process, the value of the acceleration due to gravity
will also be calculated and confirmed.
B. Equipment Used
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Computer with Excel
String
Spring scale
Stopwatch
Weights
Meter stick
Tape measure
C. Data
Three procedures were done to test and demonstrate simple harmonic motion. In all
these procedures, I made one change from the method described in the lab
description. Instead of 5 cycles, I timed 10 cycles so as to get a more accurate time for
the period.
Procedure 1
In this procedure, a fixed weight was put at the end of a string. The string was hung
above a doorway as a pendulum. In this procedure, the pendulum was swung from
increasing amplitudes and ten cycles were timed and recorded
#
Length of String: 111.8 cm = 1.118m
Amplitude
Amp.
Trial 1
Seconds
Degrees
cm
10 cycles
1
2
3
4
5
6
5˚
10˚
15˚
20˚
25˚
30˚
Procedure 2
8
16
24
32
40.5
49
21.28
21.18
21.55
21.62
21.59
21.73
Mass of bob: 37 g = .037 kg
Trial 2
Trial 3
Seconds
Seconds
10 cycles
10 cycles
21.50
21.32
21.51
21.65
21.62
21.66
21.57
21.68
21.48
21.54
21.67
21.69
Avg. Time
Period
10 cycles
1 cycle
21.45
21.39
21.51
21.60
21.63
21.69
2.145
2.139
2.151
2.160
2.163
2.169
In the second procedure, the same process was done, but with an increased amount of
weight. The lab called for one test with double the weight. As further proof of the
concept, I did 3 additional tests with increasing weight.
#
Length of String: 111.8 cm = 1.118m
Bob Weight
Trial 1 (seconds)
Grams
10 cycles
Trial 2 (seconds)
10 cycles
Trial 3 (seconds)
10 cycles
Avg. Time
10 cycles
Period
1 cycle
1
2
3
61
178
355
21.56
21.82
21.87
21.59
21.78
21.92
21.66
21.81
21.81
2.166
2.181
2.181
21.53
21.82
21.63
Procedure 3
In the last procedure, following the same process, I time the cycles with decreasing
lengths of the pendulum string.
#
Mass of bob: 61 grams
Length
Trial 1 (seconds)
Meters
10 cycles
Amplitude: 10˚
Trial 2 (seconds)
Trial 3 (seconds)
10 cycles
10 cycles
Avg. Time
10 cycles
Period
1 cycle
1
2
3
4
.247
.492
.744
1.00
10.70
14.64
17.88
20.49
10.72
14.59
17.76
20.29
1.072
1.459
1.776
2.029
10.56
14.54
17.79
20.27
10.89
14.58
17.61
20.11
Section 2: Analysis
A. Calculations
From the data recorded, it is a trivial exercise to calculate the acceleration due to gravity. The
period of a simple pendulum is calculated by:
T = 2!
L
g
In this experiment, I’ve timed the period. I’ll use this data to calculate the acceleration due to
gravity. Solving for g, I get:
4! 2 L
g= 2
t
From these equations, and observation of the data, it is clear that weight and amplitude have
no effect on the period. In fact, from this equation, the only factor that changes the period
(assuming the acceleration due to gravity is constant) is the length of the pendulum string.
So, I do several calculations, one for each length of string measured, using the average period
for that length.
For length 1.118 meters:
g=
(4! 2 )1.118
= 9.451m / s 2
2
2.161
For length 1.00 meters:
g=
(4! 2 )1.00
= 9.589m / s 2
2
2.029
For length .744 meters:
g=
(4! 2 ).744
= 9.312m / s 2
2
1.776
For length .492 meters:
g=
(4! 2 ).492
= 9.125m / s 2
2
1.459
For length .247 meters:
(4! 2 ).247
g=
= 8.485m / s 2
2
1.072
Taking the average of these calculations for g, weighted by the number of tests, I get:
(9(9.451)+9.589+9.312+9.125+8.485)/13=9.352m/s2
The standard, accepted acceleration due to gravity is 9.81m/s2. This results in a percentage
error of:
%E =
9.81! 9.352
= 4.67%
9.81
B. Graphs
No graphs are required for this experiment, but it is interesting to observe a graph of
the length of the pendulum string versus the time of the period:
C. Error Analysis
All errors are calculated using the percent difference formula:
%Error =
!
E"A
# 100%
A
There is only one place in this experiment where I can check percent error and that is
the difference between the theoretical acceleration due to gravity and that calculated.
Section 3: Discussion and Conclusions
A. Discussion
The pendulum is one of the most basic forms of simple harmonic motion. They are
used as sources of entertainment in swings, in industry in guidance systems and
aircraft instrumentation systems, in clocks and in construction as plumb bobs.
Lab Questions
A. How did the change in the weight of the bob affect the resulting period and
frequency?
As expected, the weight of the bob had no affect on the period or frequency. Since the
equation for period only contains gravity and the length of the pendulum string, the
mass of the bob has no impact on the period.
B. How did the change in amplitude affect the resulting period and frequency?
The simple answer is that the amplitude has no effect on the period or frequency.
However, the more accurate answer is that, as the amplitude increases, it increasingly
affects the amplitude. The simple formula for the period of a pendulum is predicated
on the fact that sin θ ≈ θ. As the angle of amplitude increases, this becomes less so.
The formula for period with larger angles of amplitude includes the sum of an infinite
series to compensate for this slight increase in the amplitude because of the larger
angle.
The data accumulated in this experiment illustrates the slight increase in period as the
amplitude increases. Another factor in the increase of amplitude is that, as the
amplitude increases, the maximum velocity also increases. As the velocity increases
the effects of air resistance are more pronounced. This is observed in the experiment
by the fact that, as the amplitude is increased, the percentage of amplitude lost with
each cycle is also increased. This also results in an increase in the period.
C. How did the change in length of the pendulum affect the period and frequency?
Again, as expected, the length of the pendulum had an immediate impact on the
period. As the length decreases, the period also decreases.
D. What would happen if you used very large amplitudes? Check your hypothesis by
trial. What amplitude did you use? What is the result?
This is answered extensively in question B. I did one experiment while the pendulum
length was at 1.118 meters (my initial length), using the initial mass, but using an
amplitude angle of around 50˚. This resulted in an increase in the period of about 4-
5%. This increase is caused both by the increase in amplitude and the increased
effects of air resistance. The results of the experiment show a small but steady
increase in the period as the amplitude increases.
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?
Before I try it, I will venture a guess. A magnetic field under the bob will have the
same effect as an increase in the gravitational force. Therefore, it seems like it should
result in a decrease in the period. On the other hand, since the magnetic force is only
under the center point, it might have minimal effect on the acceleration of the bob,
especially since, at the center point, the acceleration of the bob is near zero. Now, I’ll
go try it…
The test seems to give somewhat uncertain results. I ran several tests with and without
the magnet. The average period with the magnet seemed to be 1.5-2% shorter than the
period without the magnet. However, that is within the margin of error, so it is
difficult determine with the magnet I have. A stronger magnet would give a better
result.
F. How close was your calculation of the value of g at your location? What might be a
few sources of error in your experimental data and calculations?
As pointed out in the calculations section, the percent of error in the calculation of
gravity is around 4 2/3%. Sources of errors are outlined below.
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?
The acceleration due to gravity is determined by the mass of the two objects and the
distance between those objects. As the altitude increases, the distance between the
earth and the bob of the pendulum is increased, thus decreasing gravity. The
difference is minimal. For instance, at 7200 feet elevation (the elevation of my home),
the difference in gravity from sea level is around one tenth of one percent.
Under weightless conditions, the period becomes infinite. As the force of gravity is
decreased, the period increases. This becomes self-evident when one considers that,
under weightless conditions, the bob raised to, say, 30˚ amplitude, when released,
would not fall, but would remain exactly where it is released.
B. Results
The results are extremely consistent. Increase in mass has no effect on the period of
the pendulum. An increase in amplitude has a small, but consistent effect. A change in
the length of the pendulum has a marked and significant effect on the period. All of
these are expected.
C. Interpretation of Results
As previously stated, the mass of the pendulum bob has no effect on the period of the
pendulum. This is illustrated by my radical increase in mass of the bob from 37 grams
up to 355 grams. The slight increase in period can be attributed to the larger size of
the bob being more affected by air resistance. The increased amplitude resulted in
small but consistent increases in the period, as expected, both because of the formula
for amplitude and because of the increased effects of air resistance.
D. Error Sources and Why
Several sources of errors will have an impact on the results of this experiment. The
most important and, by far, most significant factor is air resistance. If we could create
a vacuum chamber with a pendulum in it, the accuracy of the test would be greatly
increased. It factor accounts for the bulk of the percent error in the calculation of
gravity.
Other contributing factors are the inaccuracy of measuring the length of the pendulum
and of timing the period.