Simple Harmonic Motion Lab Report
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School
Stony Brook University *
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Course
133
Subject
Electrical Engineering
Date
Dec 6, 2023
Type
Pages
7
Uploaded by ProfDinosaur3750
Name: Andy Kweon
Date: 11/5/2023
Course: PHY 133
TA Name: Siqi Chen
Title of Lab: Simple Harmonic Motion
Introduction
Simple harmonic motion is a type of oscillatory motion that occurs when a restoring force acts
upon a mass on a spring. Restoring force is directly proportional to the displacement of the mass from the
equilibrium position and will always move the mass back towards the equilibrium position. Using
Newton’s Second Law, the period of this motion can be derived as the following:
,
. The displacement, velocity, and acceleration of
𝐹
???
= ?𝑎 = ?
?
2
?
??
2
=− 𝑘?
?
2
?
??
2
=−
𝑘
?
? =− ω
2
?
the mass can be expressed as a function of time by solving the differential equation.
,
?(?)
=
? ???(??)
,
, where
. The relationship between
?(?) =
??
??
=−
ω? ?𝑖?(ω?)
𝑎(?) =
??
??
=− ω
2
? ???(ω?)
ω =
𝑘
?
between the angular frequency and the period can be expressed as
. Combining these two
ω = 2π? =
2π
𝑇
equations gives us
.
𝑇 = 2π
?
𝑘
This experiment will involve the usage of the Fast Fourier Transform (FFT) function, which has
many applications in engineering, math, and science. The FFT function allows us to take a signal with
time or space as its domain and transform it into a representation with frequency as its domain. The tool
has the option of numbers ranging from 256 to 4096, which represents the number of bins in which the
frequency axis is divided. We will be using the 4096 option for the lab. Fast Fourier Analysis of
sinusoidal functions with constant period will result in a peak in frequency space at the frequency of the
waves.
In this experiment, we will experimentally find the angular frequency of a simple harmonic
oscillator and compare it with the theoretical value as well as experimentally test the relationship between
the mass of an object and the angular frequency of the oscillator to see if the mass of an object impacts the
angular frequency.
Methods/Procedures
The mass of the IOLab device was found first. The screw was attached to the device and left to
rest for a few seconds. Then the device was held in the air by the screw for a few seconds before being
placed down. The force and acceleration of the device was collected for the entirety. The force due to
gravity and acceleration due to gravity were used to calculate the mass using the formula
.
𝐹
?
= ??
The long spring was attached to the screw and the device was allowed to oscillate for a minute.
The force data was collected for the entire time the device was oscillating. After, the time between 5
peaks were measured both around the beginning and the end. The time between 5 peaks would be divided
by 4 to get the period, which was used to calculate the angular frequency. The period of the end of the run
was compared to the period at the beginning of the run. The FFT function was used to find the peak
frequency, which was used to calculate the angular frequency. This angular frequency was compared to
the angular frequency found using the period.
The previous steps were repeated two more times, with each subsequent trial having an additional
mass attached to the device. In the second trial, a pack of pens was taped to the device and the entire
experiment was repeated with the additional mass.
In the third trial, a broken ruler was attached to the device and pack of pens. The entire
experiment was repeated with the additional mass.
Once three mass values and three angular frequencies were obtained, an angular frequency of the
device found from the period versus mass of device plot was created.
Results
The following are the graphs of the force data that was collected as the device was held in the air
and the acceleration data when the device was resting on the surface for all 3 separate trials. Below the
graphs is a table with all of the force and acceleration data as well as the calculated mass using those data
of each trial.
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Force (N)
Acceleration (m/s^2)
Mass (kg)
Trial 1
-1.992
0.027 N
±
-9.817
0.014 m/s^2
±
0.202913 kg
Trial 2
-2.787
0.038 N
±
-9.821
0.023 m/s^2
±
0.283780 kg
Trial 3
-2.848
0.040 N
±
-9.824
0.019 m/s^2
±
0.289902 kg
Below are the graphs showing the time between 5 peaks around the beginning and end of the
oscillation of the device as well as the peak frequency found using FFT. The table below the graphs shows
all of these data collected as well as the calculated angular frequency.
Time
between 5
peaks at
beginning
(s)
Time
between
5 peaks
at end (s)
Period at
beginnin
g (s)
Period at
end (s)
Angular
Frequency
Using
Calculated
Period
Beginning
(rad/s)
Peak
Frequency
(Hz)
Period
Using Peak
Frequency
(s)
Angular
Frequency
Using Peak
Frequency
(rad/s)
Trial
1
3.12 s
3.125 s
0.78 s
0.78125 s
8.055366
rad/s
1.270 Hz
0.787402 s
7.979641
rad/s
Trial
2
3.71441 s
3.6347 s
0.928603
s
0.908675
s
6.766277
rad/s
1.123 Hz
0.890472 s
7.056017
rad/s
Trial
3
3.7694 s
3.68441
s
0. 942350
s
0.921103
s
6.667571
rad/s
1.074 Hz
0.931099 s
6.748139
rad/s
The figure below is the plot of the angular frequency of the device that was calculated from the
period versus the mass of the device. The
, or A, was 3.484, representing the square root of the spring
𝑘
constant.
Calculations
Mass of Device
:
? =
𝐹
?
?
Trial 1:
? =
𝐹
?
?
=
−1.992 𝑁
−9.817 ?/?
2
=
0. 202913 𝑘?
Trial 2:
? =
𝐹
?
?
=
−2.787 𝑁
−9.821 ?/?
2
=
0. 283780 𝑘?
Trial 3:
? =
𝐹
?
?
=
−2.848 𝑁
−9.824 ?/?
2
=
0. 289902 𝑘?
Period
Trial 1:
𝑇 =
3.12 ?
4
=
0. 78 ?, 𝑇
=
3.125 ?
4
=
0. 78125 ?
Trial 2:
𝑇 =
3.71441 ?
4
=
0. 928603 ?, 𝑇
=
3.6347 ?
4
=
0. 908675 ?
Trial 3:
𝑇 =
3.7694 ?
4
=
0. 942350 ?, 𝑇
=
3.68441 ?
4
=
0. 921103 ?
Angular Frequency:
ω =
2π
𝑇
Trial 1:
ω =
2π
𝑇
=
2π
0.78 ?
=
8. 055366 ?𝑎?/?
Trial 2:
ω =
2π
𝑇
=
2π
0.928603 ?
=
6. 766277 ?𝑎?/?
Trial 3:
ω =
2π
𝑇
=
2π
0.942350 ?
=
6. 667571 ?𝑎?/?
Calculations Using Peak Frequency From FFT
Trial 1:
𝑇 =
1
?
=
1
1.27 𝐻?
=
0. 787402 ?, ω
=
2π
𝑇
=
2π
0.787402 ?
=
7. 979641 ?𝑎?/?
Trial 2:
𝑇 =
1
?
=
1
1.123 𝐻?
=
0. 890472 ?, ω
=
2π
𝑇
=
2π
0.890472 ?
=
7. 056017 ?𝑎?/?
Trial 3:
𝑇 =
1
?
=
1
1.074 𝐻?
=
0. 931099 ?, ω
=
2π
𝑇
=
2π
0.931099 ?
=
6. 748139 ?𝑎?/?
Mass Error
Trial 1:
0. 202913
(
0.027
−1.992
)
2
+ (
0.014
−9.817
)
2
=
0. 013629 𝑘?
Trial 2:
0. 283780
(
0.038
−2.787
)
2
+ (
0.023
−9.821
)
2
=
0. 013834 𝑘?
Trial 3:
0. 289902
(
0.040
−2.848
)
2
+ (
0.019
−9.824
)
2
=
0. 014177 𝑘?
Percent Difference
Trial 1:
8.055366 −7.979641
|
|
7.979641
· 100% = 0. 948978%
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Trial 2:
6.766277 −7.056017
|
|
7.056017
· 100% = 4. 106283%
Trial 3:
6.667571 −6.748139
|
|
6.748139
· 100% = 1. 193929%
Discussions/Conclusions
The period of each trial when the device was oscillating at the beginning of the run was very
similar to the period from the end of the run. In Figure 23, the period at the beginning is slightly longer
than the period at the end. This trend is not followed by the 1st trial as the period was shorter at the
beginning compared to the end of the run by 0.00123 s. The angular frequency values calculated from
both the periods observed in the experiment as well as the peak frequency value found using the FFT
function were also close to each other. Angular frequency values in Trial 2 had the most difference, with a
percent difference of 4.106293%. This difference could be due to the way the device was pulled at the
beginning of the run so that it can oscillate on the spring, as it may have impacted the period calculated
which resulted in a larger difference in angular frequency compared to other trials.
In Figure 24, the equation of the Angular Frequency vs. Mass plot was
. This
? = 3. 484?
−0.526
equation can be described with the equation
or
, k is the spring constant and
? = ??
?
ω = ( 𝑘
)(?)
−1/2
x is the mass. The spring constant for the long spring from the last lab was 11.5 N/m. When taking the
square root of this value, we get 3.391165, which is very similar to the value of A. This is expected as the
value of A is equal to the square root of the spring constant. Based on this information, this experiment
showed that mass does have an effect on the angular frequency. As mass increased, the angular frequency
decreased.
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