Simple Harmonic Motion Lab Report

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Stony Brook University *

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Electrical Engineering

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Dec 6, 2023

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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? = 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 = 𝑇 Trial 1: ω = 𝑇 = 0.78 ? = 8. 055366 ?𝑎?/? Trial 2: ω = 𝑇 = 0.928603 ? = 6. 766277 ?𝑎?/? Trial 3: ω = 𝑇 = 0.942350 ? = 6. 667571 ?𝑎?/? Calculations Using Peak Frequency From FFT Trial 1: 𝑇 = 1 ? = 1 1.27 𝐻? = 0. 787402 ?, ω = 𝑇 = 0.787402 ? = 7. 979641 ?𝑎?/? Trial 2: 𝑇 = 1 ? = 1 1.123 𝐻? = 0. 890472 ?, ω = 𝑇 = 0.890472 ? = 7. 056017 ?𝑎?/? Trial 3: 𝑇 = 1 ? = 1 1.074 𝐻? = 0. 931099 ?, ω = 𝑇 = 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.