HM_01_PCS-LabReport-SHM

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Department of Physics Course Number PCS 125 Course Title Physics: Waves and Fields Semester/Year Fall 2022 Instructor Yuan Xu TA Name Filip Bodera Lab/Tutorial Report No. 1 Report Title Investigation of Simple Harmonic Motion in a Spring-Mass System Section No. 45 Group No. N/A Submission Date 01/24/2022 Due Date 01/24/2022 Student Name Student ID Signature* Hamza Makrod 93122 HM (Note: remove the first 4 digits from your student ID) *By signing above you attest that you have contributed to this submission and confirm that all work you have contributed to this submission is your own work. Any suspicion of copying or plagiarism in this work will result in an investigation of Academic Misconduct and may result in a “0” on the work, an “F” in the course, or possibly more severe penalties, as well as a Disciplinary Notice on your academic record under the Student Code of Academic Conduct, which can be found online at: http://www.ryerson.ca/content/dam/senate/policies/pol60.pdf

Introduction: When a spring at rest is disturbed by adding a mass placed at different amplitudes and is released, the system will be subject to a restoring force that attempts to return the system to an equilibrium point dependent on the mass added to the system. The motion of the system will be in constant periodic oscillations that can be plotted as sine or cosine functions and utilized to calculate the angular oscillation frequency. Many other relationships can be identified in this motion as well, such as the linear proportionality of a mass’ displacement from the equilibrium position to the size of the acceleration of the system. The purpose of this experiment is to explore and apply these qualitative and quantitative properties of simple harmonic motion in tandem with Hooke’s law to predict the motion of a mass in a spring system. Theory: To theorize and conceptualize important equations of simple harmonic motion we will examine a model of the spring mass system as shown in Figure 1. Figure 1: A diagram of a spring-mass system disturbed by a mass resulting in simple harmonic motion The purpose of this experiment is predicting the motion of a disturbed spring-mass system by utilizing Hooke’s law and the theory of simple harmonic motion. So, the first priority it to ensure proper understanding of these closely related theories. The basic principle behind simple harmonic motion is that it is a unique form of motion that attempts to restore a system back to its equilibrium point when a system is disturbed by applying a force directly proportional to its displacement. A common instance in which this force at work can be observed is in a playground swing. When a swing at rest is disturbed by a swinger applying their force in a direction the swing will continue to travel periodically back and forth like a pendulum until it loses energy and comes to rest at the original position (known as the point of equilibrium). This also holds true for a spring, which when pulled away from its resting position and is released will return

to it when the spring relaxes. The restoring force has a common name in Hooke’s law which is mathematically represented as F =− k ∆ y . The negative sign indicates the direction of the force is in opposition to the displacement of the spring, ‘ k’ is the variable for the spring constant which is a measure of the stiffness of the spring, and ∆ y represents the change in displacement of the spring from the equilibrium position. A prediction of the displacement of a spring can be made based off the stiffness of the spring because the higher the spring constant means the more force required to move the spring. So, if the mass and position of releasing the spring are constant (ie.100g and maximum pull point), since thicker springs have their equilibrium position closer to an undisturbed spring than thinner ones, the higher the spring constant the higher the displacement will be. Another important property to be used to predict and evaluate simple harmonic motion is the angular frequency ω , as the shape of a periodic oscillation is sinusoidal. To find the equation for ω , the wave for an equation given as 𝑦 ( 𝑡 )= 𝐴 cos( 𝜔𝑡 + 𝜙 ) can be used and substituted back into Hooke’s law, where ‘A’ represents amplitude and 𝜙 denotes the phase constant. The derivation of ω can begin with F =− k ∆ y , the ∆ y = y ( t ) − y 0 and since y 0 = 0 because it is the equilibrium position the new equation is F =− k × y ( t ) . According to Newton’s second law of motion F =− k × y ( t ) = ma , and since acceleration is the second derivative of displacement (y(t) in this case), − k × y ( t ) = m d 2 y dt 2 . The equation given for displacement is 𝑦 ( 𝑡 )= 𝐴 cos( 𝜔𝑡 + 𝜙 ), so acceleration is y(t)’’. If, 𝑦 ( 𝑡 )= 𝐴 cos( 𝜔𝑡 + 𝜙 ) Then, 𝑦 ( 𝑡 )’=- 𝜔𝐴 sin( 𝜔𝑡 + 𝜙 ) Therefore, 𝑦 ( 𝑡 )’’=- ω 2 𝐴 cos( 𝜔𝑡 + 𝜙 ) And since, 𝑦 ( 𝑡 )= 𝐴 cos( 𝜔𝑡 + 𝜙 ), then 𝑦 ( 𝑡 )’’=- ω 2 * y(t). This means that ma =− k × y ( t ) = m ( − ω 2 × y ( t ) ) , so − k = m× − ω 2 , ∴ ω = √ k m . This essentially means that angular frequency is completely dependent on the spring constant and mass. Once angular frequency is calculated the expressions for the frequency ‘f’ and the period ‘T’ can also be derived by applying f = ω 2 π ∧ T = 1 f and substituting appropriate values, this results in f = 1 2 π × √ k m ∧ T = 2 π × √ m k . These derivations indicate that many different aspects of simple harmonic motion are dictated by the spring constant and mass. Therefore, as long as the values of these measurements are properly calculated, an accurate prediction of motion for a spring-mass system should be feasible. Since amplitude is irrelevant in the calculation of the period of the oscillation an appropriate prediction would be that as long as ‘k’ and ‘m’ remain constant changing amplitudes will have no effect on the period of oscillation. If there is any difference in ‘T’ for different amplitudes, it should be quite miniscule to the point that it can be written off as insignificant and accounted as experimental error (delay in stopwatch management). In the case that mass is changing variable than the period should in fact change,

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