# Lab_10 - Laboratory 10 Simple Harmonic Motion Introduction...

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Laboratory 10 Simple Harmonic Motion Introduction Any physical system governed by a restoring force , described (at least approximately) by a quadratic potential well , and which executes periodic motion , can be described as a simple harmonic oscillator . Simple harmonic oscillators occur in many fields, from engineering and physics to geology and biology. Therefore, understanding this concept is critical to analyzing a broad range of theoretical models describing real-world systems. One of the simplest examples of a simple harmonic oscillator is a mass oscillating due to a spring’s restoring force. A biochemical extension of this consists of two oxygen atoms (masses) oscillating under the influence of the interatomic force (spring) between them. The atoms’ positions fluctuate through their equilibrium position as they are attracted to and repelled by each other. Although in- teratomic forces are generally complex, in the region near equilibrium they are well-characterized by the linear relationship F = k ( x x 0 ) as long as the atoms’ excursions from their equilibrium separation, x 0 , are not too large. Both a mass in a mass-spring system and the oxygen atoms in the biological system will undergo oscillatory motion at some natural frequency that is a function of the system itself and does not depend on the amplitude of the oscillation. During this laboratory, we will investigate simple harmonic motion through the measurement of the natural oscillation frequencies of three different systems. Theoretical Background: Simple Harmonic Motion Simple harmonic motion is sinusoidal motion that results when an object is subject to a restoring force of the form F = k x , where x is the object’s displacement from its equilibrium position - the position where it expe- riences no net force. This formula says that no matter which direction the displacement is in, the restoring force will act to bring the object back toward its equilibrium position. A graph of force vs. displacement for simple harmonic motion is shown in Figure 1. Using Newton’s second law, F = m a , we obtain the equation of motion m a = k x , (1) 85

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Figure 1: These two graphs are defining features of simple harmonic motion. The force law is linear with respect to the displacement, x , and the potential energy function is a parabola. You can think of simple harmonic motion as a situation where a ball sitting at the bottom of the potential energy “well” is displaced slightly. What would happen? It would roll back and forth periodically, and its x -position would be a sine curve. where x is the object’s position and a is its acceleration. For those familiar with calculus, a is the second derivative of x with respect to time, so m d 2 x dt 2 = k x .
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