15.2 - Energy in Simple Harmonic Motion

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Last updated 6:19 PM on 6/3/26
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Potential Spring Energy of a Linear Oscillator

Recall that the energy for a spring-mass system is given by:

U(t) = (1/2)k(x(t))²

But because we know that x(t) = xmcos(wt + phi), we know that the potential energy of the spring-mass system is given by the following:

U(t) = (1/2)k(xm)²cos²(wt + phi)

<p>Recall that the energy for a spring-mass system is given by: </p><p></p><p>U(t) = (1/2)k(x(t))²</p><p></p><p>But because we know that x(t) = x<sub>m</sub>cos(wt + phi), we know that the potential energy of the spring-mass system is given by the following: </p><p></p><p>U(t) = (1/2)k(x<sub>m</sub>)²cos²(wt + phi) </p><p></p><p></p>
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Kinetic Energy for a Spring-Mass System

Like with the potential energy of the system, we use the differentiated form of the position function and plug it into the kinetic energy formula to find the kinetic energy of the spring-mass system itself:

<p>Like with the potential energy of the system, we use the differentiated form of the position function and plug it into the kinetic energy formula to find the kinetic energy of the spring-mass system itself: </p><p></p><p></p>
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Conservation of Mechanical Energy for a Spring-Mass Linear Oscillator System

Since the mechanical energy is the sum of the potential and kinetic energy, we can just combine the formulas before and identify the trigonometric identity necessary to simplify the expression:

This reinforces the Conservation of Mechanical Energy is also applicable of the linear-oscillator; this only works because we assume that the surface is frictionless, so there is no energy lost due to friction with each oscillation

<p>Since the mechanical energy is the sum of the potential and kinetic energy, we can just combine the formulas before and identify the trigonometric identity necessary to simplify the expression:</p><p></p><p>This reinforces the Conservation of Mechanical Energy is also applicable of the linear-oscillator; this only works because we assume that the surface is frictionless, so there is no energy lost due to friction with each oscillation</p>
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