Yo-yoing over the harmonic oscillator

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SUMMARY

The discussion centers on the harmonic oscillator (HO) equation, specifically the relationship between the cosine function and its role in the solution to the differential equation d²x/dt² = -kx/m. The user highlights that the second derivative of -cos(θ) equals cos(θ), demonstrating the periodic nature of the cosine function. Additionally, they suggest exploring the form x = a*cos(bt) to verify its compatibility with the HO equation, emphasizing the importance of proper notation in mathematical expressions.

PREREQUISITES
  • Understanding of differential equations, specifically second-order linear equations.
  • Familiarity with trigonometric functions, particularly cosine and its derivatives.
  • Basic knowledge of harmonic motion concepts in physics.
  • Experience with mathematical notation and its significance in problem-solving.
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  • Study the derivation of the harmonic oscillator equation and its solutions.
  • Learn about the properties of trigonometric functions and their derivatives.
  • Explore the application of the cosine function in modeling oscillatory motion.
  • Investigate the use of software tools like Wolfram Alpha for solving differential equations.
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Students of physics and mathematics, educators teaching harmonic motion, and anyone interested in the mathematical foundations of oscillatory systems.

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I've been looking around and trying to figure it out, but I can't seem to figure out how the cosine function get's into the solution to the HO equation d2x/dt2=-kx/m. I know this is extremely basic, but could someone indulge me?
 
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It is not so difficult to use a better notation!

Try to see whether x = acos(bt), where a and b are constants, fits with the equation

\frac{d^{2}x}{dt^{2}} = -(positive constant)x.
 

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