Differential equations - harmonic motion

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SUMMARY

The discussion focuses on the analysis of a mass-spring system undergoing damped harmonic motion, specifically a 5-kilogram mass oscillating vertically in a fluid. The governing equation of motion is derived as md²x/dt² = mg - B - bdx/dt - kx, where mg represents the weight, B the buoyant force, b the frictional constant, and k the spring constant. The problem emphasizes the need to distinguish between simple harmonic motion and forced harmonic motion, ultimately concluding that the system exhibits damped harmonic oscillation due to the presence of friction.

PREREQUISITES
  • Understanding of differential equations and their applications in physics.
  • Knowledge of concepts related to harmonic motion, including spring constants and oscillation.
  • Familiarity with forces acting on a mass, such as gravitational, buoyant, and frictional forces.
  • Ability to analyze and interpret phase space plots in the context of dynamical systems.
NEXT STEPS
  • Study the principles of damped harmonic motion and its mathematical representation.
  • Learn how to derive and solve second-order differential equations in physics contexts.
  • Explore the effects of friction on oscillatory systems and how to model them mathematically.
  • Investigate phase space analysis and its application in visualizing dynamical systems.
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Students and educators in physics, particularly those studying mechanics and oscillatory motion, as well as engineers and researchers involved in systems modeling and analysis of damped oscillations.

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Homework Statement



A mass of 5 kilograms dangles from a spring, stretching the spring centimeters when in equilibrium.
The mass oscillates vertically in a fluid, with a frictional force times its speed.
If initially it starts from rest at an extension below the equilibrium point of centimeters, describe the subsequent motion, with a plot of the extension against time and with a phase space plot.


Homework Equations



I looked through my book, but my first problem is that I can't figure out if it is Simple Harmonic Motion, or Forced Harmonic Motion. I don't see any specific equations for this that involves all of the variables I have.

The Attempt at a Solution

 
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Write the eqn of motion first. Take the downward direction as positive.

The weight mg acts downward, the buoyant force B acts upward, the frictional force bv acts opp to the velo, where b is a constant, and k is the spring constant, x is the displacement from the equilibrium posn.

ma = mg - B - bv - kx =>
md^2x/dt^2 = mg - B - bdx/dt - kx.

Sub in the values of the constants. This is damped harmonic oscillation.
 

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