Oscillation of a vertical spring

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SUMMARY

The discussion centers on the oscillation of a mass attached to a vertical spring, specifically deriving the frequency of small oscillations and the governing equation for the spring's length under external oscillation. The frequency of small oscillations is established as ##\omega = \sqrt{g/(b-a)}##, where g is the acceleration due to gravity, and b and a are the extended and natural lengths of the spring, respectively. The equation governing the spring's length under external oscillation is derived as $$\ddot{L} + \omega^2 L = \omega^2 b + cn^2 \sin(nt)$$, where c is the amplitude of the external oscillation and n is its frequency.

PREREQUISITES
  • Understanding of harmonic motion and oscillation principles
  • Familiarity with Hooke's Law and spring constants
  • Knowledge of differential equations and their applications in physics
  • Basic concepts of forces acting on a mass-spring system
NEXT STEPS
  • Study the derivation of the harmonic oscillator equation in detail
  • Learn about the effects of damping on oscillatory systems
  • Explore the concept of forced oscillations and resonance
  • Investigate the applications of oscillatory motion in engineering and physics
USEFUL FOR

Students of physics, mechanical engineers, and anyone studying dynamics and oscillatory systems will benefit from this discussion.

  • #61
See the figure: It shows the time dependence of both the length L and the position of the mass, (y) for the case when w=2pi, n=4pi, c=0.1 b=1.Note how much less the mass moves compared to the spring.

Also watch the video about an experiment with a slinky. The top of the slinky is released and falls down, while the mass at the bottom stays almost motionless except the last stage when the slinky gets relaxed.

http://www.youtube.com/watch?v=oKb2tCtpvNU&NR=1

ehild
 

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