Conservation of energy I think

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SUMMARY

The discussion focuses on the mechanics of a bead on a vertical spring, specifically analyzing the restoring force experienced by the bead when displaced from its equilibrium position. It is established that when the spring is neither stretched nor compressed, the restoring force is proportional to x³ for small displacements (x), rather than the linear x. This conclusion is derived using the approximation (1+x)ⁿ = 1+nx for x << 1, which is crucial for understanding the behavior of the system in this context.

PREREQUISITES
  • Understanding of Hooke's Law and restoring forces
  • Familiarity with potential energy (U), kinetic energy (K), and total mechanical energy (E)
  • Basic knowledge of calculus, particularly Taylor series expansions
  • Ability to draw and interpret free body diagrams
NEXT STEPS
  • Study the derivation of restoring forces in non-linear systems
  • Learn about potential energy functions and their applications in mechanics
  • Explore the implications of small angle approximations in oscillatory motion
  • Investigate the dynamics of systems with frictionless motion
USEFUL FOR

Students studying classical mechanics, physics educators, and anyone interested in the principles of oscillatory motion and energy conservation in mechanical systems.

Benevelli
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Homework Statement


PLEASE HELP: A bead at the end of a vertical spring moves without friction along a horizontal wire. WHen the spring is perfectly vertical , the spring is neither stretched or compressed. Show that the bead experiences a restoring force, and that if x is small, then the magnitude of the force is proportional to x^3 rather x^1. You will want to use the approximation (1+x)^n = 1+nx for x<<1

Homework Equations


U + K = E

The Attempt at a Solution


I tried thinking about this like a pendulum with kx as the restoring force. Other than that, my work has gotten me nowhere.
 
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Draw a free body diagram for the bead at some displacement x. What are the forces on it?
 

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