Springs and Potential Energy

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SUMMARY

The discussion centers on the relationship between gravitational potential energy and elastic potential energy in a spring-wombat-Earth system. It establishes that the loss of gravitational potential energy equals twice the gain in elastic potential energy when a wombat is lowered from a ceiling. The key equations involved are the change in gravitational potential energy, given by -mg(Xf - Xi), and the change in elastic potential energy, calculated as 0.5k(Xf² - Xi²). The discrepancy arises because the force exerted by the person lowering the wombat introduces a non-conservative force, resulting in the gravitational potential energy being greater than the elastic potential energy.

PREREQUISITES
  • Understanding of gravitational potential energy and its formula: -mg(Xf - Xi)
  • Familiarity with elastic potential energy and its calculation: 0.5k(Xf² - Xi²)
  • Knowledge of non-conservative forces and their effects on energy systems
  • Basic principles of mechanics, particularly Hooke's Law
NEXT STEPS
  • Study the implications of non-conservative forces in mechanical systems
  • Explore Hooke's Law and its applications in real-world scenarios
  • Investigate energy conservation principles in elastic systems
  • Learn about the dynamics of spring-mass systems and their oscillatory motion
USEFUL FOR

Students studying physics, particularly those focusing on mechanics and energy conservation, as well as educators seeking to explain the principles of potential energy in spring systems.

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Homework Statement


An ideal spring is suspended from a ceiling with a wombat attached to the end. The wombat is slowly lowered until the upward force exerted by the spring on the wombat balances the weight of the wombat. Show that the loss of gravitational potential energy equals twice the gain in the elastic potential energy of the spring-wombat-Earth system. WHY are these two quantities NOT equal?

Homework Equations


Change in gravitational potential = -mg(Xf - Xi)
Change in elastic potential = .5k(Xf2 - Xi2)

The Attempt at a Solution


I realize that it is due to the fact that a person slowly lowers the wombat, thus acting as a non-conservative force, but do not know how to derive the solution.
 
Last edited:
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I realize that the gravitational potential energy must be greater, but why is is exactly twice, no matter the mass of the object?
 

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