Why Does Hooke's Law Seem Contradictory in Calculating Spring Energy?

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SUMMARY

The discussion centers on the apparent contradiction between Hooke's Law, expressed as F = kx, and the energy stored in a spring, given by E = (1/2)kx². Users noted that calculating gravitational force as mg = (1/2)xk leads to confusion. The key takeaway is that the energy equation applies to variable forces, necessitating the use of calculus to accurately derive energy from force over displacement. Understanding the relationship between force and energy in springs requires a grasp of differential calculus.

PREREQUISITES
  • Hooke's Law (F = kx)
  • Energy stored in springs (E = (1/2)kx²)
  • Differential calculus
  • Basic mechanics (force, mass, and gravity)
NEXT STEPS
  • Study the application of differential calculus in physics
  • Learn about the work-energy principle in variable force systems
  • Explore advanced topics in elasticity and material science
  • Investigate practical experiments to measure spring displacement and energy
USEFUL FOR

Students of physics, mechanical engineers, and anyone interested in understanding the principles of mechanics and energy in spring systems.

tallwallyb
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Homework Statement
Given a vertical spring with spring constant k=53N/m, a 2.5kg weight is hung from it. What is the change in x of the weight. (USE HOOKE'S LAW and compare it to using ENERGY.
Relevant Equations
F = kx
PEs = (1/2)kx^2
PEg = (mgh)
On the energy part, I keep getting mg=(1/2)*x*k, which is contradictory to Hooke's law F=mg=kx. What is going on?
 
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Please show how you applied energy considerations. Suppose you were given the spring and mass and you wanted to measure "the change in x of the weight." What procedure will you follow to do that? That procedure is important because it will inform your use of energy considerations.
 
tallwallyb said:
On the energy part, I keep getting mg=(1/2)*x*k, which is contradictory to Hooke's law F=mg=kx. What is going on?
I'm guessing you divided E by x expecting to get F. But the equation energy = force x displacement only works if the force is constant.
Are you familiar with differential calculus?
 

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