Proof of the stiffness of the horizontal component of a spring

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SUMMARY

The discussion focuses on deriving the equivalent stiffness of a horizontal spring from a vertical spring using the relationship F=k*z and the angle theta. The user attempts to substitute the equations F = F(x)*cos(theta) and x = z*cos(theta) into F(x)=k(x)*x, leading to the conclusion that k(x)=k*cos^2(theta). This indicates that the horizontal equivalent stiffness is less than the original stiffness k, confirming the expected behavior of spring mechanics.

PREREQUISITES
  • Understanding of Hooke's Law (F=k*z)
  • Basic knowledge of trigonometric functions (cosine)
  • Familiarity with spring mechanics and stiffness concepts
  • Ability to manipulate algebraic equations
NEXT STEPS
  • Study the derivation of equivalent stiffness in multi-directional spring systems
  • Learn about the effects of angles on force distribution in springs
  • Explore advanced topics in mechanical vibrations and spring dynamics
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Students in mechanical engineering, physics enthusiasts, and anyone studying the principles of spring mechanics and force analysis.

jonathan308
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Homework Statement



Suppose a spring with forces F=k*z. Find the equivalent stiffness of a horizontal spring, where the spring and its horizontal equivalent is separated by theta degrees. The horizontal equivalent presumably will have a force F(x)=k(x)*x

Homework Equations



F = F(x)*cos(theta)
x = z*cos(theta)

The Attempt at a Solution



I have try to substitute F=k*z, F = F(x)*cos(theta) and x=z*cos(theta) into the equation F(x)=k(x)*x, but i only end up with a equation k = k(x)*cos^2(theta). The answer should be k(x)=k*cos^2(theta) and it only makes sense if the horizontal equivalent of stiffness is smaller than k. Please help and point out where I have made a mistake. thanks :)
 
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