How Is Work Calculated in a Stretched Spring?

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Discussion Overview

The discussion revolves around the calculation of work done on a spring when it is stretched. Participants explore the relationship between force, elongation, and work, particularly in the context of a spring's stiffness and its varying force during stretching.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant presents a derivation of work done by a force applied to stretch a spring, concluding with W = kd^2.
  • Another participant challenges this derivation, noting that the force exerted by the spring is not constant and only reaches k*d at full extension.
  • A participant suggests integrating the force over the distance to find the work done, proposing W = integral(kxdx) and arriving at W = (kd^2)/2.
  • There is a repeated affirmation of the integration approach leading to the conclusion of W = (kd^2)/2.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the initial derivation of work done. There is disagreement regarding the assumption of constant force, and the integration method is proposed as a resolution, but the discussion remains open to further clarification.

Contextual Notes

The discussion highlights the dependence on the assumption of force constancy and the need for integration to accurately calculate work done on a spring. The factor of 1/2 in the work equation is noted but not universally accepted without further exploration.

Who May Find This Useful

Students and educators in physics or engineering, particularly those studying mechanics and the behavior of springs.

MrLiou168
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Homework Statement


A spring with stiffness k and unstretched length L is stretched so the elongation is d = x2 - L. A force is applied to make the final length of the spring x2. What is the work done by the force in terms of d?

Homework Equations


W = F * d = F*dx
d = x2 - L
F = k*dx

The Attempt at a Solution


Assuming W = F*dx and F = k*dx, then I derived F = k(x2 - L) = k*d

And plugging F back into the work equation, I got W = (kd)*d which is W = kd^2.

However, isn't the actual equation for work done by a spring W = (kx^2)/2? I can't seem to find where I missed the factor of 1/2. Any help greatly appreciated!
 
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You assumed in your derivation that the force was constant and equal to its maximum value. Not so. As the spring is stretched, the force starts at zero and only reaches k*d at its full extension.
 
Thanks Doc. So in this case would I simply integrate to find W? As in W = integral (F*dx)

and then W = integral(kxdx) = (kd^2)/2 ...?
 
MrLiou168 said:
Thanks Doc. So in this case would I simply integrate to find W? As in W = integral (F*dx)

and then W = integral(kxdx) = (kd^2)/2 ...?
Exactly.
 

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