Spring
Hooke's law
Most springs (not stretched or compressed beyond the elastic limit) obey Hooke's law, which states that the force with which the spring pushes back is linearly proportional to the distance from its equilibrium length:
where
-
x is the displacement vector - the distance and direction in which the spring is deformedF is the resulting force vector - the magnitude and direction of the restoring force the spring exertsk is the spring constant or force constant of the spring.
Coil springs and other common springs typically obey Hooke's law. There are useful springs that don't: springs based on beam bending can for example produce forces that vary nonlinearly with displacement.
There are also linear springs which don't follow Hooke's law: a Negator spring (the spring that a self retracting tape measure uses) provides a constant force.[citation needed]
Simple harmonic motion
Since force is equal to mass, m, times acceleration, a, the force equation for a spring obeying Hooke's law looks like:
The mass of the spring is assumed small in comparison to the mass of the attached mass and is ignored. Since acceleration is just the second time derivative of x,
This is a second order linear differential equation for the displacement x as a function of time. Rearranging:
the solution of which is the sum of a sine and cosine:
A and B are arbitrary constants that may be found by considering the initial displacement and velocity of the mass. The graph of this function with B = 0 (zero initial position with some positive initial velocity) is displayed in the image on the right.
Theory
In classical physics, a spring can be seen as a device that stores potential energy by straining the bonds between the atoms of an elastic material.
Hooke's law of elasticity states that the extension of an elastic rod (its distended length minus its relaxed length) is linearly proportional to its tension, the force used to stretch it. Similarly, the contraction (negative extension) is proportional to the compression (negative tension).
This law actually holds only approximately, and only when the deformation (extension or contraction) is small compared to the rod's overall length. For deformations beyond the elastic limit, atomic bonds get broken or rearranged, and a spring may snap, buckle, or permanently deform. Many materials have no clearly defined elastic limit, and Hooke's law can not be meaningfully applied to these materials.
Hooke's law is a mathematical consequence of the fact that the potential energy of the rod is a minimum when it has its relaxed length. Any smooth function of one variable approximates a quadratic function when examined near enough to its minimum point; and therefore the force — which is the derivative of energy with respect to displacement — will approximate a linear function.
Force of fully compressed spring
-
Fmax = (Ed4(L − nd)) / (16(1 + nu)(D − d)3n)
where
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E - Young's modulusd - spring wire diameterL - free length of springn - number of active windingsnu - Poisson ratioD - spring outer diameter



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