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h1. Hooke's Law for elastic interactions

{excerpt}A mathematical approximation to the restoring behavior of springs and other elastic solids under small deformations.{excerpt}

h3. Motivation for Concept

Elastic objects are objects which rebound to their original shape from a temporary deformation.  All solids are basically elastic under small deformations, but "small" is a relative term.  A metal ball, for instance, is only elastic for deformations that are so small as to be invisible to the naked eye.  If you press the ball hard enough to deform it noticeably, it will retain a dent.  A rubber ball is elastic under a much wider range of deformations, and can have its shape noticeably changed without causing a permanent dent.    

The elastic properties of objects is vital to understanding the engineering of all structures, from airplanes to skyscrapers.  As such, Hooke's description of the restoring force produced by an object undergoing _elastic deformation_ is an extremely useful piece of mathematics, and has acquired the title "Hooke's Law", even though it is not a universal Law in the same sense as, e.g. [Newton's 2nd Law|Newton's Second Law] or his law of [universal gravitation|gravitation (universal)].  Hooke's "Law" is really a parameterization which is only valid for deformations small enough that the object is in the elastic regime.  This fact does not detract from its enormous utility, since keeping structural members in an elastic state is often a goal in engineering.  Objects experiencing elastic deformation are said to "obey Hooke's Law".  In introductory mechanics, Hooke's Law is most frequently used to describe the resoring force of springs, which are objects designed to "spread out" large deformations over a series of coils, so that the complete object can change shape dramatically while each portion of the coil deforms only a relatively small amount.

h3. Hooke's Law in terms of Force

h4. Mathematical Statement of the Law

As applied to springs, Hooke's Law is generally stated for a spring which has one end fixed.  For that case, the restoring force acting on the other end of the spring when it is moved by stretching or compressing the spring along its length (taken to be the {*}x{*}-direction) will be given by:

{latex}
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 Excerpt
A mathematical approximation to the restoring behavior of springs and other elastic solids under small deformations.
Motivation for Concept
Elastic objects are objects which rebound to their original shape from a temporary deformation. All solids are basically elastic under small deformations, but "small" is a relative term. A metal ball, for instance, is only elastic for deformations that are so small as to be invisible to the naked eye. If you press the ball hard enough to deform it noticeably, it will retain a dent. A rubber ball is elastic under a much wider range of deformations, and can have its shape noticeably changed without causing a permanent dent. 
The elastic properties of objects is vital to understanding the engineering of all structures, from airplanes to skyscrapers. As such, Hooke's description of the restoring force produced by an object undergoing elastic deformation is an extremely useful piece of mathematics, and has acquired the title "Hooke's Law", even though it is not a universal Law in the same sense as, e.g. Newton's 2nd Law or his law of universal gravitation. Hooke's "Law" is really a parameterization which is only valid for deformations small enough that the object is in the elastic regime. This fact does not detract from its enormous utility, since keeping structural members in an elastic state is often a goal in engineering. Objects experiencing elastic deformation are said to "obey Hooke's Law". In introductory mechanics, Hooke's Law is most frequently used to describe the resoring force of springs, which are objects designed to "spread out" large deformations over a series of coils, so that the complete object can change shape dramatically while each portion of the coil deforms only a relatively small amount.
Hooke's Law in terms of Force
Mathematical Statement of the Law
As applied to springs, Hooke's Law is generally stated for a spring which has one end fixed. For that case, the restoring force acting on the other end of the spring when it is moved by stretching or compressing the spring along its length (taken to be the x-direction) will be given by:
 Latex
\begin{large}\[ \vec{F} = -k(x-x_{0})\hat{x}\]\end{large}
{latex}

where {*}_x_~0~{*} is the natural position of the end of the spring that is being moved.  


h4. Elasticity and Simple Harmonic Motion

The elastic restoring force fulfills the conditions for [simple harmonic motion].  This can be seen for the case of a spring held fixed at one end.  The defining relationship for simple harmonic motion in the {*}_x_{*} direction is:

{latex}
where x0 is the natural position of the end of the spring that is being moved. 
Elasticity and Simple Harmonic Motion
The elastic restoring force fulfills the conditions for simple harmonic motion. This can be seen for the case of a spring held fixed at one end. The defining relationship for simple harmonic motion in the x direction is:
 Latex
\begin{large}\[ \frac{d^{2}x}{dt^{2}} = - \omega^{2}x \]\end{large}
{latex}

where {*}ω{*} 
where ω (which 
 
 is 
 
 the 
 
 angular 
 
 frequency 
 
 of 
 
 the 
 
 resulting 
 
 motion) 
 
 is 
 
 not 
 
 a 
 
 function 
 
 of 
 {*}_
x
_{*}.  

This relationship will be present for a mass attached to a spring that is fixed on the other end.  Hooke's Law tells us:

{latex}
. 
This relationship will be present for a mass attached to a spring that is fixed on the other end. Hooke's Law tells us:
 Latex
\begin{large}\[ ma_{x} = - kx \]\end{large}
{latex}


where 
 
 we 
 
 have 
 
 assigned 
 
 coordinates 
 
 such 
 
 that 
 {*}_x_~0~ = 0{*}.  

By the definition of acceleration:

{latex}
 x0 = 0. 
By the definition of acceleration:
 Latex
\begin{large}\[ a_{x} = \frac{d^{2}x}{dt^{2}} = - \frac{k}{m}x\]\end{large}
{latex}


which 
 
 is 
 
 in 
 
 the 
 
 form 
 
 of 
 
 the 
 
 simple 
 
 harmonic 
 
 motion 
 
 equation 
 
 with


{
 Latex
}
\begin{large}\[ \omega = \sqrt{\frac{k}{m}}\]\end{large}
{latex}

h3. Elastic Potential Energy

Assuming an object attached to a spring that obeys Hooke's Law with the motion confined to the {*}_x_{*} direction, it is customary to choose the coordinates such that {*}_x_ = 0{*} when the object is in a position such that the spring is at its natural length.  The force on the object from the spring is then:

{latex}
Elastic Potential Energy
Assuming an object attached to a spring that obeys Hooke's Law with the motion confined to the x direction, it is customary to choose the coordinates such that x = 0 when the object is in a position such that the spring is at its natural length. The force on the object from the spring is then:
 Latex
\begin{large}\[ \vec{F} = - kx \hat{x} \]\end{large}
{latex}


It 
 
 is 
 
 also 
 
 customary 
 
 to 
 
 make 
 
 the 
 
 assignment:


{
 Latex
}
\begin{large}\[ U(0) \equiv 0\]\end{large}
{latex}


Thus, 
 
 the 
 
 potential 
 
 can 
 
 be 
 
 defined:


{
 Latex
}
\begin{large}\[ U(x) = U(0) - \int_{0}^{x} (-kx)\:dx = \frac{1}{2}kx^{2}\]\end{large}
{latex}


For 
 
 an 
 
 object 
 
 moving 
 
 under 
 
 the 
 
 influence 
 
 of 
 
 a 
 
 spring 
 
 only, 
 
 the 
 
 associated 
 
 potential 
 
 energy 
 
 curve 
 
 would 
 
 then 
 
 be:


!epe.gif!

The graph indicates the presence of one stable equilibrium point at {*}_x_ = 0{*}.

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The graph indicates the presence of one stable equilibrium point at x = 0.