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Hooke's law: What is a Hooke spring?

Important Formula

Formula: Hooke's Law

$$\class{red}{F} ~=~ -D \, y$$ $$\class{red}{F} ~=~ -D \, y$$ $$D ~=~ \frac{\class{red}{F}}{y}$$ $$y ~=~ \frac{\class{red}{F}}{D}$$

Rearrange formula

What do the formula symbols mean?

Restoring force

$$ \class{red}{F} $$ Unit $$ \mathrm{m} $$

Spring force exerted by the spring to return the spring to its unstretched state.

The minus sign takes into account the direction of the force. The force acts against the direction of deflection. For the magnitude, the minus sign may of course be omitted.

Spring constant

$$ D $$ Unit $$ \frac{\mathrm{kg}}{\mathrm{s}^2} $$

The spring constant is a material constant and depends on the spring used. It tells how well the spring can be stretched (that is how elastic the spring is).

Material	Spring constant $D$ in $\frac{\mathrm N}{ \mathrm{m} } $
Aluminium (Al)	0.81
Aurium (Au) „Gold“	0.92
Titanium (Ti)	1.45
Platinium (Pt)	2.06

Deflection

$$ y $$ Unit $$ \mathrm{m} $$

The deflection of the spring from its neutral position. For example, if you pull on the spring, $y$ is the distance by which the spring was deflected.

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A Hooke spring is an elastic mechanical spring that builds up a restoring force $ F $ when it is stretched or compressed. The restoring force of a Hooke spring depends on the deflection $ y $.

Hooke's law - displacement of the spring + restoring force

In Hooke's law, the force $F $ and the deflection $y$ are linearly related. This means: If you double the deflection of the spring, the restoring force $F$ doubles:

$$ \begin{align} \class{red}{F} ~=~ -D \, y \end{align} $$

Here, $ D $ is a spring constant that indicates how elastic a spring is.

Table : Examples of spring constants for different materials
Material	Spring constant $D$ in $\frac{\mathrm N}{ \mathrm{m} } $
Aluminium (Al)	0.81
Aurium (Au) „Gold“	0.92
Titanium (Ti)	1.45
Platinium (Pt) „Platin“	2.06

Hooke Spring = Harmonic Oscillator

Hooke's spring is an example of a harmonic oscillator. The mass on the spring performs harmonic oscillations and can be described with a cosine or sine function:

$$ \begin{align} y(t) ~=~ x_0 \, \cos(\omega \, t ~+~ \varphi) \end{align} $$

The restoring force $ F $ is proportional to the deflection $y$, whereby the propotionality constant is the product of the mass $m$ and the frequency $ \omega $ with which the spring oscillates:

$$ \begin{align} F(t) ~=~ m \, a(t) ~=~ -m \, \omega^2 \, y(t) \end{align} $$

$$ \begin{align} \class{blue}{v(}t\class{blue}{)} ~=~ -\omega \, A \sin(\omega \, t + \varphi) \end{align} $$

$$ \begin{align} v_{\text{max}} ~=~ \sqrt{ \frac{D}{\class{brown}{m}} } \, A \end{align} $$

$$ \begin{align} a(t) ~=~ -\omega^2 \, A \, \cos(\omega \, t + \varphi) \end{align} $$

$$ \begin{align} a_{\text{max}} ~=~ \frac{D}{\class{brown}{m}}\,A \end{align} $$

$$ \begin{align} f ~=~ \frac{1}{2\pi} \sqrt{\frac{D}{\class{brown}{m}}} \end{align} $$

$$ \begin{align} T ~=~ 2\pi \, \sqrt{\frac{ \class{brown}{m} }{D}} \end{align} $$

How to measure mechanical forces?

A mechanical force $ F $ can be measured with a Hooke's spring. The change in length $ y $ of a Hooke spring is measured. If you know the spring constant $ D $, you can easily calculate the force using Hooke's law.