<p>If we sidestep your typo (the last term has one absolute too much), both formulations are correct. They just express different things. The $k$ in Hooke's law is for a particular spring. $k_s$ is the siffness for a paricular material. </p>

<p>Now in the linear portion there is a direct relationship betwen these the material stffness is directly propotional to the spring factor and length of member. So for example when the rest length of a different spring with same material is 2 times as long its 1/2 as stiff.</p>

CC BY-SA 3.0
2017-10-13T17:04:32.547Z
I add an example could you please see that?
shashack
CC BY-SA 3.0
2017-10-13T17:56:46.187Z
@shashack obviously they model a  different thing. The other models springs that can be any size but equally elastic and the other only works for springs that you haev separately tested. All is as i explained and expected.
joojaa
CC BY-SA 3.0
2017-10-13T12:34:37.820Z
2017-10-13T12:34:37.820Z
joojaa
<p>I'm studying spring model. There is a suggested equation (Hooke's Law vector form) But, I couldn't understand how to derive that equation. </p>

<p>I'm reading 'Computer Animation Algorithms and Techniques Third Edition.
In the page 241, (7.88)
The equation is as follows: </p>

<p>($|v_{1}^* - v_{2}^*|$: the rest length)</p>

<p>($|v_{1} - v_{2}|$: the current length)</p>

<p>$F_s = (\frac{k_s|v_1 -v_2| - |v_{1}^* - v_{2}^*|}{{|v_{1}^* - v_{2}^*|}})\frac{(v_1 -v_2)}{|v_1 -v_2|}$ // I think the $k_s$ is typo. </p>

<p>(I referred to <a href="http://www2.cs.uregina.ca/~anima/408/Notes/Cloth/Cloth.htm" rel="nofollow noreferrer">http://www2.cs.uregina.ca/~anima/408/Notes/Cloth/Cloth.htm</a>)</p>

<p>So, maybe $F_s = k_s(\frac{|v_1 -v_2| - |v_{1}^* - v_{2}^*|}{\color{red}{|v_{1}^* - v_{2}^*|}})\frac{(v_1 -v_2)}{|v_1 -v_2|}\cdot\cdot$(1) is correct. </p>

<p>However, I wonder why the red part divides the equation?</p>

<p>If I know correctly about Hooke's Law, $F=-k_s(current_{lenght} - rest_{length})\cdot\cdot$(2).</p>

<p>It means that the red part should be removed like that $F_s = k_s(|v_1 -v_2| - |v_{1}^* - v_{2}^*|)\frac{(v_1 -v_2)}{|v_1 -v_2|}\cdot\cdot$(3).</p>

<p>For example, in 3D space, there is a spring that has the spring term as $k_s$. </p>

<p>The rest length is '3', so let's assume the left rest point, $v_2^*$, is fixed at (0,0,0) and the right rest point, $v_1^*$, is (3,0,0).</p>

<p>And then we moved $v_1^*$ to $v_1 $=(5, 0, 0) with some force.
We can calculate the force with (2), and the force is $-2k_s = -k_s(|(5,0,0) - (0,0,0)| - |(3,0,0) - (0,0,0)|)$</p>

<p>But if we use equation (1), the result is different.</p>

<p>$F_s = -k_s(\frac{|v_1 -v_2| - |v_{1}^* - v_{2}^*|}{{|v_{1}^* - v_{2}^*|}})\frac{(v_1 -v_2)}{|v_1 -v_2|} \\\quad= -k_s(\frac{|(5,0,0) -(0,0,0)| - |(3,0,0) - (0,0,0)|}{{|(3,0,0) - (0,0,0)|}})\frac{((5,0,0) -(0,0,0))}{|(5,0,0) -(0,0,0)|} \\\quad= -\frac{2}{3}k_s(1,0,0)$ </p>

<p>The force amount of $-\frac{2}{3}k_s(1,0,0)$ is totally different from $-2k_s$</p>

<p>Could you explain what I'm wrong?</p>

CC BY-SA 3.0
2017-10-13T09:54:58.607Z
The name of the guy is Hooke threfore it is Hooke's law
joojaa
CC BY-SA 3.0
2017-10-13T16:28:04.613Z
Oh, sorry about that! I edited my typos.
shashack
<mathematics><physically-based><vector-graphics><physics><model>
Hooke's Law vector form
shashack
# Hooke's Law vector form

(Asked by: username_0 on 2017-10-13)

I'm studying spring model. There is a suggested equation (Hooke's Law vector form) But, I couldn't understand how to derive that equation.

I'm reading 'Computer Animation Algorithms and Techniques Third Edition.
In the page 241, (7.88)
The equation is as follows:

($|v\_{1}^\* - v\_{2}^\*|$: the rest length)

($|v\_{1} - v\_{2}|$: the current length)

$F\_s = (\frac{k\_s|v\_1 -v\_2| - |v\_{1}^\* - v\_{2}^\*|}{{|v\_{1}^\* - v\_{2}^\*|}})\frac{(v\_1 -v\_2)}{|v\_1 -v\_2|}$ // I think the $k\_s$ is typo.

(I referred to <http://www2.cs.uregina.ca/~anima/408/Notes/Cloth/Cloth.htm>)

So, maybe $F\_s = k\_s(\frac{|v\_1 -v\_2| - |v\_{1}^\* - v\_{2}^\*|}{\color{red}{|v\_{1}^\* - v\_{2}^\*|}})\frac{(v\_1 -v\_2)}{|v\_1 -v\_2|}\cdot\cdot$(1) is correct.

However, I wonder why the red part divides the equation?

If I know correctly about Hooke's Law, $F=-k\_s(current\_{lenght} - rest\_{length})\cdot\cdot$(2).

It means that the red part should be removed like that $F\_s = k\_s(|v\_1 -v\_2| - |v\_{1}^\* - v\_{2}^\*|)\frac{(v\_1 -v\_2)}{|v\_1 -v\_2|}\cdot\cdot$(3).

For example, in 3D space, there is a spring that has the spring term as $k\_s$.

The rest length is '3', so let's assume the left rest point, $v\_2^\*$, is fixed at (0,0,0) and the right rest point, $v\_1^\*$, is (3,0,0).

And then we moved $v\_1^\*$ to $v\_1 $=(5, 0, 0) with some force.
We can calculate the force with (2), and the force is $-2k\_s = -k\_s(|(5,0,0) - (0,0,0)| - |(3,0,0) - (0,0,0)|)$

But if we use equation (1), the result is different.

$F\_s = -k\_s(\frac{|v\_1 -v\_2| - |v\_{1}^\* - v\_{2}^\*|}{{|v\_{1}^\* - v\_{2}^\*|}})\frac{(v\_1 -v\_2)}{|v\_1 -v\_2|} \\\quad= -k\_s(\frac{|(5,0,0) -(0,0,0)| - |(3,0,0) - (0,0,0)|}{{|(3,0,0) - (0,0,0)|}})\frac{((5,0,0) -(0,0,0))}{|(5,0,0) -(0,0,0)|} \\\quad= -\frac{2}{3}k\_s(1,0,0)$

The force amount of $-\frac{2}{3}k\_s(1,0,0)$ is totally different from $-2k\_s$

Could you explain what I'm wrong?

**Tags:** mathematics, physically-based, vector-graphics, physics, model

> **Comments:**
> **username_1**: The name of the guy is Hooke threfore it is Hooke's law  
> **username_0**: Oh, sorry about that! I edited my typos.  

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## 1 answers

### Answer by username_1 on 2017-10-13. Score: 1 | Accepted answer

If we sidestep your typo (the last term has one absolute too much), both formulations are correct. They just express different things. The $k$ in Hooke's law is for a particular spring. $k\_s$ is the siffness for a paricular material.

Now in the linear portion there is a direct relationship betwen these the material stffness is directly propotional to the spring factor and length of member. So for example when the rest length of a different spring with same material is 2 times as long its 1/2 as stiff.

> **Comments:**
> **username_0**: I add an example could you please see that?  
> **username_1**: @username_0 obviously they model a  different thing. The other models springs that can be any size but equally elastic and the other only works for springs that you haev separately tested. All is as i explained and expected.  

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