Help With Understanding Hooke's Law

[tomtomtom1]

Hi all

I was hoping someone could help me understand Hookes Law.

Hooke’s Law, states that Stress is proportional to Strain.

The bit I am struggling to understand is that Stress is F/A and Strain is Change in Length/Original Length, so if Stress is proportional to Strain then shouldn't F/A be proportional to Change in Length/Original Length?

I am getting confused because Hookes Law is given by the equation F=Kx, where F, is the force, K is a constant and X is the change in the length - How do you drive F=Kx from F/A and Change in Length/Original Length?

Can anyone help?

Thank you.

 

[RPinPA]

The original length is a constant of the spring.

So let's say we have $(F/A) = R * (x/L)$ where R = some constant of proportionality, x = change in length and L = original length.

Then $F = (RA/L)x$ and since R, A and L are constants of the spring, we can treat the whole expression in parentheses as a constant. Call it $k$, so $F = kx$.

The fact that the "spring constant" depends on the original length of the spring tells you something about how the spring will behave, what the new constant of proportionality will be, if you cut the spring in half.

 

[tomtomtom1]

RPinPA

Thank you for your response, from your comment I was able to follow you calculations.

If I can add the following, Hookes Law states that Applied force is proportional to the change in Length which I have interpreted as saying if I apply a force of 5N on a rod which results in the rod extending by 30mm then if I double the applied force then the change in the rods length will also double so by applying 10N of force the rod would extend by 60mm - this is what I interpret as proportional.

The point I am trying to make is that applied force is proportional to the change in length not applied force Equal to change in length, in your response you have added R which is a constant and then changed proportional to Equal - where does R come from?

Am I being stupid?

I understand the math just don't understand the concept and it really bugs me.

 

[RPinPA]

If I can add the following, Hookes Law states that Applied force is proportional to the change in Length which I have interpreted as saying if I apply a force of 5N on a rod which results in the rod extending by 30mm then if I double the applied force then the change in the rods length will also double so by applying 10N of force the rod would extend by 60mm - this is what I interpret as proportional.

That is exactly correct. $F = kx$ says that applied force is proportional to the amount of stretch. $k$ is the constant of proportionality. For your example, $k = (5 \text { N}) / (0.030 \text{ mm}) = 167 \text{ N/m}$. Every N of applied force will stretch it by 1/167 of a meter or 6 mm. But $k$ is a property of this particular spring. Make it out of a different material, or use a thicker wire or a thinner one, or a longer or shorter piece of the same wire, and you'll have different a $k$.

The point I am trying to make is that applied force is proportional to the change in length not applied force Equal to change in length,

That's correct, force is proportional to the change in length. I thought you were asking how you go from force being proportional to $(x/L)$ to force being proportional to $x$ and my answer is that if it's proportional to $x$ times a constant $(1/L)$, it's proportional to $x$.

Force can't be equal to change in length anyway, since they have different units. There HAS to be a proportionality constant.

in your response you have added R which is a constant and then changed proportional to Equal - where does R come from

I didn't add R. You did. You implied there's a constant. I'm just giving a name to the proportionality constant in your original statement:

Hooke’s Law, states that Stress is proportional to Strain.

Stress is proportional to strain. That means stress equals a constant times strain. When you have a proportion, you have a proportionality constant. I gave that constant the name $R$ but in fact I think it's the thing called Young's Modulus and there's some different symbol which is standard.

Every material will have a different Young's Modulus. Rubber is stretchier than steel, and steel is probably stretchier than wood.

I didn't "change proportional to equal". Saying "a is proportional to b" and saying "a equals some constant times b" are equivalent statements. "a equals some constant times b" is the same as saying "a/b is always the same number" which is the same as saying "a is proportional to b".

I understand the math just don't understand the concept and it really bugs me.

You seem to understand the concept of "proportional" just fine, but feel free to ask more questions. Physics is full of these kinds of proportions, and when we have a proportion, that requires a proportionality constant, which we often give a name.

For instance, Newton figured out that gravitational force between two masses $m_1$ and $m_2$ at distance $r$ was proportional to $m_1 m_2 / r^2$. He didn't say it's equal, he said it's proportional. So we multiply that thing by a proportionality constant, which we call G, which is a property of the universe and which depends on what units we're using. $F = G * (m_1 m_2 / r^2)$. He just knew it was proportional, he didn't know what G was. That had to be measured, much later.

Physics is full of these kinds of constants that have to be measured, many of them depending on the type of material and tabulated, and the vast majority of them are just names given to a proportionality constant in some proportion. It's a lot more common to have a proportion than to have an equality.

 

[CWatters]

Hooks law basically describes the elastic properties of a particular object like a wire.

However sometimes you want to know the properties of the material the wire is made of so you can calculate the elastic properties of any wire of different sizes. The /A and the /L in Stress and Strain essentially eliminate the size of the wire as a factor leaving you with just the material properties.

Bit like the difference between the heat capacity of a tank of water and the specific heat capacity of water in general.