# Is There A Relationship Between Young's Modulus And Spring Constant?

• FredericChopin
In summary, the Young's modulus and spring constant of a string or rod-like object are directly related and can be expressed by the formula k = Y*A/L. However, this formula may not hold true for more complex elastic systems where multiple dimensions are involved. The shape and dimensions of the object must also be considered in determining the relationship between Young's modulus and the spring constant.
FredericChopin
We know that the Young's modulus of an object is defined as the ratio between its stress and strain:

Y = σ/ε

, or:

Y = F*L/A*ΔL

We also know that Hooke's law, which can be applied to any linear elastic object, can find spring constant:

F = k*x

, rearranged:

k = F/x

But is there a relationship between the Young's modulus and the spring constant of an object? Is there a formula that shows this relationship?

Thank you.

Sure, we can make a closer comparision
Y = F*L/A*ΔL

Considering the ideal springs, when you put two in parallel the strength would double, whereas series would half. This means that k is proportional to A/L. Put it like k=cA/L where c is another constant, then F=c*A*ΔL/L, rearrange into c=F*L/A*ΔL which is analogous to the previous one.

However Young's modulus is applied to string or rod like material, which means the c would equal to the Young's modulus when k is for a string or rod. Otherwise, you need to consider the shape of the spring.

Just to clarify, from reading vanhees71's weblink and your post, the relationship between the Young's modulus and the spring constant of a string or rod-like object is:

k = Y*A/L ?

Also, would this formula only hold true up to the proportional limit of an object (since Hooke's law also states this limit)?

Last edited:
k = Y*A/L

Yes that's good.

Youngs modulus is simply for stress (loading) in a single dimension, such as stretching a string or compressing a column. So a single dimension (length) is involved.

Vanhees reference refers to a more general elastic system that might have loading in two or three dimensions and so area or volume might be involved. In this case the response is still proportional to the load, by spring constant but this does not relate so simply to youngs modulus.

This is what zealscience also meant by considering the shape.

Excellent! Thank you.

Last edited:

## 1. What is Young's modulus and spring constant?

Young's modulus is a measure of the stiffness or elasticity of a material, while the spring constant is a measure of the stiffness of a spring. Young's modulus is measured in units of stress divided by strain, while the spring constant is measured in units of force divided by distance.

## 2. Is there a mathematical relationship between Young's modulus and spring constant?

Yes, there is a mathematical relationship between Young's modulus and spring constant. The spring constant is equal to the Young's modulus multiplied by the cross-sectional area of the material divided by its length.

## 3. How does changing the Young's modulus affect the spring constant?

Changing the Young's modulus will directly affect the spring constant. A higher Young's modulus will result in a higher spring constant, meaning that the material or spring will be stiffer and require more force to be stretched or compressed.

## 4. Is the relationship between Young's modulus and spring constant the same for all materials?

No, the relationship between Young's modulus and spring constant can vary depending on the material. Different materials have different Young's moduli and cross-sectional areas, which will affect the spring constant.

## 5. Can Young's modulus and spring constant be used interchangeably?

No, Young's modulus and spring constant are two different measurements and cannot be used interchangeably. They describe different properties of materials and have different units of measurement.

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