Comparing Modulus of Elasticity in Maths vs Physics

[GeneralOJB]

I'm learning about Hooke's law and modulus of elasticity (also known as youngs modulus) but it seems I am being taught it differently in maths and physics.

In maths I am taught that T=λx/l and λ is the modulus of elasticity, measured in Newtons.

In physics I am taught that T=λAx/l where λ is measured in Pascals.

What's going on?

Just to clarify, T is the tension in the spring, x is the extension, l is the natural unstretched length of the spring and A is the cross sectional area.

 

[AlephZero]

The physics formula is usually written T=EAx/l where E is Young's modulus, and as you said it is measured in Pascals. For a rod or a straight wire, A is the cross section area of the wire. That makes sense, because E is a property of the material (steel, aluminum, rubber, nylon, etc), not the shape of any particular piece of the material like a wire or a rod. The formula for the force also includes the shape of the object, that is its length and cross section area.

Physically, E is the (negative) pressure you would need to apply to the end of the rod, to double its length. That is not a practical thing to do for most materials, because they would break long before the length had doubled,) and E is usually a big number. For steel, for example, it is about 2 x 10¹¹ Pascals. But since E is a property of the material, and not just something to do with springs, it appears in many other situations in mechanics which you will probably learn about later.

The physics formula T=EAx/l only applies to a straight piece of wire or a rod. If you have something like a coil spring, there is a complicated formula that involves the radius of the wire the spring is made from, the radius of the coils of the spring, the number of turns per unit length of the spring, etc but that is not very practical. Instead you use the "maths" formula. In that formula λ is not the elastic modulus (or Youngs modulus) of the material. λ describes how a particular design of spring behaves. It is the force (in Newtons) required to double the length of the spring (assuming it will stretch that much without damaging it, or course).

Often, you use a formula that doesn't even include the length of the spring, T = kx. In that formula k is the stiffness (in Newtons/meter) of the spring.

 

[GeneralOJB]

The physics formula is usually written T=EAx/l where E is Young's modulus, and as you said it is measured in Pascals. For a rod or a straight wire, A is the cross section area of the wire. That makes sense, because E is a property of the material (steel, aluminum, rubber, nylon, etc), not the shape of any particular piece of the material like a wire or a rod. The formula for the force also includes the shape of the object, that is its length and cross section area.

Physically, E is the (negative) pressure you would need to apply to the end of the rod, to double its length. That is not a practical thing to do for most materials, because they would break long before the length had doubled,) and E is usually a big number. For steel, for example, it is about 2 x 10¹¹ Pascals. But since E is a property of the material, and not just something to do with springs, it appears in many other situations in mechanics which you will probably learn about later.

The physics formula T=EAx/l only applies to a straight piece of wire or a rod. If you have something like a coil spring, there is a complicated formula that involves the radius of the wire the spring is made from, the radius of the coils of the spring, the number of turns per unit length of the spring, etc but that is not very practical. Instead you use the "maths" formula. In that formula λ is not the elastic modulus (or Youngs modulus) of the material. λ describes how a particular design of spring behaves. It is the force (in Newtons) required to double the length of the spring (assuming it will stretch that much without damaging it, or course).

Often, you use a formula that doesn't even include the length of the spring, T = kx. In that formula k is the stiffness (in Newtons/meter) of the spring.

Thanks for that, I understand now. What is the correct word to describe λ then? My textbooks refer to λ as the modulus of elasticity of the spring, rather than the modulus of elasticity of the material.

 

[AlephZero]

I don't like your textbook calling in a "modulus of elasticity", because that should have units of stress/strain (Pascals), not force/strain (Newtons).

And λ is a poor choice of symbol, because the standard definition of λ is a different way to measure the elastic modulus (called Lamé's first parameter, but don't worry about exactly what that is).

But if that is what your textbook uses, I guess you will have to use it, until you move on to another textbook.

In "real life" engineering, the most common formulas use the spring stiffness k, or Young's modulus E.

 

[PJC01]

It is not uncommon for concepts and equations to be taught differently in different subjects, even if they are related. In this case, the modulus of elasticity is a property that describes the stiffness of a material and it is used in both mathematics and physics. However, the units and equations used to represent it may vary.

In mathematics, the modulus of elasticity is typically denoted by λ and is measured in Newtons. This is because in mathematical models, the modulus of elasticity represents the force required to produce a given amount of deformation in a material.

In physics, the modulus of elasticity is denoted by λ as well, but it is measured in Pascals. This is because in physics, the modulus of elasticity represents the stress required to produce a given amount of strain in a material. Stress is measured in Pascals, which is a unit of pressure, while strain is a dimensionless quantity.

So, in essence, the difference lies in the perspective from which the modulus of elasticity is being viewed - a force perspective in mathematics and a stress perspective in physics. Both approaches are valid and useful in their respective fields.

It is important to understand the context in which the modulus of elasticity is being used and to pay attention to the units and equations being used. This will help you to make connections between the two subjects and deepen your understanding of the concept.