How Is the Spring Constant Derived from Young's Modulus, Length, and Area?

[nickb145]

Homework Statement

Consider a metal bar of initial length L and cross-sectional area A. The Young's modulus of the material of the bar is Y. Find the "spring constant" k of such a bar for low values of tensile strain.


Express in terms of Y, L, and A.

Homework Equations

I know the hooks law equation and the modulus but i can't figure out the answer it wants me ot get

Maybe I'm doing it wrong but i am getting Y/(L*A)...

The Attempt at a Solution

 

[CWatters]

Spring constant = F/ΔL .....(1)

Youngs Mod. Y = F/ΔL * L/A ......(2)

Rearrange (2) to give an equation for F/ΔL

Substitute into 1

 

[nickb145]

im getting F=(Y*A)/L

 

[CWatters]

You are close but the answer isn't the force F = (Y*A)/L

The problem asks you to work out an equation for the spring constant k.

 

[Nickie]

Your attempt at a solution is correct. The "spring constant" k can be calculated using the formula k = Y/(L*A). This formula represents the relationship between the applied force (F) and the resulting elongation (x) of the bar, which is given by Hooke's law: F = k*x. In this case, the applied force is the tensile force, and the resulting elongation is the tensile strain, which is small for low values of strain. Therefore, for small values of strain, the spring constant can be approximated by the Young's modulus divided by the product of the bar's length and cross-sectional area. This relationship is known as Hooke's law and is a fundamental principle in the study of materials and their mechanical behavior. It allows us to predict the response of a material to an applied force and is essential in designing and engineering structures and devices.