A rod with cross-sectional area A and length L is made from a material with a Young's modulus of Y. This rod could be considered an extremely stiff spring with some spring constant k. Find an expression for k in terms of the other quantities.
To find an expression for the spring constant k, we need to relate it to the other quantities given: the cross-sectional area A, the length L, and the Young's modulus Y.
The Young's modulus Y relates the stress (force per unit area) applied to a material to the strain (fractional change in length) it undergoes. In the case of this rod, if we apply a force F along its length, it will cause the rod to extend by some amount ΔL.
The stress (σ) applied to the rod is given by:
σ = F / A
And the strain (ε) experienced by the rod is given by:
ε = ΔL / L
According to Hooke's Law, the strain is proportional to the stress within the elastic limit of the material. This can be written as:
ε = (1 / Y) * σ
where Y is the Young's modulus.
Since the strain (ε) can also be written as ΔL / L, we can rearrange the equation to solve for ΔL:
ΔL = (L / Y) * σ
Now let's consider the force applied to the rod (F). The force required to extend a spring is given by Hooke's Law:
F = k * ΔL
where k is the spring constant.
Substituting the expression for ΔL from earlier, we have:
F = k * (L / Y) * σ
Rearranging this equation to solve for the spring constant k, we find:
k = (Y * F) / (L * σ)
The expression for the spring constant k in terms of the other quantities is:
k = YF / (L * σ)
Therefore, the spring constant k is equal to the Young's modulus Y times the applied force F, divided by the product of the length L and the stress σ.