Deriving expressions for Linear Expansivity and Young's Modulus

[Jappey101]

Homework Statement

The equation of state of a system consisting of a cylinder of a certain perfectly elastic material, subjected to an axial load F, is

F= CT(L-Li)^2

where Li is the unstretched length, L is the length under a load F when the temperature is T on a selected empirical scale and C is a constant.

DERIVE EXPRESSIONS FOR THE LINEAR EXPANSIVITY AND YOUNG'S MODULUS OF THE MATERIAL.

Homework Equations

dW=FdL

dL=(∂L/∂θ)- F held constant+(∂L/∂F)-θ held constant

linear expansivity=1/L(∂L/∂θ)- F held constant

young's modulus= (L/A)(∂F/∂L)-θ held constant

The Attempt at a Solution

I am lost. I tried integrating the formula, i tried expanding the equation of state, but all my attempts left me nowhere. I just need the first step because I am lost. Thank you!

 

[mark726]

The first step in deriving expressions for the linear expansivity and Young's modulus of the material is to understand the physical meaning and significance of these quantities. The linear expansivity is a measure of how much a material expands or contracts when subjected to a change in temperature. It is defined as the change in length per unit original length per unit change in temperature. On the other hand, Young's modulus is a measure of the stiffness or rigidity of a material, and it is defined as the ratio of stress to strain in a material under tension or compression.

To derive the expressions for these quantities, we can start by considering the formula for the work done on the cylinder, dW=FdL. This formula represents the small amount of work done on the cylinder when it is subjected to a small change in length, dL, under a constant load, F. We can rewrite this formula in terms of the variables in the equation of state, F= CT(L-Li)^2, as follows:

dW= CT(L-Li)^2 dL

Now, we can use the chain rule to express dL in terms of the variables in the equation of state, as shown in the Homework Equations section. This will give us the following expression:

dW= CT(L-Li)^2 [(∂L/∂θ)- F held constant] dθ

Next, we can integrate this expression to find the total work done on the cylinder between two temperatures, θ1 and θ2, as follows:

W= ∫dW= ∫CT(L-Li)^2 [(∂L/∂θ)- F held constant] dθ

= C∫(L-Li)^2 [(∂L/∂θ)- F held constant] dθ

= C(L-Li)^2 ∫[(∂L/∂θ)- F held constant] dθ

= C(L-Li)^2 [(∂L/∂θ)(θ2-θ1)- F(θ2-θ1)]

= C(L-Li)^2 [(L2-Li)-(L1-Li)]

= C(L2-L1)(L-Li)^2

= CΔθ(L-Li)^2

where Δθ=θ2-θ1 is the change in temperature and L2 and L1 are the lengths of the cylinder at temperatures θ