When relaxed, an elastic cord has length L, cross section area A, mass M, and Young modulus Y.
An object of mass m is hung from the ceiling using the cord. The system reaches a steady state.
What is the longitudinal mass distribution lambda(z) of the stretched cord as a function of distance z from the object?
Hint: consider the stress-strain relation only for an infinitesimal segment of the cord. You may assume that the cross section remains A.
As usual, don't forget to explore the limits of your result.
Looks like a MIT 8:01 problem to me :)
It is too much for me at present but this link might help a little:
https://books.google.com/books?id=PTj2eEbT_jQC&pg=PA67&dq=elongation+of+long+hanging+rod+due+to+own+weight&hl=en&sa=X&ei=7TGfVI_7NcufgwSx-YKYCg&ved=0CGwQ6AEwCQ#v=onepage&q=elongation%20of%20long%20hanging%20rod%20due%20to%20own%20weight&f=false
try deleting the s on https
http://books.google.com/books?id=PTj2eEbT_jQC&pg=PA67&dq=elongation+of+long+hanging+rod+due+to+own+weight&hl=en&sa=X&ei=7TGfVI_7NcufgwSx-YKYCg&ved=0CGwQ6AEwCQ#v=onepage&q=elongation%20of%20long%20hanging%20rod%20due%20to%20own%20weight&f=false
Note equation 1.78 seems to be missing an = sign just left of the integral sign
To determine the longitudinal mass distribution lambda(z) of the stretched cord as a function of distance z from the object, we can consider the stress-strain relation for an infinitesimal segment of the cord.
Let's start by considering a small segment of the cord of length dz, located at a distance z from the object. The segment has a mass dm and a cross-sectional area A.
We can assume that the cord is in static equilibrium, which means that the forces acting on the segment are balanced. The weight of the segment dm is acting downward, and the tension in the cord is acting upward.
The weight of the segment dm is given by dm = (m/L) * dz, where m is the mass of the object and L is the relaxed length of the cord.
The tension in the cord can be determined using the stress-strain relation. In this case, we can assume that the cross-sectional area remains constant, so we can use the formula for the stress in terms of the Young modulus.
The stress (force per unit area) is given by σ = F/A, where F is the tension in the cord and A is the cross-sectional area.
The strain (change in length normalized by the original length) is given by ε = ΔL/L, where ΔL is the change in length of the cord segment.
According to Hooke's law, the stress is proportional to the strain, with the proportionality constant being the Young modulus. So we can write σ = Y * ε.
For an infinitesimal segment of the cord, the change in length ΔL is related to the elongation ϵ by ΔL = ϵ * dz.
Combining these relations, we have:
σ = Y * ε = Y * (ΔL/L) = Y * (ϵ * dz/L)
The force acting on the cord segment is F = σ * A = Y * (ϵ * dz/L) * A.
Now, since the cord is in static equilibrium, the tension in the cord must balance the weight of the segment dm. So we have:
Y * (ϵ * dz/L) * A = (m/L) * dz
Simplifying this equation, we find:
Y * ϵ * A = (m/L)
Rearranging, we obtain:
ϵ = (m/L) / (Y * A)
Now, we can substitute ϵ = ΔL / dz, and rearrange the equation to solve for ΔL:
ΔL = (m/L) * (dz / (Y * A))
The longitudinal mass distribution lambda(z) is defined as the mass per unit length. So we can write:
lambda(z) = dm / ΔL = (m/L) * (Y * A) / (m/L) * (dz / (Y * A))
Simplifying, we find:
lambda(z) = 1 / dz
Therefore, the longitudinal mass distribution lambda(z) is a constant and equal to 1/dz. This means that the mass per unit length is constant along the cord, regardless of the distance from the object.
It's important to note that this result holds as long as we assume the cross-sectional area A remains constant and the cord is in static equilibrium. If the assumptions are not valid, the longitudinal mass distribution may not be constant along the cord.