- #1
gabee
- 175
- 0
Homework Statement
I was thinking over lunch today about this old problem from intro physics but I can't remember whether or not this is the correct solution.
What is the final length L of a spring of mass M and spring constant k, initially of length L0 after it is left to hang under its own weight?
Homework Equations
The Attempt at a Solution
Divide the unstretched spring into infinitesimal segments dl. Each of these segments will stretch an infinitesimal distance dx under the weight of an infinitesimal mass dm by the relation g dm = k dx. Let [tex]\lambda[/tex] be the linear density of the spring, so that [tex]dm = \lambda \,dl[/tex]. Then,
[tex]g \lambda \,dl = k \,dx[/tex], and
[tex]dx = \frac{g}{k}\lambda dl[/tex].
Integrating LHS from 0 to X and RHS from 0 to L0, we find that [tex]X = \frac{g}{k}\lambda L_0[/tex], so the final length of the spring is L = L0 + X or
[tex]L = L_0(1 + \frac{g}{k}\lambda) = \frac{g}{k}M + L_0[/tex].
Is that right?
Last edited: