# Kinetic energy of a continuum system

1. Feb 17, 2010

### topcomer

1. The problem statement, all variables and given/known data

Consider a rubber band living in the plane and having a uniform density, whose endpoints are V1 and V2. The rubber band passes through a runner without mass, inside which it can slide in a frictionless manner. The runner can assume any arbitrary position E in the plane, hence inducing a stretching in the rubber band.

2. Relevant equations

Write the kinetic energy of the system as a function of the coordinates V1,V2,E.

3. The attempt at a solution

I have two ideas but I got stuck with both. One is to map the material (undeformed) and the spatial (deformed) configurations to a reference one, to split the motion of material and spatial points, and try to do a kinematic analysis with convective terms.

The other is a calculus of variations approach, i.e. to consider infinitesimal motion of each one of the points V1,V2,E, and to compute the corresponding material movement at every point, from which the local kinetic energy is obtained, and then integrate up over the segments to obtain the total one.

Which one is more reasonable?

Last edited: Feb 17, 2010