Kinetic energy of a continuum system

[topcomer]

Homework Statement

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.

Homework Equations

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

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?

 

[anathema]

Thank you for your interesting question. Both of your ideas seem reasonable and could potentially lead to a solution. However, I would suggest taking a closer look at the second approach, using a calculus of variations. This method is often used in mechanics to obtain the equations of motion for a system. In this case, you could consider the rubber band as a continuous system and use the calculus of variations to find the equations of motion for the material points V1 and V2, as well as the position of the runner E. From there, you can derive the kinetic energy of the system as a function of these coordinates.

To do this, you can start by defining the Lagrangian of the system, which is the difference between the kinetic and potential energies. In this case, the potential energy can be assumed to be zero since the rubber band is uniform and there is no external force acting on it. Then, using the calculus of variations, you can derive the equations of motion for each point and solve them simultaneously to obtain the positions of V1, V2, and E as a function of time. Finally, you can plug these positions into the expression for kinetic energy to obtain the desired function.

I hope this helps. Good luck with your calculations!