(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a spring of natural length L_0 with constant k which rests on a horizontal frictionless surface. The spring is attached at one end to a fixed post and at the other end to a mass m. Suppose the spring is rotating around the post in a circle with angular velocity w. What is the new length of the spring?

The above problem is simple enough, and I have found a solution to it. Now consider the following related problem: you have a circular elastic band of natural radius R_0, mass m, and spring constant k. It is rotating on frictionless surface about its center with angular velocity w. What is the new radius of the rubber band?

I am trying to use my ideas from the first problem to solve the second problem.

2. Relevant equations

F=-Δx

F_c=mw^2R

3. The attempt at a solution

First I will show my solution to the first problem. It is as simple as setting the centripetal force equal to the restoring force of the spring: F_s=kΔx=F_c=mw^2L, where Δx=L-L_0. Solving I find,

L=(kL_0)/(k-mw^2)

As for the second problem, the main difficulty is how to model an elastic band. One idea is to think about it as the sum of many small springs attached to eachother at the center of the circle, and extending to the boundary of the circle, with a small mass element on the other end (in other words, each mass element dm of the band can be thought of as attached to it's own spring).

I am not really sure how to proceed this way, so another idea is to use Young's modulus and think about the stress/strain of the rubber band as it is deformed by the centrifugal force.

Another idea is to use energy. The rotational energy of the band is 1/2Iw^2, where I=mR^2 (the band can be thought of as a hoop). Not sure about the elastic potential energy of the band.

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# Homework Help: Centrifugal force and elastic deformation

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