Two rotating masses attached by a spring

[forceface]

Homework Statement

Two pucks of mass m slide freely on a horizontal plane. They are connected by a
spring (constant k and negligible un-stretched length) and set in circular motion
with angular momentum L. The pucks are given a small, simultaneous radial
poke. What is the frequency of subsequent radial oscillations?

Homework Equations

Lagrangian? -> Equations of motion?

The Attempt at a Solution

I thought of setting up the Lagrangian of the system and finding the Equations of Motion and then some how apply the small radial pertabation and from that the radial frequency should just pop out. My Lagrangian is
L=1/2m(r1'^2+r2'^2)+1/2m((r1*θ1')^2+(r2*θ2')^2)-1/2k(r1-r2)^2. Since the system is set in circular motion all θ' are constant such that θ'=ω for both theta. With that said the equations of Motions are
-m*r1''+m*w^2*r1-k(r1-r2)=0
&
-mr2''+m*w^2*r2+k(r1-r2)=0
Now how apply the small radial perturbation? is r1-r2<<r1 or r2?
(Note that all time derivatives are denoted by a ', i.e. v(t)=x'(t))

 

[Simon Bridge]

... or you could set up the 1D system in rotating coordinates.

 

[vanhees71]

The Lagrangian looks fine, although with LaTeX it would be somwhat easier to read ;-).

I'd, however, use different generalized coordinates, namely center-of-mass and relative coordinates, because the use of coordinates adapted to the symmetries of the problem is always of advantage. The "little radial kick" has simply to be implemented into the initial conditions, although it's not very clearly stated what's meant by that "kick" precisely.

Is this really the complete problem statement given to you?

 

[Oxvillian]

Since the system is set in circular motion all θ' are constant such that θ'=ω for both theta. With that said the equations of Motions are

Hi forceface - I think you're doing things kind of backwards here. You need to get the equations of motion first before you start thinking about specific solutions (like the solution where the masses spin round in a circle). That will lead you to an "effective potential", from which you can extract your answer.