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.

 

[paranormal]

I would approach this problem by first clarifying some assumptions and simplifications in the given scenario. For example, are the pucks sliding on a frictionless surface? Is the spring massless? Are the pucks considered point masses or do they have a finite size?

Once these details are clarified, I would proceed with the approach of setting up the Lagrangian and finding the equations of motion. However, instead of directly applying a small radial perturbation, I would first analyze the system's stability and determine whether it is in stable or unstable equilibrium. This can be done by finding the effective potential energy of the system and analyzing its minima and maxima.

If the system is in stable equilibrium, then a small radial perturbation would result in oscillations around the equilibrium point. The frequency of these oscillations can be found by linearizing the equations of motion around the equilibrium point and solving for the natural frequency of the resulting linear system.

If the system is in unstable equilibrium, then a small radial perturbation would result in the pucks moving further away from each other. In this case, the frequency of the subsequent radial oscillations would not be well-defined as the pucks would continue to move away from each other.

In summary, as a scientist, I would approach this problem by first clarifying any assumptions and simplifications, analyzing the stability of the system, and then determining the frequency of subsequent radial oscillations based on the system's behavior.