SHM: Impulse applied to two masses connected by spring

[A17]

I've come across the following problem:

Two otherwise isolated, equal masses m are at rest and connected by a spring with constant k. An impulse is applied to one of the masses along the direction of the spring connecting them. What happens?

Qualitatively, I think that the force will both accelerate the whole system and compress the spring, resulting in translational motion of the whole system and SHM about the centre of mass. The proportion of these two types of motion will depend on k and the time over whoch the force is applied.

However, I can't quite put this into mathematics. Given the impulse, m and k, is there a way to calculate the c-of-m speed and the chacteristic properties of the oscillation exactly?

Any help woild be much appreciated!

 

[maverick280857]

Is the surface frictionless?

Well, qualitatively if the surface is not frictionless then an impulse takes some time to get communicated through the spring to the other mass because the mode of communication is the spring force which develops as the spring gets elongated or compressed.

If you know the nature of the impulse $F(t)$ then you can set up the differential equations of motion and then use the Laplace Transform to solve them. The impulse could be defined in several ways. Do you know the transform method?

In the absence of specific information, this problem can become quite complicated if you want to know the exact parameters (position/velocity) of the two masses. But in cases where the force is zero and the system responds to an initial condition (elongated or compressed spring for example) the solution is fairly simple.

Can you show us your work (the equations you have set up so far)?

 

[Meir Achuz]

The impulse J gives the mass you strike an initial velocity v_0=J/m.
Just apply this initial condition.

 

[Doc Al]

The applied impulse also allows you to calculate the center of mass speed of the system. Assuming no other external forces act on the system, that speed will remain constant throughout the motion.