Distribution of Energy when work is done on a system of 2 masses connected by a spring

[MattGeo]

Suppose there is a spring-mass system arranged as shown in my crude drawing. This occurs on a frictionless surface. The spring is 0.5 meters long and is at its natural length. The 2 masses are initially at rest and the left mass is 1 kg and the right mass is 3 kg. If a 10 N force is applied leftward as shown on the 3 kg mass, then what will be the kinetic energy of the system once the force has been applied through a displacement of 10 meters? Additionally, what will be the potential energy stored in the spring due to its compression, assuming spring constant is 0.5 N/m? What will be the spring force at maximum compression?

My first inclination was to calculate the spring force at maximum compression and the magnitude of its compression. I arrived at this calculating the acceleration of the center of mass of the system.

Applying Newton's second law to the entire system the block system should accelerate at 2.5 m/s^2 and then using this fact you can deduce that the net force on the 1 kg1 kg block is 2.5 N and the net force on the 3 kg block is 7.5 N, meaning that the spring force at maximum compression is 2.5 N.

By considering the average force exerted by the spring to be 1.25 N I concluded that there must be 6.25 J of energy stored in the spring and 93.75 J of energy showing up as kinetic energy of the 2-mass system after 10 m of displacement. I feel like this must be incorrect though. I can't tell if I am even thinking about the displacement correctly. The spring is being compressed while the system begins to move.

Application of the force to the 3 kg mass will cause it to begin to accelerate but the moment it does, it will start applying a force to the other block, causing it to accelerate, with some tiny lag before it starts moving of course.

As long as the 10 N force is continuously applied and the acceleration is constant, would the spring compress to a stable compressed length or would it actually undergo oscillations while being accelerated?

Upon further thought I think I should be considering the displacement of the center of mass but also the relative displacement between the 2 masses and assume that the center of mass moves in proportion to m2/(m1+m2)

I keep confusing myself with this problem but if anyone could help to walk me through it, it would be great.

 

[ergospherical]

The net force on the whole system ($F=-10\mathrm{N}$) governs the acceleration of the centre of mass, which is $a_{\mathrm{com}} = F/(m_1 + m_2)$ and a constant.

You should transform into the (accelerating) centre of mass frame. Although the length of the spring can vary with time, $l = l(t)$, the centre of mass of the system -- i.e. the point that is a fraction $m_2/(m_1 + m_2)$ along the length of the spring, can be considered fixed in this accelerating frame. So you now essentially have two independent springs on either side of this point.

Now you can analyze the accelerations of the two masses from within this accelerating frame, remembering that an "inertial" force $-ma_{\mathrm{com}}$ also acts on both masses.

 

[MattGeo]

I honestly don't think I can figure this out without further assistance and insight. It's been quite a long time since I took an advanced classical mechanics course. I just enjoy thinking about physics and made up this problem. Could you please prompt me with any further hints or important information?

 

[ergospherical]

In the centre of mass frame, you have two "independent" springs on either side of the centre of mass. (You do need to think about what the "effective" spring constants of these two springs are. How do springs "in series" combine?).

The right-hand mass is acted upon by spring tension, the external force $F$ and the inertial force $-ma_{\mathrm{com}}$. The left-hand mass is acted upon by spring tension and the inertial force $-ma_{\mathrm{com}}$.

In either case, the problem is just that of a mass on a spring, subject to some constant external force. That is probably a problem that you have solved before -- what is the motion?

If any of these sub-problems are unfamiliar, then it might be worth refreshing your memory on them before attempting a more complex combined problem.