What Determines the Maximal Elongation of a Spring in a Pulley System?

[snate]

Homework Statement

2 bricks of mass m are located near the edge of a smooth horizontal surface, they are connected by an ideal weightless ,unstretched spring with a length of l₀ and spring constant k. The brick which is closer to the edge is connected to another brick of the same mass by an ideal unstretchable rope going through a pulley. The lowest brick is held so that the rope above it hangs vertically.
The lowest brick is released
What's the minimal time τ after which the elongation of the spring ΔL will be maximal. Find ΔL.

Homework Equations

The Attempt at a Solution

Here's the solution until the point where I'm stuck. Sorry for the Russian text, it does not contain crucial information, please ignore it

a2-a1 is the acceleration of one end of the spring from the frame of reference of another end.
Then these equations are given (y is vertical coordinate)

I understand why w² is 3k/2m but why is the amplitude(A0) mg/3k? Can someone please explain? I only need an explanation for A₀
The next equations are

At the beginning the system is in equilibrium and then x(0)=0, that leads to B being equal to 0 and and A₀+A=0 because the spring isn't stretched at the beginning.

Thanks.

 

[haruspex]

I have not tried to follow your equations, but I can suggest an easier way via a sequence of simplifications.
First, model it as all happening on a horizontal surface, with a horizontal force mg applied to the rightmost block.
Next, we can merge the two blocks connected by the inextensible string into a single block mass 2m.
Clearly the average acceleration is g/3 to the right, so take a noninertial frame accelerating that way. In this frame, the left-hand block has a pseudoforce ... to the left, and the 2m block has a pseudoforce ... to the left, leaving it with a net force ... to the right. (Fill in the blanks.)
Finally, consider the common mass centre of the two blocks. How does this move in the reference frame? How do the two blocks move in relation to it?

 

[snate]

I have not tried to follow your equations, but I can suggest an easier way via a sequence of simplifications.
First, model it as all happening on a horizontal surface, with a horizontal force mg applied to the rightmost block.
Next, we can merge the two blocks connected by the inextensible string into a single block mass 2m.
Clearly the average acceleration is g/3 to the right, so take a noninertial frame accelerating that way. In this frame, the left-hand block has a pseudoforce ... to the left, and the 2m block has a pseudoforce ... to the left, leaving it with a net force ... to the right. (Fill in the blanks.)
Finally, consider the common mass centre of the two blocks. How does this move in the reference frame? How do the two blocks move in relation to it?

Thanks, your explanation hinted me why A₀ is mg/3k. A₀=a_max/w². And maximal acceleration a_max after infinitesimal moment of time since the rightmost block has been released is gm/2m=g/2 because at that moment the spring is not deformed, so F=kΔx=k0=0, so we don't have to consider the spring and the leftmost block at that moment. So A₀ =g*2m/(2*3*k)=mg/3k. Am I correct?

 

[haruspex]

Thanks, your explanation hinted me why A₀ is mg/3k. A₀=a_max/w². And maximal acceleration a_max after infinitesimal moment of time since the rightmost block has been released is gm/2m=g/2 because at that moment the spring is not deformed, so F=kΔx=k0=0, so we don't have to consider the spring and the leftmost block at that moment. So A₀ =g*2m/(2*3*k)=mg/3k. Am I correct?

Looks good.