How to Determine Max Extension of a Spring in a Frictionless System?

[Avi1995]

The situation is shown :-

All surfaces are frictionless. I am trying to find the max extension of the spring as the force is applied. Spring is massless.
F=maI tried drawing FBDs for block A and B:-

F-kx=Ma(from fbd of A)
kx=ma(from fbd of B)
, we get
x=Fm/k(M+m)
Is this correct??
Also can anyone explain how to visualize when spring if stretched from both sides? How do you then apply Hooke's law?

 

[Dickfore]

F-kx=Ma(from fbd of A)
kx=ma(from fbd of B)
, we get
x=Fm/(M+m)
Is this correct??

You need a k on the l.h.s.

 

[Avi1995]

OOps, I didnt type it.But, The problem is asking the max extension of the spring, but this is half of answer given in book.

 

[Dickfore]

Does the system start from rest?

 

[Avi1995]

Does the system start from rest?

Yes.

 

[Dickfore]

The stretching of the spring changes with time. In the initial moment, it is unstretched, and the force acting on the left body m is zero, whereas the acceleration of the right body M is F/M.

As the right body gets accelerated, it acquires a velocity and moves further from the left body, thus stretching the spring.

As the spring stretches, it starts accelerating the left body, and decelerating the right body, thus tending to bring their velocities to become equal and stop its stretching.

But, due to inertia, the right spring will keep on approaching the right body and there will be an oscillatory motion of the spring. You are required to find the maximum stretching, not the equilibrium one.

I think you should go to a non-inertial reference frame moving with the same acceleration as the one you should calculate as if the bodies were rigidly connected. Then, what are the equations of motion for each body?