What is the velocity of the center of inertia after bar 1 breaks off the wall?

[Saitama]

Homework Statement

Two bars of masses $m_1$ and $m_2$ connected by a weightless spring of stiffness $k$ rest on a smooth horizontal plane. Bar 2 is shifted a small distance $x$ to the left and then released. Find the velocity of the centre of inertia of the system after bar 1 breaks off the wall.

Homework Equations

The Attempt at a Solution

The bar 1 breaks off from the wall when the bar 2 just passes its original position but I don't have any idea about how to begin making the equations here.

 

[consciousness]

What will be the velocity of 1 when it breaks off the wall?
What will be the velocity of 2 when 1 breaks off the wall?(conserve energy)

 

[verty]

What can you tell us about the momentum of the system at various times?

 

[Saitama]

What will be the velocity of 1 when it breaks off the wall?
What will be the velocity of 2 when 1 breaks off the wall?(conserve energy)

Conserving energy,
m_1v_1^2+m_2v_2^2=kx^2
where $v_1$ and $v_2$ are the velocities of 1 and 2 when 1 breaks off the wall.

What can you tell us about the momentum of the system at various times?

Do you ask me to conserve linear momentum?

 

[Tanya Sharma]

Conserving energy,
m_1v_1^2+m_2v_2^2=kx^2
where $v_1$ and $v_2$ are the velocities of 1 and 2 when 1 breaks off the wall.

What will be the velocity of block 1 when it breaks off the wall?

Do you ask me to conserve linear momentum?

No need to conserve momentum.Linear momentum will not be conserved till block 1 remains in contact with the wall.

 

[Saitama]

What will be the velocity of block 1 when it breaks off the wall?

If I assume $v_1=0$, I get the right answer. Thanks!

 

[Tanya Sharma]

If I assume $v_1=0$, I get the right answer. Thanks!

Great ...

But do you understand why should you do so ?

 

[Saitama]

But do you understand why should you do so ?

As the question asks $v_{CM}$ at the instant 1 breaks off the wall, so we can assume $v_1=0$ and I don't see how will $v_1$ reach a finite value in an instant, right?

 

[Tanya Sharma]

When block 2 is released,it will move towards right,but block 1 will remain pushed against the wall.Block 2 will reach a point A such that the spring is in its natural uncompressed length.The force on block 1 at this point will be zero .

Now due to the momentum gained,Block 2 will continue moving towards right.Just as block 2 crosses point A,there will be a stretching in the spring which will pull the block 1 off the wall with an initial speed zero.

 

[Saitama]

When block 2 is released,it will move towards right,but block 1 will remain pushed against the wall.Block 2 will reach a point A such that the spring is in its natural uncompressed length.The force on block 1 at this point will be zero .

Now due to the momentum gained,Block 2 will continue moving towards right.Just as block 2 crosses point A,there will be a stretching in the spring which will pull the block 1 off the wall with an initial speed zero.

Thanks!

I had somewhat similar scenario in my mind when I began with this question. I figured out that 1 will separate from wall when 2 just passes A. I did not think of $v_1=0$. Should have been more careful.