Proving Two Bodies Moving Together are One System

[Kartik.]

How do we say(or prove) that when two bodies are moving together , they can be considered as one system?

Suppose we have two blocks of wood kept on a friction less surface side by side and you apply a force on anyone of them. I suppose from left to right, then the force equations for each block would be

F - F_BA -F₁= M_AA_A
F_AB= M_BA_B

And we say that both blocks will move with same accelerations.(Is this right?)

F₁ is the force or say resistance which block B gives to A when a Force F is applied(does this force exist?)

Can we prove the accelerations equal? Or its something else?

 

[jtbell]

If the applied force F is small, the blocks stick together (static friction), and their accelerations are equal. If you know the coefficient of static friction μ_s, you can calculate the maximum force for which this is true.

If the applied force is larger than this, the blocks start to slide against each other (kinetic friction), and their accelerations are not equal.

 

[Kartik.]

If the applied force F is small, the blocks stick together (static friction), and their accelerations are equal. If you know the coefficient of static friction μ_s, you can calculate the maximum force for which this is true.

If the applied force is larger than this, the blocks start to slide against each other (kinetic friction), and their accelerations are not equal.

How do you relate friction with the movement of the blocks?
The case i took is on a friction less plane and assuming no friction between the blocks
And i would really like you to answer the few questions i wrote in those brackets

 

[HallsofIvy]

If the applied force F is small, the blocks stick together (static friction), and their accelerations are equal. If you know the coefficient of static friction μ_s, you can calculate the maximum force for which this is true.

If the applied force is larger than this, the blocks start to slide against each other (kinetic friction), and their accelerations are not equal.

You seem too be assuming that one of the blocks is on top of the other. If two blocks are side by side (as said in the initial post), and you push on one, so that it pushes on the other (not said in the initial post but if not, one block would have moved independently of the other) then, yes, the two blocks move together, at the same acceleration. If one block has mass m and the other mass M, then together they will move with acceleration a= F/(m+ M).

 

[Kartik.]

You seem too be assuming that one of the blocks is on top of the other. If two blocks are side by side (as said in the initial post), and you push on one, so that it pushes on the other (not said in the initial post but if not, one block would have moved independently of the other) then, yes, the two blocks move together, at the same acceleration. If one block has mass m and the other mass M, then together they will move with acceleration a= F/(m+ M).

Like for the two blocks if you write the force equations the equations would be

For both the blocks considered as a single system -
F = (M+m)a

And for just the block on which the force is applied
F-N = ma₁

and we say that a = a₁; my questions is why?

 

[nasu]

It is actually an approximation good for rigid bodies.
If the accelerations were different, on of two things will happen:
If the pushed body has lower acceleration, the first one will travel a larger distance than the first one. The pushed body being in the way, it (or both) will have to stretch to make room
If the pushed body has higher acceleration, it will loose contact with the pushing body.

Same idea applies to a rope. We assume the rope is inextensible so the bodies connected with a rope under tension have same acceleration (otherwise the rope will have to change its length).
If you allow for elasticity of rope or bodies, then the accelerations are not necessarily equal.

 

[jtbell]

If two blocks are side by side (as said in the initial post)

Sorry, I missed that.

Carry on...

 

[HallsofIvy]

Like for the two blocks if you write the force equations the equations would be

For both the blocks considered as a single system -
F = (M+m)a

And for just the block on which the force is applied
F-N = ma₁

and we say that a = a₁; my questions is why?

Because they both represent the same thing- the acceration of that block!