Proving Two Bodies Moving Together are One System

In summary, when two bodies are moving together, they can be considered as one system. This can be demonstrated by applying a force to one of the bodies and observing that both bodies move with the same acceleration. This is true as long as the applied force does not exceed the maximum force allowed by the coefficient of static friction between the two bodies. This concept also applies to objects connected by a rope, assuming the rope is inextensible. However, if the objects or rope have elasticity, the accelerations may not be equal.
  • #1
Kartik.
55
1
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 - FBA -F1= MAAA
FAB= MBAB

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?
 
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  • #2
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 μs, 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.
 
  • #3
jtbell said:
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 μs, 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
 
  • #4
jtbell said:
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 μs, 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).
 
  • #5
HallsofIvy said:
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?
 
  • #6
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.
 
  • #7
HallsofIvy said:
If two blocks are side by side (as said in the initial post)

Sorry, I missed that. :blushing:

Carry on...
 
  • #8
Kartik. said:
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!
 

1. How can I prove that two bodies are moving together as one system?

To prove that two bodies are moving together as one system, you can use the principle of superposition, which states that the total displacement of a system is equal to the sum of the individual displacements of its components. This means that if the two bodies have the same displacement, velocity, and acceleration, they can be considered as one system.

2. What evidence can I gather to show that two bodies are part of the same system?

To gather evidence that two bodies are part of the same system, you can measure their velocities and accelerations and see if they are equal or if they change in the same way. You can also observe their motion and see if they move in the same direction and at the same time.

3. Can two bodies moving together still be considered as separate systems?

Yes, two bodies moving together can still be considered as separate systems if their motion is not perfectly synchronized. For example, two cars driving next to each other on the highway can be considered as separate systems, even though they are moving together.

4. Is there a specific mathematical equation to prove that two bodies are moving together as one system?

Yes, the equation to prove that two bodies are moving together as one system is the principle of superposition. This equation states that the total displacement of a system is equal to the sum of the individual displacements of its components.

5. Why is it important to prove that two bodies are moving together as one system?

Proving that two bodies are moving together as one system is important because it allows us to analyze and understand their motion as a whole. This can help us predict their future positions and movements, and also make calculations and solve problems related to their motion more accurately.

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