Linear Momentum of system of two blocks

• RoboNerd
In summary: They say the speed of the slab and block is the same, so the magnitudes of their velocities are the same, but one should have a plus sign, but the other should have a negative one.
RoboNerd
1. Homework Statement

The problem is shown in the photo above. I would like to discuss part B.

Homework Equations

conservation of linear momentum

The Attempt at a Solution

the solutions say that we should have the following approach:
Mb*vb = [Mb + Ms] * vFinal.

Plug in the numbers and get vF = 0.57 m/s

However, I do not understand this. I thought that momentum is a vector and that while the block is moving forward, the slab is moving backward, so I would need to have the following:

Mb*vb = Mb*vF + (Ms* -vF)

notice the negative vF for the final momentum term of the slab. I put a negative because the slab would be moving backwards with a negative velocity.

Why is my approach not correct? I got a different vF than 0.57.

Thanks

RoboNerd said:
However, I do not understand this. I thought that momentum is a vector and that while the block is moving forward, the slab is moving backward, so I would need to have the following:

Mb*vb = Mb*vF + (Ms* -vF)

notice the negative vF for the final momentum term of the slab. I put a negative because the slab would be moving backwards with a negative velocity.

Why is my approach not correct? I got a different vF than 0.57.

Thanks
i think the first part is important -when you draw the FBD for the two bodies your confusion will get settled- remember when two surfaces in contact move relative to each other the frictional forces on the bodies also act equally but in opposite direction. if one is pulling behind the other will be pushing ahead. let us draw the part a, that is the forces.

RoboNerd said:
View attachment 98864 1. Homework Statement

The problem is shown in the photo above. I would like to discuss part B.

Homework Equations

conservation of linear momentum

The Attempt at a Solution

the solutions say that we should have the following approach:
Mb*vb = [Mb + Ms] * vFinal.

Plug in the numbers and get vF = 0.57 m/s

However, I do not understand this. I thought that momentum is a vector and that while the block is moving forward, the slab is moving backward, so I would need to have the following:

Mb*vb = Mb*vF + (Ms* -vF)

notice the negative vF for the final momentum term of the slab. I put a negative because the slab would be moving backwards with a negative velocity.

Why is my approach not correct? I got a different vF than 0.57.

Thanks

Why do you think the slab would move backwards?

I did the first part allright. For both of those objects, I will have a normal and gravitational force. the block will have a leftwards frictional force and the slab will have a rightwards frictional force.

The block is going to be going forward until the frictional force stops it.
The slab will thus have to go backwards to conserve momentum

RoboNerd said:
I did the first part allright. For both of those objects, I will have a normal and gravitational force. the block will have a leftwards frictional force and the slab will have a rightwards frictional force.

The block is going to be going forward until the frictional force stops it.
The slab will thus have to go backwards to conserve momentum

The slab moves forward because there is nothing pushing the slab to the left. All the momentum is in the forward direction.

The block has ƒ = 0.5 × 10 × 0.2 = 1 N, so it has a = -2 m/s2
And the slab has same ƒ, so it has a = ⅓ m/s2

Then v1 block = v2 slab

Sorry if I only up to here

PeroK said:
The slab moves forward because there is nothing pushing the slab to the left. All the momentum is in the forward direction.

How is that possible? I know that when I stand on a piece of cardboard and I move forward, the board will move backwards to conserve momentum [but I know that there is friction force here]. Are you saying that there is no friction between the block and slab that will cause the slab to move in reverse? [this is false because there is friction]

They say the speed of the slab and block is the same, so the magnitudes of their velocities are the same, but one should have a plus sign, but the other should have a negative one.

RoboNerd said:
when I stand on a piece of cardboard and I move forward, the board will move backwards to conserve momentum
I assume you are thinking of standing still on the cardboard then walking forwards. That is not the same as set up here. The right analogy would be to get up speed on a (frictional) floor then jump onto the cardboard and try to stop on it. Which way will the cardboard move now?

I and the cardboard will move forward. But how do we know if this analogy is valid, especially with all those frictional forces we have drawn in the free body diagram?

What makes my approach wrong?

First and foremost
The frictional force acting on the small block is in the backward direction (I assume the right direction to be the forward direction and the left direction accordingly)
The frictional force (which equals mu times N and is purely kinetic accelerates the slab
And the same frictional force which acts on the small block decelerates it to a certain speed vf
And the final velocity of the slab also becomes vf

You're lucky
The problem doesn't involve the length of the slab (over which the block moves)
Otherwise it would have been too cumbersome

No!
Why would the slab have a backward velocity?
When you run over a floor and gain speed
And then jump on a cardboard, the cardboard moves along with you
Why?
It's because of the points which are in contact (here, they're your legs)
Considering the cardboard offers friction
Your feet are moving forward w.r.t to the cardboard
However, consider the motion from your frame
When you move forward
Don't you momentarily see the cardboard moving behind you in the other direction? (However, this isn't the final speed of the cardboard, just the negative of the initial speed you possessed
And that too for a small instant
And now since w.r.t your feet, the cardboard tries to move backward, what will friction do?
It will oppose relative motion at the "point of contact" and push the cardboard forward!:)UchihaClan13

Since there's no external force acting on the slab-block system (friction is an internal non conservative force)
Gravity is an external force here
But its component in the horizontal direction is m×g×cos 90=0
So using the principle of conservation of momentum
The answer you got is correct
Nothing wrong!

UchihaClan13

RoboNerd said:
I and the cardboard will move forward. But how do we know if this analogy is valid, especially with all those frictional forces we have drawn in the free body diagram?

What makes my approach wrong?
Friction acts to oppose relative motion (actual or potential) of the two surfaces in contact. Note that exact wording. So it acts on each of the two surfaces in opposition to its motion relative to the other surface.
The initial state is that the block is moving and the slab is not. Which two contacting surfaces are in relative motion? Which way does friction act on each to oppose that relative motion?

Let's see. The slab and the block are in contact and are in relative motion.

Ahh... I think I see now.

My force diagram dot for the slab shows the friction force going rightwards, so the friction force must accelerate the block. What an interesting question this is.

Thanks a lot for the help!

You are welcome!

1. What is linear momentum?

Linear momentum is a physical quantity that describes the motion of an object in a straight line. It is the product of an object's mass and its velocity, and is represented by the symbol p.

2. How is linear momentum of a system of two blocks calculated?

The linear momentum of a system of two blocks can be calculated by adding together the individual momenta of each block. This can be represented as p = p1 + p2, where p1 and p2 are the momenta of the two blocks.

3. Is linear momentum conserved in a system of two blocks?

Yes, linear momentum is conserved in a system of two blocks as long as there is no external force acting on the system. This means that the total linear momentum before and after the interaction of the two blocks will remain the same.

4. How does the mass and velocity of each block affect the linear momentum of the system?

The mass of each block directly affects the linear momentum of the system, as momentum is directly proportional to mass. However, the velocity of each block has a larger impact on the overall momentum, as momentum is directly proportional to velocity squared.

5. Can the linear momentum of the system be negative?

Yes, the linear momentum of the system can be negative if the direction of motion is opposite to the chosen positive direction. This indicates that the system is moving in the opposite direction with the same amount of momentum.

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