Two questions regarding the conservation of linear momentum

In summary: The only way to have both linear and angular momentum is for the center of mass to have a velocity while spinning, which is the case when it touches the ground. This is due to the reaction force from the ground, which allows the wheel to transfer its angular momentum into linear momentum. In summary, the conservation of linear momentum can be seen in both cases discussed. In the first case, the momentum of the ball is transferred to the wall and the Earth, keeping the total linear momentum constant. In the second case, the spinning wheel has both angular and linear momentum, but when it touches the ground, the reaction force allows for the transfer of angular momentum into linear momentum, keeping the total linear momentum
  • #1
hideelo
91
15
Lately I have been thinking about the conservation of linear momentum, and there are two cases that I can think of that seem to violate the conservation principle.

1. Throwing a rubber ball against a wall. Imagine you threw a rubber ball in what we will call the positive x direction, if the ball were to hit a wall we all know from experience what will happen next, the ball will bounce off the wall. However, the ball started off with linear momentum in the plus x direction and now it has its momentum in the negative x direction. How in this case is momentum conserved?

2. Imagine you had a spinnng wheel which you gently lower down onto the ground, as soon as it touches the ground it will start moving away from you. Once again, before it touches the ground it has only angular momentum and no linear momentum, yet as soon as it touches the ground it will move away from you and therefore must have linear momentum. Where is the momentum coming from?Thank you
 
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  • #2
Direction is not considered in momentum calculation. Momentum = Mass * Velocity. This satisfies the first question.

Secondly, overall momentum is conserved. Changing "types" of momentum does not violate the Law of Conservation of Momentum.
 
  • #3
I disagree on both counts, first of all momentum is a vector, direction does matter, and I'm pretty sure linear and angular momentum are both conserved independently.
 
  • #4
You can disagree all you like. Is there such thing as 'negative momentum?' If it suits you, consider the equation for momentum to be such: |v| * |m| = momentum.

Also, let's pretend that they are conserved independently. You're forgetting that it does in fact have linear momentum, even while spinning. v = rw

Velocity (v) equals the radius (r) of rotating object times the angular speed (omega).
 
  • #5
mrnike992 said:
Direction is not considered in momentum calculation. Momentum = Mass * Velocity. This satisfies the first question.


Huh? This is patently false. Of course momentum depends on direction, it is a vector quantity, just like the velocity! ##\vec{p}=m\vec{v}##

@ Hideelo:

1) The wall (if it's attached to the Earth, then the Earth) will move, just very very little because it's very very massive. This effect will assure momentum conservation between the ball+wall system. You have to consider the wall as well in the calculation for momentum conservation.

2) In a perfectly symmetrical system, the spinning wheel would not move linearly, but just continue to spin. The fact that real tops do move linearly have to do with the fact that in the real world, we cannot make the initial conditions perfectly symmetric. This is essentially an example of spontaneous symmetry breaking! And again, in order to consider the total linear momentum, one would have to consider the ground into this calculation. The ground moves VERY VERY little, but it does move!
 
  • #6
hideelo said:
1. Throwing a rubber ball against a wall. Imagine you threw a rubber ball in what we will call the positive x direction, if the ball were to hit a wall we all know from experience what will happen next, the ball will bounce off the wall. However, the ball started off with linear momentum in the plus x direction and now it has its momentum in the negative x direction. How in this case is momentum conserved?
The wall + Earth receive twice the original momentum of the ball in positive direction, so total linear momentum remains constant.

hideelo said:
2. Imagine you had a spinnng wheel which you gently lower down onto the ground, as soon as it touches the ground it will start moving away from you. Once again, before it touches the ground it has only angular momentum and no linear momentum, yet as soon as it touches the ground it will move away from you and therefore must have linear momentum. Where is the momentum coming from?
Here again the Earth receives the opposite linear momentum that the wheel receives, so total linear momentum remains constant.
 
  • #7
mrnike992 said:
Direction is not considered in momentum calculation. Momentum = Mass * Velocity. This satisfies the first question.

Wrong. Total momentum is a vector and is conserved as a vector, including its direction

mrnike992 said:
Secondly, overall momentum is conserved. Changing "types" of momentum does not violate the Law of Conservation of Momentum.

Wrong. Linear and angular momenta are conserved individually not "overall". There is no "overall momentum", because linear and angular momenta have different units and cannot be added.

mrnike992 said:
Is there such thing as 'negative momentum?'
Its a vector that might have a negative component.

mrnike992 said:
If it suits you, consider the equation for momentum to be such: |v| * |m| = momentum.
Wrong. This is not the equation for momentum.

mrnike992 said:
You're forgetting that it does in fact have linear momentum, even while spinning. v = rw
Wrong. If the center doesn't move the total linear momentum of the wheel is zero.
 
Last edited:
  • #8
A.T. said:
Wrong. Total momentum is a vector and is conserved as a vector, including its direction

Okay, yes, we've covered this. Then the rebound is not due to it's momentum, but rather to Newton's third law of motion.

A.T. said:
Wrong. Linear and angular momenta are conserved individually not "overall". There is no "overall momentum", because linear and angular momenta have different units and cannot be added.

What? This doesn't quite make sense to me. Also, it does have linear momentum even when only spinning.
 
  • #9
mrnike992 said:
Then the rebound is not due to it's momentum, but rather to Newton's third law of motion.
Newton's third law is closely related to momentum conservation.

mrnike992 said:
Also, it does have linear momentum even when only spinning.
If the center doesn't move the total linear momentum of the wheel is zero.
 
  • #10
A.T. said:
Wrong. If the center doesn't move the total linear momentum of the wheel is zero.

Obviously. But, if there is angular momentum, there is linear momentum. Period.
 
  • #11
mrnike992 said:
But, if there is angular momentum, there is linear momentum.
Wrong. If it spins in place the total angular momentum of the wheel is not zero, but the total linear momentum of the wheel is zero.
 
  • #12
A.T. said:
Wrong. If it spins in place the total angular momentum of the wheel is not zero, but the total linear momentum of the wheel is zero.
Actually, THAT'S wrong. If it has a linear velocity and a mass, then it has a linear momentum.
 
  • #13
mrnike992 said:
If it has a linear velocity ...
It doesn't. It spins in place.
 
  • #14
Yes it does. I believe I posted the equation earlier, but I will repeat myself: V=R*w
 
  • #15
Linear velocity equals the radius of the rotating object times the angular velocity.
 
  • #16
Therefore, if it has a linear velocity and a mass, then it must have a linear momentum.
 
  • #17
mrnike992 said:
Linear velocity equals the radius of the rotating object times the angular velocity.

The linear velocity of a point on a rotating object is equal to the vector cross product of the radius vector from the center to that point times the angular velocity of the object.

That linear velocity is different for every point because it points in a different direction. If you add up the momenta of all the points on a suitably symmetric rotating object, the total comes to zero.
 
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  • #18
Okay, that makes sense.
 
  • #19
hideelo said:
Lately I have been thinking about the conservation of linear momentum, and there are two cases that I can think of that seem to violate the conservation principle.

1. Throwing a rubber ball against a wall. Imagine you threw a rubber ball in what we will call the positive x direction, if the ball were to hit a wall we all know from experience what will happen next, the ball will bounce off the wall. However, the ball started off with linear momentum in the plus x direction and now it has its momentum in the negative x direction. How in this case is momentum conserved?

2. Imagine you had a spinnng wheel which you gently lower down onto the ground, as soon as it touches the ground it will start moving away from you. Once again, before it touches the ground it has only angular momentum and no linear momentum, yet as soon as it touches the ground it will move away from you and therefore must have linear momentum. Where is the momentum coming from?


Thank you
Remember that momentum is only conserved for an isolated system where there are no external forces acting on it. Neither the ball nor the wheel is isolated. The ball has a force from the wall and the wheel has a force from the ground.

You can include the whole Earth in the system to get rid of external forces (or at least ignore them) and then you get momentum conservation.
 

1. What is the law of conservation of linear momentum?

The law of conservation of linear momentum states that in a closed system, the total momentum of all objects remains constant before and after any interactions occur. In other words, the total momentum before a collision or interaction is equal to the total momentum after the collision or interaction.

2. How is the conservation of linear momentum related to Newton's third law?

Newton's third law states that for every action, there is an equal and opposite reaction. This means that in a collision between two objects, the total momentum of the objects before and after the collision must be equal and opposite. This demonstrates the conservation of linear momentum, as the total momentum of the system is conserved even if individual momentums change.

3. Can the law of conservation of linear momentum be violated?

No, the law of conservation of linear momentum is a fundamental law of physics and has been extensively tested and proven to be true. It holds true in all known physical interactions and cannot be violated.

4. How is the law of conservation of linear momentum applied in real-world situations?

The law of conservation of linear momentum is applied in many real-world situations, such as collisions between objects, explosions, and rocket propulsion. It is also used in engineering and design to ensure that the momentum of a system is conserved and the desired outcome is achieved.

5. What is the relationship between the mass and velocity of objects in the conservation of linear momentum?

The conservation of linear momentum states that the total momentum of a system is constant. This means that if the mass of an object increases, its velocity must decrease in order to maintain the same total momentum. Similarly, if the velocity of an object increases, its mass must decrease to maintain the same total momentum.

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