Wet wheel and conservation of momentum

[jartsa]

If there's no friction between mug and table and you aren't requiring that the bicycle moves at a constant speed, then yes. That's a different setup with a different mechanism for transferring momentum between the various components of the system so can lead to different momentum transfers.

Well I guess I need to make some design changes:

A cyclist has a paint roller attached to the end of the handlebar, he drives past a wall onto which the roller paints a line.

Now I claim this bike-paint roller device is a kind of a rocket, the propellant is the paint, the momentum of the propellant goes to the rest of the rocket and nowhere else.

 

[Nugatory]

Well I guess I need to make some design changes:

A cyclist has a paint roller attached to the end of the handlebar, he drives past a wall onto which the roller paints a line.

Now I claim this bike-paint roller device is a kind of rocket, the propellant is the paint, the momentum of the propellant goes to the rest of the rocket and nowhere else.

If I’m understanding you properly: the roller is loaded up with paint and at rest relative to the bicycle. It starts turning when it contacts the wall, and as it rolls it unloads its paint onto the wall to leave a line. Can we assume for simplicity that the roller itself is massless (so we don’t need to calculate its moment of inertia or mess with its rotational kinetic energy and it can spin up instantaneously) and that it rolls without slipping (so we don’t have to mess with friction) and that the paint doesn’t spatter?

With these assumptions, the interaction between the bicycle and the wall reduces the mass of the bicycle and increases the mass of the earth; the change in each is of course equal to the mass of the paint transferred from one to the other. The paint starts out moving at the initial speed of the bicycle and is slowed to the speed of the wall; therefore the wall exerts a force on the paint to decelerate it, and by Newton’s second law the paint exerts an equal and opposite force on the wall.

Working in a frame in which the wall is initially at rest: the speed of the bicycle does not change. The wall starts at rest and is very slightly accelerated. The paint starts out with some momentum because it’s moving at the same speed as the bicycle; this momentum is transferred to the wall with a line of paint on it.

 

[jartsa]

If I’m understanding you properly: the roller is loaded up with paint and at rest relative to the bicycle. It starts turning when it contacts the wall, and as it rolls it unloads its paint onto the wall to leave a line. Can we assume for simplicity that the roller itself is massless (so we don’t need to calculate its moment of inertia or mess with its rotational kinetic energy and it can spin up instantaneously) and that it rolls without slipping (so we don’t have to mess with friction) and that the paint doesn’t spatter?

Yes those sound like good assumptions.We could also put motors, brakes, sensors, and computers on the roller, surely that technology can eliminate forces on the wall. ... No let's put the sensors on the wall, and a wireless communication system between the sensors and the paint roller.
The paint tool must be a roller, not a brush. Spray can works too, if correctly aimed. Brush does not exert a force on the paint, so paint has to exert a force on the wall.

 

[jbriggs444]

@Dale seems to have hinted pretty strongly at the situation in #27. The elephant in the room is the mistaken assumption that the water on the wheel or the paint on the roller is uniformly distributed around the circumference of the device.

If the wheel is rolling and losing water/paint, this assumption is clearly counter-factual. There is less paint/water on the wheel behind the contact point than on the wheel in front of the contact point.

You can try to repair this problem by losing water very slowly or applying a very thin coat of paint. But that does not help. You reduce the imbalance and the rate at which momentum is transferred due to the imbalance. But you increase the length of time for which the imbalance is present.

As has been suggested, if you find yourself in a situation where conservation laws are being violated, you've made a mistake. You need to find that mistake.

 

[russ_watters]

Isn't this scenario exactly the same as picking up a bucket of water, moving it 2m to the right and putting it back down?

 

[jbriggs444]

Isn't this scenario exactly the same as picking up a bucket of water, moving it 2m to the right and putting it back down?

I have not seen it nailed down carefully. My take was a wet wheel (or saturated paint roller) losing water (or paint) as it rolls on a dry surface at what is expected to be an unvarying speed while subject to no net thrust.

The puddle that the wheel starts in seems to be window dressing only -- intended to motivate an initial condition of a moving wheel saturated with water.

 

[russ_watters]

The puddle that the wheel starts in seems to be window dressing only -- intended to motivate an initial condition of a moving wheel saturated with water.

Ahh. That's...odd.

 

[Dale]

I claim this bike-paint roller device

I thought that your interest was in the paint/water and explicitly not in the combined system. I have to say that this is pretty annoying. I made a valid point for which you complained that it was not relevant, only to have you now make it explicitly relevant.

Please review and respond to my previous comment which you had previously dismissed as it is now directly relevant.

 

[jbriggs444]

Well I guess I need to make some design changes:

A cyclist has a paint roller attached to the end of the handlebar, he drives past a wall onto which the roller paints a line.

Now I claim this bike-paint roller device is a kind of a rocket, the propellant is the paint, the momentum of the propellant goes to the rest of the rocket and nowhere else.

Since you seem to be attempting to eliminate friction with the wall, why not eliminate the wall?

You are a passenger in a weightless environment of some sort. You have a long roll of transparent wrap which you have removed from the box. You look at the roll end on and see that the wrap winds in a long clockwise spiral path from the center toward the outside. You give the roll a hard spin in the clockwise direction. The right hand edge is moving "down" and the left hand edge is moving "up".

You trigger a release of some sort exactly when the leading edge of the plastic is at the right hand side of the roll. The freed wrap proceeds in a straight line downward.

You watch as the cardboard tube inside the plastic roll rises slowly in the air despite the absence of any external force on the tube+wrap system.

Is that a scenario you are after? Did you want to claim any violations of conservation of momentum, angular momentum or energy based on its predicted behavior?

 

[jartsa]

In the ideal case, during the time that the water interacts with the Earth its momentum does not change. It is only during the time that the water interacts with the wheel that its momentum changes. The thing that gets the water’s momentum should be clear.

So I guess you are hinting that the Earth does not get the momentum, but instead the bike gets the momentum.

Well that sounds sounds plausible.

(What else should I say? I can't say if it's right or wrong, I don't have such authority here)I have been wondering the following:

Energy has momentum v*E/c². So the battery of a speeding electric car is losing momentum. That momentum has to go to the Earth or to the car. Well maybe it goes to the car. So the device that takes the momentum of the energy and puts it to the car is the wheel, because that's where the energy is going, right?

 

[Nugatory]

So the battery of a speeding electric car is losing momentum. That momentum has to go to the Earth or to the car. Well maybe it goes to the car.

It depends on whether the car is changing speed or not. And there are two ways you can define the speed of the car: Relative to the earth, which is assumed to be immobile; and relative to an arbitrary fixed point in space. The first way is a really excellent approximation and it's easy to calculate with, so it's how we usually work problems involving things moving around on the surface of the Earth - but it is an approximation, and momentum is not exactly conserved by that approximation because it neglects the (very very small) changes in the Earth's momentum.

Either way, in any reasonable electrical vehicle the battery and the rest of the car are solidly connected to one another so move together. Thus, thinking about them as separate systems only complicates things: you have to consider the second-law force pair between the car and the battery, but these forces are internal to the car+battery system so do not change its momentum.

 

[Dale]

Energy has momentum v*E/c2. So the battery of a speeding electric car is losing momentum. That momentum has to go to the Earth or to the car. Well maybe it goes to the car. So the device that takes the momentum of the energy and puts it to the car is the wheel, because that's where the energy is going, right?

Assuming no dissipation and assuming the car’s mass is negligible compared to the earth, then the energy from the battery goes into the KE of the car. The total energy does not go down, and the momentum increases as v increases.

 

[jartsa]

It depends on whether the car is changing speed or not. And there are two ways you can define the speed of the car: Relative to the earth, which is assumed to be immobile; and relative to an arbitrary fixed point in space. The first way is a really excellent approximation and it's easy to calculate with, so it's how we usually work problems involving things moving around on the surface of the Earth - but it is an approximation, and momentum is not exactly conserved by that approximation because it neglects the (very very small) changes in the Earth's momentum.

Either way, in any reasonable electrical vehicle the battery and the rest of the car are solidly connected to one another so move together. Thus, thinking about them as separate systems only complicates things: you have to consider the second-law force pair between the car and the battery, but these forces are internal to the car+battery system so do not change its momentum.

Why is change in Earth's momentum very very small? Is your car moving at speed very close to c on the surface of the earth?

At speeds where Newtonian physics is a good approximation the magnitudes of momentum changes of the car and the Earth are almost the same.

 

[Dale]

Why is change in Earth's momentum very very small?

He meant that the change in the earth’s speed is small.

 

[jartsa]

Assuming no dissipation and assuming the car’s mass is negligible compared to the earth, then the energy from the battery goes into the KE of the car. The total energy does not go down, and the momentum increases as v increases.

So the energy gives the Earth impulse x, and to the car the energy gives an impulse -x + the momentum of the energy itself. This conserves momentum.

 

[jartsa]

Is that a scenario you are after? Did you want to claim any violations of conservation of momentum, angular momentum or energy based on its predicted behavior?

Yes to the first question. No the the rest.

 

[Nugatory]

Why is change in Earth's momentum very very small? Is your car moving at speed very close to c on the surface of the earth?

At speeds where Newtonian physics is a good approximation the magnitudes of momentum changes of the car and the Earth are almost the same.

This has absolutely nothing to do with any relativistic effects, so we can take Newtonian physics to be exact.

The change in the earth’s momentum is exactly equal to and opposite to the change in the car’s momentum. And that change in the earth’s momentum is very small, by many decimal orders of magnitude, compared with what it would it take for us to notice. Thus we normally ignore it and recognize that because we’re not counting it we will seem to lose it somewhere.Only when someone starts a “where did the momentum go?” thread do we point out that this tiny change in the earth’s speed is being ignored.