The inverse-square law: Gravitational force on two falling marbles

[Isambard]

Imagine making a hole in the ground, about a mile deep, with a large and square diameter. In the middle of the hole, there is a hollow and narrow tube with all air sucked out. Next to one of the walls, so close that it's touching, there is another hollow tube without air inside. Two identical marbles are dropped simultaneously, with one falling through the tube in the middle, and the other through the tube close to the wall.

Since one of them is close to one of the walls, it is closer to the mass surrounding them. The other tube is in the middle, and so the distance to the walls around it is the same, and the gravitational pull should even each other out.

Will one of the marbles hit the bottom before the other, even if it is a very tiny difference between them if that's the case?

 

[Ibix]

What do you think and why?

 

[Isambard]

Personally I feel it can be compared with sliding across a slippery surface. The more friction you experience, the more it will slow you down.
Which is why it sounds most likely that the marble falling next to the wall will fall slightly slower than the marble in the middle of the hole. Just curious if that's the correct interpretation or not.

 

[A.T.]

Personally I feel it can be compared with sliding across a slippery surface. The more friction you experience, the more it will slow you down.

1) Your title says "inverse-square law", hinting at Newtonian gravity, so I assume this is what you ask about despite posting it in the Relativity forum.

2) Newtonian gravity is a conservative force, which doesn't have any dissipative effects like friction.

3) The marbles might have slightly different fall types, due to the in-homogeneous gravitational field in the hole.

But It's not a simple analysis. It would be somewhat simpler to compare the fall in the center of a very narrow cylindrical hole with fall in the center of a wide cylindrical hole. For Newtonian gravity one could use the principle of superposition to subtract the gravity of the missing cylinder from the gravity of the planet.

My intuitive guess: The fall time in the narrow hole will be shorter, because initially you have more near mass pulling down. So even if there is less acceleration at the end due to more mass pulling up from above, the average speed should be higher.

 

[A.T.]

If a marble is dropped in the center of the hole, and the walls around it has the same distance, wouldn't the gravitational attraction towards the walls, even if it's microscopic, even each other out?

In the center of the hole only the horizontal gravity force components cancel.

If you drop a marble and it falls very close to one of the walls, that attraction should be slightly stronger.

Near a wall the horizontal gravity force components don't cancel, so it will drift towards the wall during the fall.

But what does this have to do with the fall duration, which depends on the vertical acceleration, not the horizontal one?

 

[PeterDonis]

@Isambard ChatGPT is not a valid reference here. We can't discuss what it says. Do you have any other basis for what you think is the answer to your question?

 

[PeterDonis]

Moderator's note: Thread moved to Classical Physics since it is based on Newtonian gravity.

 

[jbriggs444]

Use the principle of superposition. You have an earth with a positive density attracting the objects. And a cylinder with a negative density repelling them.

The Earth's attraction is at a slight diagonal with respect to the outer object while it attracts the inner object straight down.

So... the bottom of this "cylindrical" hole. Is it slightly dished in so that it is following the countour of a spherical shell? Same for the dished out top? Properly tapered sides?

 

[Baluncore]

The marbles initial velocity is downwards, but as the Earth rotates, the pipe also rotates, so the initial accumulated velocity component is no longer coaxial with the pipe. The marbles will hit the Eastern side of the tube after they are released, due to Earth rotation.

 

[jbriggs444]

The marbles initial velocity is downwards, but as the Earth rotates, the pipe also rotates, so the initial accumulated velocity component is no longer coaxial with the pipe. The marbles will hit the Eastern side of the tube after they are released, due to Earth rotation.

We can put some approximate numbers on this effect. I'll assume an equatorial drop.

A one mile drop is about 1600 meters. The SUVAT equation for final velocity comes from $KE=\frac{1}{2}mv^2 = mgh$. We divide out the $m$ and solve for $v = \sqrt{2gh}$. $gh$ is about $16000 \text{ meter}^2/\text{sec}^2$. Double that to 32000 and take the square root yielding about 178 meter/sec.

Coriolis acceleration is given by $2\omega \times v$. Omega here is $2 \pi$ radians per sidereal day of about 23 hours and 56 minutes (86164 seconds). That works out to 0.000729 radians per second. Multiply by the final velocity of 178 meters per second and we have a final horizontal acceleration of about 0.26 meters per sec².

The 1.6 km fall will take place at an average velocity of 178/2 = 89 meters per second. That will take just about 18 seconds during which the coriolis acceleration will be ramping up linearly.

We integrate once from horizontal coriolis acceleration to get velocity. Average velocity is half of final.
We integrate again from horizontal velocity to get horizontal displacement. This time we're integrating a quadratic, so we lose a factor of three.

So we expect $\frac{1}{6}a_\text{final}t^2 = \frac{1}{6}(0.26)(18^2) = 14 \text{ meters}$

So I get a coriolis deflection of 14 meters on a one mile drop.

Goodness knows, I may have screwed that calculation up, but it is in the right ballpark.

Edit: I did indeed screw up. Missed the factor of 2 in the coriolis formula the first time through.

 

[Baluncore]

Goodness knows, I may have screwed that calculation up, but it is in the right ballpark.

Is it towards the East?

 

[jbriggs444]

Is it towards the East?

Lol, yeah. Earth spins toward east. Higher moves faster than lower. So an eastward drift. I always have to talk that part through in exactly this way.

 

[wizwom]

Imagine making a hole in the ground, about a mile deep, with a large and square diameter. In the middle of the hole, there is a hollow and narrow tube with all air sucked out. Next to one of the walls, so close that it's touching, there is another hollow tube without air inside. Two identical marbles are dropped simultaneously, with one falling through the tube in the middle, and the other through the tube close to the wall.

Since one of them is close to one of the walls, it is closer to the mass surrounding them. The other tube is in the middle, and so the distance to the walls around it is the same, and the gravitational pull should even each other out.

Will one of the marbles hit the bottom before the other, even if it is a very tiny difference between them if that's the case?

The differential in gravitational force direction will be about 12 orders of magnitude less than the central earth gravitation force. So much less it may safely be ignored. They will hit the ground at the same time to anyone measuring.

 

[jbriggs444]

The differential in gravitational force direction will be about 12 orders of magnitude less than the central earth gravitation force. So much less it may safely be ignored. They will hit the ground at the same time to anyone measuring.

I think that we all agree that any difference will be "close enough for government work".

The question in the OP is whether the various tiny discrepancies make a "very tiny difference".

Will one of the marbles hit the bottom before the other, even if it is a very tiny difference between them if that's the case?

 

[sophiecentaur]

And a cylinder with a negative density

I'd like to buy some of that stuff. Which shop sells it?

 

[Ibix]

I'd like to buy some of that stuff. Which shop sells it?

A drill allows you to make your own negative density shapes. But it's a boring process.

 

[A.T.]

And a cylinder with a negative density

I'd like to buy some of that stuff. Which shop sells it?

I'd like to see that money with negative values on it, that the bank claims I have in my account.

 

[sophiecentaur]

A drill allows you to make your own negative density shapes. But it's a biring process.

Zero?

 

[A.T.]

Zero?

Superposition principle means that you can add up fields from different sources. If you know the field of whole object, then you can add the negated field of the materiał that would be in a hole, to compute the field of an object with a hole.

 

[Ibix]

Zero?

As @A.T. says, removing a lump of matter of density $\rho$ is the same as adding a same-shaped lump of matter of density $-\rho$ in the same place - at least as far as the mathematics of Newtonian gravity is concerned. So the drill "adds negative density matter", at least mathematically.

 

[sophiecentaur]

As @A.T. says, removing a lump of matter of density $\rho$ is the same as adding a same-shaped lump of matter of density $-\rho$ in the same place - at least as far as the mathematics of Newtonian gravity is concerned. So the drill "adds negative density matter", at least mathematically.

. . . . sort of. But that volume still has zero mass. When we do shcool 'moments' calculation we would safely use a "-M" on the see saw but that negative mass is only 'virtual' and can't exist on its own.

 

[A.T.]

. . . . sort of. But that volume still has zero mass. When we do shcool 'moments' calculation we would safely use a "-M" on the see saw but that negative mass is only 'virtual' and can't exist on its own.

For a long time, understanding of negative numbers was delayed by the impossibility of having a negative-number amount of a physical object, for example "minus-three apples", and negative solutions to problems were considered "false"...

https://en.wikipedia.org/wiki/Negative_number#History

 

[sophiecentaur]

For a long time, understanding of negative numbers was delayed by the impossibility of having a negative-number amount of a physical object, for example "minus-three apples", and negative solutions to problems were considered "false"...

https://en.wikipedia.org/wiki/Negative_number#History

There should always be a caveat when negative mass involved. There can be a temptation to conclude that things could end up falling' upwards' although, in an appropriate frame they could.

 

[A.T.]

There should always be a caveat when negative mass involved. There can be a temptation to conclude that things could end up falling' upwards'

And with a negative account balance there can be a temptation to conclude that one can withdraw cash with negative values on it, and burn it to get rid of debt.