What would happen....

[Polymath257]

Now, let's go back to the vacuum tube. And we extend the tube to space. Low Earth Orbit altitude. You drop a rock down the tube at the same time another rock in a perfectly circular orbit goes zipping by. Also the Earth's density is constant all the way through. Which rock gets to the other side of the Earth first? The one in orbit and moving at about 4.76 miles per second or the rock that starts at zero but speeds up as it falls (and slows down again on the other side)?

Yep, an amusing problem.

 

[Subduction Zone]

Yep, an amusing problem.

But sadly no takers. Do you think that a link to the formula for the period of an elliptical orbit would help? I think that it would help:

Elliptic orbit - Wikipedia

 

[fantome profane]

If it was not a vacuum air resistance would slow it and the rock would never quite reach the center of the Earth.

When an object is falling towards the earth the air resistance does not slow the object, it decreases the rate of acceleration. That is very different from saying the object slows. The falling object, depending of the size and shape and the height it is dropped from, will eventually reach terminal velocity when it is no longer accelerating. But it is still falling. Zero acceleration does not mean zero velocity.

Now that is for an object falling to the earth, and object falling through the earth would be more complicated. I think gravity would be decreasing as the object travelled due to more and more mass being behind the object and less being in front of it. It would eventually reach a point of zero acceleration, and zero gravity, but still have a considerable velocity. I still think the momentum would carry it past the centre. At which point the velocity would start to slow. When the velocity reaches zero gravity would take control again, the object would switch directions and so on. Like a pendulum.

 

[Polymath257]

When an object is falling towards the earth the air resistance does not slow the object, it decreases the rate of acceleration. That is very different from saying the object slows. The falling object, depending of the size and shape and the height it is dropped from, will eventually reach terminal velocity when it is no longer accelerating. But it is still falling. Zero acceleration does not mean zero velocity.

Now that is for an object falling to the earth, and object falling through the earth would be more complicated. I think gravity would be decreasing as the object travelled due to more and more mass being behind the object and less being in front of it. It would eventually reach a point of zero acceleration, and zero gravity, but still have a considerable velocity. I still think the momentum would carry it past the centre. At which point the velocity would start to slow. When the velocity reaches zero gravity would take control again, the object would switch directions and so on. Like a pendulum.

A couple of quibbles:

1. The acceleration due to gravity would decrease as we approach the center. The force from the mass outside of a sphere (assuming spherical symmetry) cancels out, so only the mass inside the current sphere centered at the origin contributes. This is why the falling body would show simple harmonic motion in the absence of air resistance. The force due to gravity would be zero at the center.

2. The gravity does not 'take control again' when the velocity reaches zero. The force from gravity is actually increasing after the object goes past the center, which does slow the object.

3. Air resistance would be an additional force that serves to slow the object from one period to the next. Eventually the object would settle at the center if there is air resistance.

 

[Subduction Zone]

When an object is falling towards the earth the air resistance does not slow the object, it decreases the rate of acceleration. That is very different from saying the object slows. The falling object, depending of the size and shape and the height it is dropped from, will eventually reach terminal velocity when it is no longer accelerating. But it is still falling. Zero acceleration does not mean zero velocity.

Now that is for an object falling to the earth, and object falling through the earth would be more complicated. I think gravity would be decreasing as the object travelled due to more and more mass being behind the object and less being in front of it. It would eventually reach a point of zero acceleration, and zero gravity, but still have a considerable velocity. I still think the momentum would carry it past the centre. At which point the velocity would start to slow. When the velocity reaches zero gravity would take control again, the object would switch directions and so on. Like a pendulum.

I disagree. Yes, if an object starts at zero, for short falls air resistance will lower the rate of acceleration so that an object will slow the rate of acceleration. The acceleration will keep dropping until it hits terminal velocity. But if an object starts off with a speed higher than terminal velocity the air will slow it. Another factor is that terminal velocity decreases as pressure increases. There is a real life example of that. Felix Baumgartner is the man that did a freefall "from space". You can read more about him here, and in the embedded video you can see the velocity and g forces of the jump.


Felix Baumgartner's Jump From The Edge Of Space

He fell through the thin air and surpassed Mach 1. As he continued to fall the air resistance increased until it hit one g. It only went slightly above that as he continually slowed. Eventually he opened his chute and you see a spike. And then there is a second spike when he lands.

There are two factors that would make it so that someone falling through the Earth would never reach the core. An ever increasing density of air, which would mean that the terminal velocity would keep decreasing. And second and far more important the force from gravity pulling him to the center would keep decreasing.


Before I forget, terminal velocity is tied to the force of gravity. Think in about it.

 

[Subduction Zone]

A couple of quibbles:

1. The acceleration due to gravity would decrease as we approach the center. The force from the mass outside of a sphere (assuming spherical symmetry) cancels out, so only the mass inside the current sphere centered at the origin contributes. This is why the falling body would show simple harmonic motion in the absence of air resistance. The force due to gravity would be zero at the center.

2. The gravity does not 'take control again' when the velocity reaches zero. The force from gravity is actually increasing after the object goes past the center, which does slow the object.

3. Air resistance would be an additional force that serves to slow the object from one period to the next. Eventually the object would settle at the center if there is air resistance.

I would say that his claim that terminal velocity does not slow someone is false. NASA relies on that fact during reentry. Also gravity is a factor in terminal velocity. Terminal velocity occurs when the force from gravity equals the force from air resistance. A falling object, falling into an ever denser air that may be as dense as soup, under ever decreasing force from gravity would never make it quite to the center.

 

[fantome profane]

A couple of quibbles:

1. The acceleration due to gravity would decrease as we approach the center. The force from the mass outside of a sphere (assuming spherical symmetry) cancels out, so only the mass inside the current sphere centered at the origin contributes. This is why the falling body would show simple harmonic motion in the absence of air resistance. The force due to gravity would be zero at the center.

2. The gravity does not 'take control again' when the velocity reaches zero. The force from gravity is actually increasing after the object goes past the center, which does slow the object.

3. Air resistance would be an additional force that serves to slow the object from one period to the next. Eventually the object would settle at the center if there is air resistance.

I would accept all these quibbles, but you said it again. In point #3 you said air resistance slows the object. It doesn’t, it decreases the acceleration. Unless the object changes shape and size (like deploying a parachute) air resistance does not slow a falling object.

But that being understood I accept your quibbles. But with all that I think my pendulum model still holds, and yes it would eventually settle at the centre just as a pendulum will eventually settle at the lowest point,

What I am not sure of is how much of a pendulum effect we will have. I don’t know who far the object will travel past the centre, if it will travel almost all the way back to the surface and repeat several times before settling at the centre, or if it will travel just barely past the centre before returning and stopping there.


(Remember also, as we are trying to grabbed with decreasing gravity, that the force if air resistance is also dependent on velocity)

 

[fantome profane]

I would say that his claim that terminal velocity does not slow someone is false. NASA relies on that fact during reentry. Also gravity is a factor in terminal velocity. Terminal velocity occurs when the force from gravity equals the force from air resistance. A falling object, falling into an ever denser air that may be as dense as soup, under ever decreasing force from gravity would never make it quite to the center.

I am assuming that the object does deploy a parachute.

 

[Subduction Zone]

I am assuming that the object does deploy a parachute.

No. Air resistance will slow a body. If an object is falling faster than terminal velocity it will slow. Baumgartner's speed decreased when he hit denser air.

Baumgartner did not use a drogue chute. His descent shows the g force building until it was slightly higher than one g. Then there is a large spike in g force when his chute opens. And then another when he hits the ground..

 

[Polymath257]

I would accept all these quibbles, but you said it again. In point #3 you said air resistance slows the object. It doesn’t, it decreases the acceleration. Unless the object changes shape and size (like deploying a parachute) air resistance does not slow a falling object.

Read what I wrote. Without air resistance, each cycle would be the same with the same velocities. With air resistance, the velocities would be smaller because of smaller acceleration. The object will be going slower than it would without the air resistance.

But that being understood I accept your quibbles. But with all that I think my pendulum model still holds, and yes it would eventually settle at the centre just as a pendulum will eventually settle at the lowest point,

What I am not sure of is how much of a pendulum effect we will have. I don’t know who far the object will travel past the centre, if it will travel almost all the way back to the surface and repeat several times before settling at the centre, or if it will travel just barely past the centre before returning and stopping there.

In the absence of air resistance, it would travel all the way back to the surface and repeat. That much is clear because of energy conservation. It would, in fact, exhibit simple harmonic motion (assuming a uniform sphere for the Earth--otherwise something close to simple harmonic motion), which a pendulum only approximates.

With air resistance, the specifics would depend on the density of the air. But the resistance would manifest as an energy loss, so the object would NOT rise to the same level as before.

(Remember also, as we are trying to grabbed with decreasing gravity, that the force if air resistance is also dependent on velocity)

Understood. The decreased gravity is easy enough to work with and is why there is simple harmonic motion in the absence of air resistance.

Air resistance is usually proportional to the square of velocity, which makes the equations non-linear and thereby more difficult to solve, but the overall effect is clear enough.

 

[fantome profane]

No. Air resistance will slow a body. If an object is falling faster than terminal velocity it will slow. Baumgartner's speed decreased when he hit denser air.

Baumgartner did not use a drogue chute. His descent shows the g force building until it was slightly higher than one g. Then there is a large spike in g force when his chute opens. And then another when he hits the ground..

Yes, I agree with that. For example, if someone on top of a very high building fires an gun straight down the bullet will slow. That bullet will reach the same terminal velocity that it would if he simply dropped a bullet. At that point the physics is the same.

You are correct I was wrong. But my pendulum model still works, terminal velocity is the same regardless of if it is a dropped object that accelerates till it reaches terminal velocity, or a fired object that decelerates till it reaches terminal velocity.

 

[Subduction Zone]

Read what I wrote. Without air resistance, each cycle would be the same with the same velocities. With air resistance, the velocities would be smaller because of smaller acceleration. The object will be going slower than it would without the air resistance.



In the absence of air resistance, it would travel all the way back to the surface and repeat. That much is clear because of energy conservation. It would, in fact, exhibit simple harmonic motion (assuming a uniform sphere for the Earth--otherwise something close to simple harmonic motion), which a pendulum only approximates.

With air resistance, the specifics would depend on the density of the air. But the resistance would manifest as an energy loss, so the object would NOT rise to the same level as before.



Understood. The decreased gravity is easy enough to work with and is why there is simple harmonic motion in the absence of air resistance.

Air resistance is usually proportional to the square of velocity, which makes the equations non-linear and thereby more difficult to solve, but the overall effect is clear enough.

I contend that even if air pressure was magically kept to one atmosphere it would never quite reach the center.

 

[Subduction Zone]

Yes, I agree with that. For example, if someone on top of a very high building fires an gun straight down the bullet will slow. That bullet will reach the same terminal velocity that it would if he simply dropped a bullet. At that point the physics is the same.

You are correct I was wrong. But my pendulum model still works, terminal velocity is the same regardless of if it is a dropped object that accelerates till it reaches terminal velocity, or a fired object that decelerates till it reaches terminal velocity.

I agree. But what causes the terminal velocity that we see? It is gravity. Terminal velocity occurs when mass times the acceleration due to gravity is equal to the force from air resistance. Drop mg by decreasing g and the terminal velocity drops since the object is not being pulled down as hard. If everything at the surface of the Earth was the same, except that gravity was suddenly half what it is now the terminal velocity would drop too.An object would not be pulled down as strongly so a weaker force would be needed to balance a falling object.

 

[fantome profane]

I agree. But what causes the terminal velocity that we see? It is gravity. Terminal velocity occurs when mass times the acceleration due to gravity is equal to the force from air resistance. Drop mg by decreasing g and the terminal velocity drops since the object is not being pulled down as hard. If everything at the surface of the Earth was the same, except that gravity was suddenly half what it is now the terminal velocity would drop too.An object would not be pulled down as strongly so a weaker force would be needed to balance a falling object.

What causes terminal velocity is the balancing of forces, it is when the downward force of gravity is matched by the upward force of air resistance. When the forces acting on a moving object are balanced, that object will continue moving in a straight line at a constant speed.

As you correctly pointed out, if an object is fired downward, the force provided by air resistance would be greater than the force of gravity and the object would slow. But it the object slows the force of air resistance will decrease until the point where gravity and air resistance will be balanced. If the object is dropped then then the force of gravity is greater than the force of air resistance, the object will accelerate, the force of air resistance will increase until it balances the gravity.

Ok, now for an object falling through the earth.

If the object is dropped through the hole, the force of gravity is greater than the air resistance. The object accelerates. Eventually the gravity will decrease, the object will slow, but the air resistance will decrease as well. At no point does the air resistance overcome the momentum of the object. Gravity will continue to move the object downward (towards the centre). At the point the object reaches the centre, gravity is zero, but the object is still moving, it still has momentum. How fast the object is moving would depend on the shape and density of the object, but it would still have momentum. Because of that momentum it will travel past the centre point. At this point air resistance and gravity are acting in the same direction and the object slows, but it has passed the centre point.

The pendulum swings.

 

[Subduction Zone]

What causes terminal velocity is the balancing of forces, it is when the downward force of gravity is matched by the upward force of air resistance. When the forces acting on a moving object are balanced, that object will continue moving in a straight line at a constant speed.

As you correctly pointed out, if an object is fired downward, the force provided by air resistance would be greater than the force of gravity and the object would slow. But it the object slows the force of air resistance will decrease until the point where gravity and air resistance will be balanced. If the object is dropped then then the force of gravity is greater than the force of air resistance, the object will accelerate, the force of air resistance will increase until it balances the gravity.

Ok, now for an object falling through the earth.

If the object is dropped through the hole, the force of gravity is greater than the air resistance. The object accelerates. Eventually the gravity will decrease, the object will slow, but the air resistance will decrease as well. At no point does the air resistance overcome the momentum of the object. Gravity will continue to move the object downward (towards the centre). At the point the object reaches the centre, gravity is zero, but the object is still moving, it still has momentum. How fast the object is moving would depend on the shape and density of the object, but it would still have momentum. Because of that momentum it will travel past the centre point. At this point air resistance and gravity are acting in the same direction and the object slows, but it has passed the centre point.

The pendulum swings.

Not in this case. The object will start with some velocity, yes, but the force against it will always be slightly larger than the air resistance at terminal velocity. That is why Baumgartner's example is very educational. I am just dealing with the part of his fall before he opened his chute. He built up an enormous head of speed until the atmosphere thickened enough to meet it. From then on he was essentially right at terminal velocity. The same would happen in a jump into a hole into the Earth. At first one would drop and build up velocity. But soon one would be at terminal velocity. That velocity would decrease for two reasons. First a lessening of the force of gravity (well not if we get real picky. In the real world if such a tube were possible the force of gravity would increase for the first half of the journey) But we are using the simplified model where it decreases. It would also decrease because the air density would be continually increasing no matter which model that one used. At the center of the Earth we would be deep into supercrtical fluid territory. But the important thing is that the object will slow as the terminal velocity decreases because there would be a lower gravitational force on it. At the core the terminal velocity is zero. The object may still have a tiny fraction of its velocity, but in an undisturbed container it might not even approach it in our lifetime. Its velocity would be less than that of a snail before then.

 

[beenherebeforeagain]

Hey, wouldn't such a hole through the middle of the Earth suck most or all of the atmosphere into it? Killing all life on the planet?

 

[Subduction Zone]

Hey, wouldn't such a hole through the middle of the Earth suck most or all of the atmosphere into it? Killing all life on the planet?

How big is the hole? In other words what is it's diameter?

 

[beenherebeforeagain]

How big is the hole? In other words what is it's diameter?

I dunno...let's start with 100 feet, then try 1,000...

Once upon a time I was able to quickly look up the relevant factors and make a calculation of my own...not so much these days, unfortunately...

 

[Subduction Zone]

I dunno...let's start with 100 feet, then try 1,000...

Once upon a time I was able to quickly look up the relevant factors and make a calculation of my own...not so much these days, unfortunately...

Yeah, my calculus is a bit rusty too. And this is a rather difficult problem. It would deal with some very high pressures and I do not know if the Ideal Gas Law would still hold. As to size I was thinking of one meter tops. I don't now how strong you are but the word "dropped" was used by the OP so I was thinking of something the size of a hammer at most.

 

[beenherebeforeagain]

Yeah, my calculus is a bit rusty too. And this is a rather difficult problem. It would deal with some very high pressures and I do not know if the Ideal Gas Law would still hold. As to size I was thinking of one meter tops. I don't now how strong you are but the word "dropped" was used by the OP so I was thinking of something the size of a hammer at most.

Oh.

I was looking for destruction and mayhem!