Silly physics question from ds

[elegantlion]

We're reading Simply Einstein and ds posed this question today. Where does gravity end in the Earth?

 

If you dug a hole through the center of the earth and installed an elevator shaft starting in the air where would gravity cease? Say you started way up in the sky in the elevator and only allowed the force of gravity to pull you down how far into the hole would go?

 

Gravity doesn't cease in the cave? Does it extend to the Earth's core?

 

What would happen? Ds is curious, I have no clue and told him I'd ask y'all.

[MyThreeSons]

Here's my understanding:

 

The force due to gravity acting on you -- otherwise known as your weight -- would increase as you drew closer and closer to the center of the earth. And in the absence of any external force, you would be drawn to the center of the shaft and then stay there.

 

Waiting to hear others' thoughts.

[regentrude]

This is an extremely interesting question and not silly at all!

The force of gravity between two point masses has, according to Newton's law of universal gravitation, the form GmM/r^2 where G is a universal constant, m and M are the two masses, and r the distance between them.

A uniform spherical object acts on the outside as if all the mass was concentrated in the center, so we can use this law to describe the interaction between roughly spherical objects such as moon and Earth, for example. This works because of the special 1/r^2 dependence which causes material of a spherical shell to not exert a net gravitational force on anything inside (this is called Newton's Shell Theorem and you need calculus to prove it).

 

Now lets look inside the Earth and assume an object of mass m is some distance r from the center of the Earth, but r<R_Earth. Because of Newton's Shell theorem, all the mass of the Earth that sits outside a radius r does not exert a net gravitational force on the object, only the material inside that radius. The mass of this material is only a fraction of the mass of the Earth, namely M= M_Earth*(r/R_E)^3.

This small sphere of radius r and mass M now acts as a uniform sphere with the force of gravity being equal to GmM/r^2. If you put in what the mass M is at that radius, you find that the force of gravity inside the Earth behaves as proportional to r! So, inside the Earth, the force is like a giant spring that wants to pull the object towards the center of the Earth.

 

If you dig a hole through the Earth, and threw a ball into it, it would feel pulled towards the center of the Earth by gravity, would speed up, have its highest speed at the center, overshoot the center and move towards the other end of the hole, becoming slower (because gravity is now pulling it backwards towards the center). It will stop on the other surface, turn around and go back. It will continue to oscillate, such as a mass that you hang at the end of a spring and let oscillate up and down. The force at the center will be zero, but the object will have its highest speed and continue to move.

Edited October 28, 2011 by regentrude

[elegantlion]

Thank you. I shall forward this to ds, with a warning not to attempt such an experiment. Yes, in our previous house he has been known to dig holes in the back yard simply because he could.

[elegantlion]

He was delighted to read the answers. Thank you again.

[Julieofsardis]

My daddy has a theory that Regentrude's description is the bottomless pit from Bible fame.

[MyThreeSons]

Yes, in our previous house he has been known to dig holes in the back yard simply because he could.

 

I remember my Mom visiting our house once and asking me why we had a bunch of holes dug in the back yard. My response was simple: "Because we have three sons!"

[Teachin'Mine]

This is an extremely interesting question and not silly at all!

The force of gravity between two point masses has, according to Newton's law of universal gravitation, the form GmM/r^2 where G is a universal constant, m and M are the two masses, and r the distance between them.

A uniform spherical object acts on the outside as if all the mass was concentrated in the center, so we can use this law to describe the interaction between roughly spherical objects such as moon and Earth, for example. This works because of the special 1/r^2 dependence which causes material of a spherical shell to not exert a net gravitational force on anything inside (this is called Newton's Shell Theorem and you need calculus to prove it).

 

Now lets look inside the Earth and assume an object of mass m is some distance r from the center of the Earth, but r<R_Earth. Because of Newton's Shell theorem, all the mass of the Earth that sits outside a radius r does not exert a net gravitational force on the object, only the material inside that radius. The mass of this material is only a fraction of the mass of the Earth, namely M= M_Earth*(r/R_E)^3.

This small sphere of radius r and mass M now acts as a uniform sphere with the force of gravity being equal to GmM/r^2. If you put in what the mass M is at that radius, you find that the force of gravity inside the Earth behaves as proportional to r! So, inside the Earth, the force is like a giant spring that wants to pull the object towards the center of the Earth.

 

If you dig a hole through the Earth, and threw a ball into it, it would feel pulled towards the center of the Earth by gravity, would speed up, have its highest speed at the center, overshoot the center and move towards the other end of the hole, becoming slower (because gravity is now pulling it backwards towards the center). It will stop on the other surface, turn around and go back. It will continue to oscillate, such as a mass that you hang at the end of a spring and let oscillate up and down. The force at the center will be zero, but the object will have its highest speed and continue to move.

 

Fascinating!!! What a great question!!!

[Another Lynn]

This is an extremely interesting question and not silly at all!

The force of gravity between two point masses has, according to Newton's law of universal gravitation, the form GmM/r^2 where G is a universal constant, m and M are the two masses, and r the distance between them.

A uniform spherical object acts on the outside as if all the mass was concentrated in the center, so we can use this law to describe the interaction between roughly spherical objects such as moon and Earth, for example. This works because of the special 1/r^2 dependence which causes material of a spherical shell to not exert a net gravitational force on anything inside (this is called Newton's Shell Theorem and you need calculus to prove it).

 

Now lets look inside the Earth and assume an object of mass m is some distance r from the center of the Earth, but r<R_Earth. Because of Newton's Shell theorem, all the mass of the Earth that sits outside a radius r does not exert a net gravitational force on the object, only the material inside that radius. The mass of this material is only a fraction of the mass of the Earth, namely M= M_Earth*(r/R_E)^3.

This small sphere of radius r and mass M now acts as a uniform sphere with the force of gravity being equal to GmM/r^2. If you put in what the mass M is at that radius, you find that the force of gravity inside the Earth behaves as proportional to r! So, inside the Earth, the force is like a giant spring that wants to pull the object towards the center of the Earth.

 

If you dig a hole through the Earth, and threw a ball into it, it would feel pulled towards the center of the Earth by gravity, would speed up, have its highest speed at the center, overshoot the center and move towards the other end of the hole, becoming slower (because gravity is now pulling it backwards towards the center). It will stop on the other surface, turn around and go back. It will continue to oscillate, such as a mass that you hang at the end of a spring and let oscillate up and down. The force at the center will be zero, but the object will have its highest speed and continue to move.

 

Will the ball eventually stop at the core? We recently checked out a DVD from the library about gravity that depicted a hypothetical (cartoon) tunnel through the Earth's core, and an object oscillating back and forth through it until it eventually stopped at the core - is this correct?

[8filltheheart]

Paula,

 

It is an awesome question! Regentrude, thank you for answering it. I'm going to have ds read it b/c I know he will enjoy it!

 

To answer a question you asked on the NASA thread that I didn't respond to (I am lucky to have 2 mins to myself these days!!), one of the many things it has is a discussion forum where these kinds of questions can be asked. This is the type of question my 15 yos proposes multiple times/day except his are in terms of particle physics or astronomical phenomena.......he fries my brains and most of the time I don't even understand the question, let alone having answers. I'm glad he has a place to hang out where STEM is everyone's passion.

 

If your ds would like to ask my ds any questions about INSPIRE, he can send ds a PM via my acct w/his email address and ds would be happy to email him back. We are going to be crazy busy today and tomorrow, but he would probably be able to answer him tomorrow night.

[regentrude]

Will the ball eventually stop at the core? We recently checked out a DVD from the library about gravity that depicted a hypothetical (cartoon) tunnel through the Earth's core, and an object oscillating back and forth through it until it eventually stopped at the core - is this correct?

 

If we take air resistance into account, then yes, the oscillation will get smaller and smaller and eventually die down, just as a pendulum eventually comes to rest as well. The object will end up sitting at the center of the Earth.