Would a Body at the Center of the Earth Have Weight?

[stevepark]

A science mag article I read recently claimed that a body at the center of the Earth would have a very high weight. I don't see how a body at the center of the Earth would have any weight as any given mass has an equal opposite cancelling it out.
Explainations?

 

[Loren Booda]

stevepark,

Compressed by cumulative matter, it would have a high density (mass/length³), but as you point out, a symmetric cancellation in gravity making a near-zero weight~(mg_r)-(mg_r).

 

[stevepark]

gravity at center of large mass

Thanks for your response.

 

[way2go]

According to Newton's law F=GmM/r^2, a body of mass at a distance r from the center of the Earth (assumed that all mass of the Earth has been concentrated in the center) experiences a weight F. So, as r is very small near the center of the earth, the body will experience a very large weight force F.

 

[Adrian Baker]

Originally posted by way2go
According to Newton's law F=GmM/r^2, a body of mass at a distance r from the center of the Earth (assumed that all mass of the Earth has been concentrated in the center) experiences a weight F. So, as r is very small near the center of the earth, the body will experience a very large weight force F.

No it won't. Where is the mass you are attracted by? All around you if you are at the Earth's centre. You can only assume the Earth acts as a point mass IF you are on or above the surface.

Read Loren's reply above.

 

[way2go]

Originally posted by Adrian Baker
No it won't. Where is the mass you are attracted by? All around you if you are at the Earth's centre. You can only assume the Earth acts as a point mass IF you are on or above the surface.

Read Loren's reply above.

I think Loren is quite right, but for a body at the center of the Earth r will tend to 0. According F=GmM/r^2 F will tend to infinity no matter what value M has. That's my opinion how the article should be interpreted, although Loren is also quite right.

 

[Adrian Baker]

Originally posted by way2go
I think Loren is quite right, but for a body at the center of the Earth r will tend to 0. According F=GmM/r^2 F will tend to infinity no matter what value M has. That's my opinion how the article should be interpreted, although Loren is also quite right.

So you are both right huh? Your weight will be both zero and infinite!
If you insist on thinking that the Earth's mass only acts like a point mass on you, DESPITE, you being in the centre of it, then your 'logic' works.
You are though wrong.

 

[himanshu121]

No it won't. Where is the mass you are attracted by? All around you if you are at the Earth's centre. You can only assume the Earth acts as a point mass IF you are on or above the surface

I agree with Adrian

 

[way2go]

Originally posted by Adrian Baker
So you are both right huh? Your weight will be both zero and infinite!
If you insist on thinking that the Earth's mass only acts like a point mass on you, DESPITE, you being in the centre of it, then your 'logic' works.
You are though wrong.

Yeah, OK, I admit. That makes more sense.

 

[stevepark]

grtavity

If the Earth were two halves, M1 & M2, gravity at the center distance betweent them, (C=1/2D), would be 0.

M1----------C----------M2
But:
If the two halves were together, with virtually no D,

M1CM2

then the pressure of M1<-->M2 at the center would be the max the two bodies could produce, would it not?

 

[Arcon]

Originally posted by stevepark
A science mag article I read recently claimed that a body at the center of the Earth would have a very high weight. I don't see how a body at the center of the Earth would have any weight as any given mass has an equal opposite cancelling it out.
Explainations?

Weight is defined as the force required to support a body in a gravitational field so that it remains at rest. Suppose you hollowed out a spherical cavity centered at the center of the Earth. Then the gravitational field inside the cavity would not only vanish at the center but everywhere inside the cavity.

If the cavity is not centered at the center of the Earth then the field is uniform and the gravitational acceleration is proportional to the distance from the center of the Earth.

See - http://www.geocities.com/physics_world/gr/grav_cavity.htm

Arcon

 

[Doc Al]

Originally posted by way2go
According to Newton's law F=GmM/r^2, a body of mass at a distance r from the center of the Earth (assumed that all mass of the Earth has been concentrated in the center) experiences a weight F.

This formula will only give the weight of an object for distances (from the Earth's center) greater than the radius of the earth.

 

[GRQC]

The gravitational field inside a spherical body of mass M is
a linear function of the distance from the center,

$$F = \frac{GMmr}{R^3}$$

where R is the radius of the sphere and r is the distance of the object (mass m) from the center. As r goes to 0, the force will vanish.

What's interesting about this is: if you had a tunnel dug through the Earth (from one side to the other), an object dropped in the hole would eventually come back out (since the force is like that of a spring, a restorative force), and so the object would exhibit simple harmonic motion.