Unveiling the Mysteries of Gravity at the Centre of the Earth

[khurram usman]

gravity at centre of earth...?

first of all as we move towards the Earth centre does the gravity and the acceleration due to it increase?
secondly what is the affect on weight as we move towards the centre?
is 'g' zero at centre?
if we dig a tunnel across the Earth passing through the centre from one pole toward the other and let a body freefall through it what will happen?will the body accelerate or not? will it stop at center or continue on towards other pole?

 

[Dale]

Do you want to assume a hypothetical Earth with perfect spherical symmetry and uniform density or do you want to discuss the actual Earth which is non spherical and non uniform density?

 

[khurram usman]

Do you want to assume a hypothetical Earth with perfect spherical symmetry and uniform density or do you want to discuss the actual Earth which is non spherical and non uniform density?

assume a perfectly symmetrical hypothetical earth

 

[harrylin]

first of all as we move towards the Earth centre does the gravity and the acceleration due to it increase?
secondly what is the affect on weight as we move towards the centre?
is 'g' zero at centre?
if we dig a tunnel across the Earth passing through the centre from one pole toward the other and let a body freefall through it what will happen?will the body accelerate or not? will it stop at center or continue on towards other pole?

https://www.physicsforums.com/showthread.php?t=207148


Harald

 

[Dale]

assume a perfectly symmetrical hypothetical earth

Then g will decrease as you go down and it will be 0 at the center. If a body free-falls in a tunnel then when it reaches the center it will have some velocity and no force acting on it (assuming no air resistance) therefore it will continue towards the other pole.

 

[Naty1]

first of all as we move towards the Earth centre does the gravity and the acceleration due to it increase?

As Dalespam notes, g decreases. It's not difficult to visualize since as you proceed closer to the center there is less gravitational mass pulling you to the center...because as you descend increasing amounts of mass are attracting you in other directions. At the very center you are being pulled equalled in all directions and so the forces all cancel.

However, gravitational potential is not zero at the center, so time IS dilated...it passes more slowly than on the surface.

 

[khurram usman]

Then g will decrease as you go down and it will be 0 at the center. If a body free-falls in a tunnel then when it reaches the center it will have some velocity and no force acting on it (assuming no air resistance) therefore it will continue towards the other pole.

ok...i see now how g varies
as the body falls g decreases till the centre...so i don't suppose the body will acelerate during this fall?

 

[khurram usman]

As Dalespam notes, g decreases. It's not difficult to visualize since as you proceed closer to the center there is less gravitational mass pulling you to the center...because as you descend increasing amounts of mass are attracting you in other directions. At the very center you are being pulled equalled in all directions and so the forces all cancel.

However, gravitational potential is not zero at the center, so time IS dilated...it passes more slowly than on the surface.

by gravitational potential you mean that work will need to be done in order to make the body move ...right?
and how is time dilated?

 

[Dale]

ok...i see now how g varies
as the body falls g decreases till the centre...so i don't suppose the body will acelerate during this fall?

The body will accelerate as long as g is non-zero (neglecting air resistance), which is everywhere except at the very center point.

 

[Superstring]

Inside the Earth (assuming it is a sphere of uniform density):

$$g = g_0 \frac{r}{R}$$

where:

$g_0 = -GM / R^2$ which is the acceleration due to gravity at the surface of the Earth ($M$ is Earth's mass)
$r$ is the distance from the center of the earth
$R$ is the radius of the earth.

 

[sophiecentaur]

The body will accelerate as long as g is non-zero (neglecting air resistance), which is everywhere except at the very center point.

There is less and less acceleration as you go down but there is always some, until you are at the centre - then you are at your maximum speed and the force is then against your motion. The situation is actually the same as for a mass on a spring. The restoring force to the centre of the Earth is proportional to the distance away from the centre in exactly the same way as the restoring force towards the equilibrium position for a mass on a spring. In both cases, you get what is called simple harmonic motion. Interestingly, the period of oscillation is exactly the same for large or small amplitudes of oscillation and, in the case of the hole through the Earth, the time is the same as the time for a satellite in low Earth orbit to go round the Earth once.

 

[Lsos]

Also interestingly, any straight tunnel that goes through the Earth (but not necessarily through the center) will also have the same period of oscillation. So, if you dug a tunnel straight to China and jumped in, you will come out in China at the same time (about 42 minutes...42 minutes and 12 seconds if Earth was a perfect sphere) no matter where you dug that tunnel from.

Assuming perfect conditions, of course...

 

[cjl]

Yep. Interestingly enough, that's also the time it would take to go halfway around the Earth in an orbital trajectory at very low altitude (such that R_orbit ~ R_earth).

 

[A.T.]

Yep. Interestingly enough, that's also the time it would take to go halfway around the Earth in an orbital trajectory at very low altitude (such that R_orbit ~ R_earth).

If an astronaut wants to commit suicide on the Moon he could shoot his railgun horizontally away from himself and just wait a while...