Gravity at centre of earth .?

1. Aug 19, 2011

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?

2. Aug 19, 2011

Staff: Mentor

Re: gravity at centre of earth.....?

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?

3. Aug 19, 2011

khurram usman

Re: gravity at centre of earth.....?

assume a perfectly symmetrical hypothetical earth

4. Aug 19, 2011

harrylin

Re: gravity at centre of earth.....?

Best,
Harald

5. Aug 19, 2011

Staff: Mentor

Re: gravity at centre of 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.

6. Aug 19, 2011

Naty1

Re: gravity at centre of earth.....?

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.

7. Aug 19, 2011

khurram usman

Re: gravity at centre of earth.....?

ok...i see now how g varies
as the body falls g decreases till the centre....so i dont suppose the body will acelerate during this fall?

8. Aug 19, 2011

khurram usman

Re: gravity at centre of earth.....?

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?

9. Aug 19, 2011

Staff: Mentor

Re: gravity at centre of earth.....?

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

10. Aug 19, 2011

Superstring

Re: gravity at centre of earth.....?

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.

11. Aug 19, 2011

sophiecentaur

Re: gravity at centre of earth.....?

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.

12. Aug 19, 2011

Lsos

Re: gravity at centre of earth.....?

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...

13. Aug 19, 2011

cjl

Re: gravity at centre of earth.....?

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 Rorbit ~ Rearth).

14. Aug 19, 2011

A.T.

Re: gravity at centre of 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...