Gravitational force at the Earth's centre

[Grahame1]

One of my students asked if the graviational force at the Earth's centre is zero. The question was thrown open to the other students and two possible answers were given: zero or infinite! One way of reasoning is to say that the graviational force sums to zero as (assuming uniformity of the Earth) for every force pulling one way is 'cancelled' by and equal and opposite force. But I then imagined modelling the pull of gravity by tension in a rope. Suppose you hold two ropes, one in each hand and the ropes pull in opposite directions with the same magnitude of force. It is true that I wouldn't accelerate as the forces 'cancel' (note the quotation marks), but I certainly wouldn't feel zero force! So the question is: would an observer at Earth's centre feel zero gravity, or would that observer feel the gravitational pull outwards (in all dirctions) towards the surrounding mass?

 

[Ibix]

Zero gravity. Every particle of your body is pulled in both directions (every direction, in fact) and the forces cancel on every particle, so you feel nothing. That's not the same as having a rope tied to each arm.

The "shell theorem" is the thing to look up.

 

[jbriggs444]

Google spherical shell theorem.

A flip answer is that you cannot "feel" gravity at all, as long as it is uniform. It pulls on all parts of your body equally. What you "feel" instead is the contact force of your feet on the floor or your butt in your chair and the various internal compression and tension forces that keep you standing upright on your feet or keep you from slumping out of your chair.

Edit to add: An astronaut in a spacecraft orbiting in low Earth orbit about 200 miles up is subject to a force of gravity that is about 90% as strong as on the Earth's surface. The astronaut feels weightless because nothing is pushing on him, not because there is no gravity.

 

[Lance219]

0. Suppose you have non zero gravity, and at moment 0 you are at rest, by Newton's second law, you will have a non-zero acceleration, but which way? The globe is symmetrical, no direction is more privileged than others. Hence the gravity by Earth at Earth center should be 0.

 

[Ryan Kayan]

Though the Earth's gravity would no longer be acting on you (since presumably you would already be at it's center of mass), you would still be subject to the gravitation force of other matter and energy in our solar system and beyond, given sufficient time for the space-time curvature (i.e., gravity) to reach you.