Rikki Menzies, of Hastings, asks :-
Does one become lighter or heavier as one moves towards the centre of the earth? At what distance from the centre is one heaviest?
Malcolm Ingham, a geophysicist at Victoria University of Wellington, responded.
Your weight, or how heavy you feel, is actually a force, the force of gravity, with which the Earth pulls you downwards. This force of gravity depends on your mass and also on how far you are from the centre of the Earth.
Your mass is constant (sorry, 75 kg is 75 kg wherever you are), but only the part of the Earth that is closer to the centre than you are contributes to your weight force. As you drop down an imaginary drill-hole towards the centre of the Earth, the mass of the Earth between you and the centre gets less and less, until at the centre of Earth there is nothing left.
But there’s another factor too: your weight force also depends inversely (in fact on the inverse square) of your distance from the centre of the Earth. The amazing thing is that, as you travel through most of the Earth's mantle (the more-or-less solid part down to 2900 km depth, made up largely of a material called olivine) these two effects balance, and your weight hardly changes at all.
Only just as you approach the boundary between the mantle and Earth's dense iron-rich core would you experience an increase in weight. Even then your weight is only about 5% more than it was at Earth’s surface. However, once you are in the core, although as you move further inwards your distance from the centre decreases, the amount of Earth’s mass between you and the centre falls off even more rapidly and your weight decreases steadily with depth. Should you get to the centre of the Earth you will find yourself weightless!
Of course, as is well known to miners, temperature increases with depth into the Earth. At the centre of the Earth it is estimated to be about 5000 degrees Centigrade, so you would also be starting to feel a little warm!