You also have to take into account that as you penetrate to the centre of the earth, the effective mass of the earth changes.
This is because you are attracted to the mass above you and around you as well as that below you.
You can demonstrate this with your karting lead.
Suspend two lumps of lead with a pith ball or expanded styrene between them. Move one of the lumps away from the ball and the ball will move towards the nearer lump. Move the other lump back and the ball will move back again. It doesn't change if you temporarily connect the two lumps together With a little lead rod, one block doesn't suddenly get all the gravity.
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