At what depth below Earth's surface does a body's weight become half of its weight on the surface? (Treat Earth as a uniform sphere of radius R.)
Inside a uniform Earth, only the mass closer to the centre than you pulls on you, and that mass grows as the cube of your distance from the centre. The result is that g falls in a straight line with depth.
So the weight halves at a depth of R/2, about 3,200 km down. Going up, it halves at only 0.41R, because outside the Earth gravity falls as the inverse square, faster than the straight-line fall inside.
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