HOW SMALL WOULD THE EARTH HAVE TO BE FOR US TO FEEL IT SPINNING?

We can’t feel the rotation of the Earth directly, because we are rotating with it. But if the Earth somehow compressed itself without losing any mass, it would have to spin faster to conserve its angular momentum – like the pirouetting ice skater so beloved of physics teachers. This would increase the centrifugal force acting on us, and because this force acts radially outwards, it would partly cancel out the force of gravity, and our weight would decrease. Halving the diameter of the Earth would reduce our weight by around 1.2kg, which is probably not enough to notice. But centrifugal force doesn’t increase linearly, and on a quarter-sized Earth your weight would drop by 15kg in total, so you would feel lighter and could jump higher.

Whether this counts as feeling the rotation itself is debatable. On a merry-go-round, you feel the spinning because the radius of the ride is so small that the centrifugal force varies noticeably across the length of your body. For this to happen, our miniature Earth would have to be so small that your own height was a significant proportion of the planet’s radius. Long before we reached that point, though, the Earth would have disintegrated from its own centrifugal force.