At what height above the earth must an artificial satellite orbit if it is to remain at the same place relative to the earth?

The anglular velocity of the satellite must be same as the angular velocity of the spin of the earth. Let the mass of the satellite be m and the mass of the earth be M and the angular velocity of earth be w (w = 2*pi/24 hours). The force balance eqn

mrw^2 = GMm/r^2


Solving you get r = (GM/w^2)^(1/3)