Question: When a body is near the surface of the earth, we usually assume that the force due to gravity on it is a constant mg, where m is the mass of the body and g is the acceleration due to gravity at sea level. For a body at a distance h above the surface of the earth, a more accurate expression for the force F is
F = (mgR2)/(R + h)2
where R is the radius of the earth. We will consider the situation in which the body is close to the surface of the earth so that h is much smaller than R.
(a) Show that F ≈ mg.
(b) Express F as mg multiplied by a series in h/R.
(c) The first-order correction to the approximation F ≈ mg is obtained by taking the linear term in the series but no higher terms. How far above the surface of the earth can you go before the first-order correction changes the estimate F ≈ mg by more than 10%? (Assume R = 6400 km.)