When a body is near the surface of the earth, we usuallyassume that the force due to

Chapter 10, Problem 51

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When a body is near the surface of the earth, we usuallyassume that the force due to gravity on it is a constantmg, where m is the mass of the body and g is the accelerationdue to gravity at sea level. For a body at a distanceh above the surface of the earth, a more accurate expressionfor the force F isF = mgR2(R + h)2where R is the radius of the earth. We will consider thesituation in which the body is close to the surface of theearth 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 seriesbut no higher terms. How far above the surfaceof the earth can you go before the first-order correctionchanges the estimate F mg by more than10%? (Assume R = 6400 km.)