Question

(6 (You must show all your work). The force due to gravity on an object with mass m at a height h above the surface of the earth is _mg R F = (R+h) where is R the radius of the earth and g is the acceleration due to gravity. (a) (8 points). Express F as a series in powers of h/R. (b) (8 points). Observe that if we approximate F by the first term in the series, we get the expression Fe mg that is usually used when is much smaller than. Use the Alternating Series Estimation Theorem to estimate the range of values of h for which the approximation F mg is accurate to within one percent. (Use R = 6400 km.)

Answer 1

Consider the force due to gravity on an object with mass mat a height h above the surface of the earth is, FMGR (R+h)? mg = mg ( 1 ) =ms ( 2 ) By use the Binomial series. 0 (b) Fas a series in power of " then, F = mg (22) 69 = me 1–2A+CA- The above equation as an alternating series so by the Estimation theorem the error in the approximation F = mg is less than "8" and the accuracy within 1% then, 2mgh R <0.01 mgR || (R+h)) 2h(R+h) <0.01 R3 The above inequality should be difficult solve for h. So R = 6400 km 2h(6400 + h) <0.01 (6400) h<31.68548279 Therefore, the approximation is accurate to within 1% for h<31.68548279