(a) Assuming Earth can be a homogeneous sphere that has a radius rand a mass m,

Question

(a) Assuming Earth can be a homogeneous sphere that has a radius rand a mass m, show that the period T (time for one daily rotation)of Earth's rotation about its axis is related to its radius by T =br2, where b= (4/5) m/L. Here L is themagnitude of the spin angular momentum of Earth. (b) supposethat the radius r changes by a very small amount, r, due tosome internal cause, such as thermal expansion. Show that thefractional change in the period T is given approximately byT/T = 2r/r. (c) By how many kilometers would rneed to increase for the period to change by 0.25 d/y, (so thatleap years would not longer by necessary)?

Explanation / Answer

L = I = 2 M R2 / 5 T = 1 / f = 2 / T = 4 M R2 /(5 L) = b R2 dT = 2 b R dR T / T = 2 b R R / b R2 = 2 R/ R T = .25 d T = 365.25 d R = (T / T) * R / 2 = (.25 / 365.25) * 6370/ 2 = 2.2 km dT = 2 b R dR T / T = 2 b R R / b R2 = 2 R/ R T = .25 d T = 365.25 d R = (T / T) * R / 2 = (.25 / 365.25) * 6370/ 2 = 2.2 km

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