Show that the centrifugal force -mo x (eo x r) at a colatitude (polar angle) can be written -m vo,, where 0,=-1|xr|=- sin 0 07540 (b) If, is the earth's exterior gravitational potential, prove that a thin surface layer of water (the "ocean") would assume the shape of an equipotential surface , + D, = const (known as the geoid). (c) Suppose that the earth has a small quadrupole deformation, with . Gr- |1 J(-) P (cos e)] P(cos $= - M.Gr- Here P(cos 0) [3 (cos e)- 1] and J 1.083 x 103 characterizes the earth's deviation from sphericity (see Prob. 5.7). Obtain the approximate relation AR R =+ = 1 wR 2 M.G open to only si zilT ha Mio 8 apos nadw noit where AR is the difference between the geoid's equatorial and polar radii. Compare the resulting value of AR/R, with (R.-R,)/R, 3.35 x 10 determined from the measured value of the earth's equator- ial and polar radii. 19030 of noies (d) Explain why - V(0, +) on the geoid is the local acceleration g due to gravity at sea level, and show that its magnitude is given approximately by g =a(0, +)/ar. Hence derive Clairaut's formula g=g.[1 + (cos e) MG (5 wR AR 2 R - Use the measured values g. 978.03 cm s-2 and g, *983.20 cm s-2 to estimate AR/R,, and compare with the previous value.