Gravity - The Solid Earth - page 227

mantle of reasonably constant viscosity: the models imply an upper mantle with
viscosity 10 21 Pa s and a lower mantle with viscosity in the range 10 22 -10 23 Pa s.

Figure 5.22 shows a series of estimates of detailed mantle viscosity structure based
on sea-level and gravity data from Europe and Hudson Bay. It appears that the
viscosity of the mantle increases somewhat with depth and that a low-viscosity
asthenosphere is not a universal feature - beneath old continental regions the
viscosity of the upper mantle seems to be fairly high.

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Problems

1. Calculate the mass of the Earth.
2. Calculate the maximum gravitational attraction between Newton and the apple, assum-
ing that (a) it passed 30 cm from his centre of mass and (b) it hit him.
3. Calculate the weight of a 60-kg astronaut on the Moon. How much less is this than
the astronaut's weight on the Earth? (The Moon's mass is 7.4 × 10 22 kg and its radius
is 1738 km.)
4. (a) Calculate the radius of orbit of a geostationary satellite (a satellite whose orbit is
such that it remains above the same point on the Earth's surface).
(b) Calculate the period of a satellite orbiting 200 km above the Earth.
(c) Calculate the period of a satellite orbiting 200 km above the Moon.
5. What is the radius of the Moon's orbit about the Earth? Use the mass of the Earth
as calculated in Problem 1 and assume that the period of rotation is 28 days. Explain
whyitisnot necessary to know the mass of the Moon to perform the calculation.
6. Calculate the mass of the Sun, assuming the Earth's orbit to be circular with a radius
of 1.5 × 10 8 km.
7. A small moon of mass m and radius a orbits a planet of mass M ,keeping the same
face towards the planet. Show that, if the moon approaches the planet closer than
r c = a (3 M / m ) 1 / 3 , loose rocks lying on the surface of the moon will be lifted off.
(From Thompson (1987).)
8. What is the difference in length, in kilometres, between a degree of latitude at the
equator and at the pole?
9. Explain (qualitatively and quantitatively) how the source of the Mississippi River can
be about 5 km closer to the centre of the Earth than its mouth. (From Thompson
(1987).)
10. It is observed that the acceleration g due to gravity is greater down a mine than it
is at the Earth's surface. Show that this result can be explained if the Earth's density
increases sufficiently rapidly with depth. (From Thompson (1987).)
11. What is the difference between the value of the Earth's gravitational acceleration at
the equator and the poles? In what units would this difference usually be quoted?
12. (a) Verify the isostatic-equilibrium calculation for continents and ocean basins, using
densities of sea water, crust and mantle of 1.03 × 10 3 , 2.9 × 10 3 and 3.3 ×
10 3 kg m − 3 , respectively, an ocean-basin depth of 5 km and an oceanic crustal
thickness of 6.6 km.