Problem 1
For a point on the surface of the Moon determine the ratio of the acceleration of gravity due to the mass of the Earth to the acceleration of gravity due to the mass of the Moon.
Problem 2
Determine the ratio of the centrifugal acceleration to the gravitational acceleration at the Earth's equator.
Problem 4
Assume a large geoid anomaly with a horizontal scale of several thousand kilometers has a mantle origin and its location does not change. Because of continental drift the passive margin of a continent passes through the anomaly. Is there a significant change in sea level associated with the passage of the margin through the geoid anomaly? Explain your answer.
Problem 8
Consider a spherical body of radius \(a\) with a core of radius \(r_{c}\) and constant density \(\rho_{c}\) surrounded by a mantle of constant density \(\rho_{m}\). Show that the moment of inertia \(C\) and mass \(M\) are given by $$\begin{aligned}C &=\frac{8 \pi}{15}\left[\rho_{c} r_{c}^{5}+\rho_{m}\left(a^{5}-r_{c}^{5}\right)\right] \\\M &=\frac{4 \pi}{3}\left[\rho_{c} r_{c}^{3}+\rho_{m}\left(a^{3}-r_{c}^{3}\right)\right] \end{aligned}$$ Determine mean values for the densities of the Earth's mantle and core given \(C=8.04 \times 10^{37}\) \(\mathrm{kg} \mathrm{m}^{2}, M=5.97 \times 10^{24} \mathrm{~kg}, a=6378 \mathrm{~km},\) and \(r_{c}=3486 \mathrm{~km}\)
Problem 9
Assuming that the difference in moments of inertia \(C-A\) is associated with a near surface density \(\rho_{m}\) and the mass \(M\) is associated with a mean planetary density \(\bar{\rho}\), show that $$J_{2}=\frac{2}{5} \frac{\rho_{m}}{\bar{\rho}} f$$ Determine the value of \(\rho_{m}\) for the Earth by using the measured values of \(J_{2}, \bar{\rho},\) and \(f .\) Discuss the value obtained.
Problem 18
A volcanic plug of diameter \(10 \mathrm{~km}\) has a gravity anomaly of \(0.3 \mathrm{~mm} \mathrm{~s}^{-2}\). Estimate the depth of the plug assuming that it can be modeled by a vertical cylinder whose top is at the surface. Assume that the plug has density of \(3000 \mathrm{~kg} \mathrm{~m}^{-3}\) and the rock it intrudes has a density of \(2800 \mathrm{~kg} \mathrm{~m}^{-3}\).
Problem 23
Consider the formation of a sedimentary basin on the seafloor. Suppose isostatic compensation is achieved by the displacement of mantle material of density \(\rho_{m}\). Show that sediment thickness \(s\) is related to water depth \(d\) by $$s=\frac{\left(\rho_{m}-\rho_{w}\right)}{\left(\rho_{m}-\rho_{s}\right)}(D-d)$$ where \(D\) is the initial depth of the sediment-free ocean. What is the maximum possible thickness of the sediment if \(\rho_{s}=2500 \mathrm{~kg} \mathrm{~m}^{-3}, \rho_{m}=\) \(3300 \mathrm{~kg} \mathrm{~m}^{-3},\) and \(D=5 \mathrm{~km} ?\)
