Problem 1
An average thickness of the oceanic crust is \(6 \mathrm{~km}\). Its density is \(2900 \mathrm{~kg} \mathrm{~m}^{-3}\). This is overlain by \(5 \mathrm{~km}\) of water \(\left(\rho_{w}=1000 \mathrm{~kg} \mathrm{~m}^{-3}\right)\) in a typical ocean basin. Determine the normal force per unit area on a horizontal plane at the base of the oceanic crust due to the weight of the crust and the overlying water.
Problem 2
A mountain range has an elevation of \(5 \mathrm{~km}\). Assuming that \(\rho_{m}=3300 \mathrm{~kg} \mathrm{~m}^{-3}, \rho_{c}=2800 \mathrm{~kg} \mathrm{~m}^{-3}\), and that the reference or normal continental crust has a thickness of \(35 \mathrm{~km}\), determine the thickness of the continental crust beneath the mountain range. Assume that hydrostatic equilibrium is applicable. A MATLAB code for solving this problem is given in Appendix \(D\).
Problem 3
There is observational evidence from the continents that the sea level in the Cretaceous was \(200 \mathrm{~m}\) higher than today. After a few thousand years, however, the seawater is in isostatic equilibrium with the ocean basins. What was the corresponding increase in the depth of the ocean basins? Take \(\rho_{w}=1000 \mathrm{~kg} \mathrm{~m}^{-3}\) and the density of the displaced mantle to be \(\rho_{m}=\) \(3300 \mathrm{~kg} \mathrm{~m}^{-3}\)
Problem 4
A sedimentary basin has a thickness of \(4 \mathrm{~km}\). Assuming that the crustal stretching model is applicable and that \(h_{c c}=35 \mathrm{~km}, \rho_{m}=3300 \mathrm{~kg} \mathrm{~m}^{-3}\), \(\rho_{c c}=2750 \mathrm{~kg} \mathrm{~m}^{-3},\) and \(\rho_{s}=2550 \mathrm{~kg} \mathrm{~m}^{-3},\) determine the stretching factor. A MATLAB code for solving this problem is given in Appendix \(D\).
Problem 5
A sedimentary basin has a thickness of \(7 \mathrm{~km}\). Assuming that the crustal stretching model is applicable and that \(h_{c c}=35 \mathrm{~km}, \rho_{m}=3300 \mathrm{~kg} \mathrm{~m}^{-3}\), \(\rho_{c c}=2700 \mathrm{~kg} \mathrm{~m}^{-3},\) and \(\rho_{s}=2450 \mathrm{~kg} \mathrm{~m}^{3},\) determine the stretching factor. A MATLAB code for solving this problem is given in Appendix \(D\).
Problem 18
Consider a simple two-layer model of a planet consisting of a core of density \(\rho_{c}\) and radius \(b\) surrounded by a mantle of density \(\rho_{m}\) and thickness \(a-b\). Show that the gravitational acceleration as a function of radius is given by $$\begin{aligned} g(r) &=\frac{4}{3} \pi \rho_{c} G r \quad 0 \leq r \leq b \\ &=\frac{4}{3} \pi G\left[r \rho_{m}+b^{3}\left(\rho_{c}-\rho_{m}\right) / r^{2}\right] \quad b \leq r \leq a \end{aligned}$$ and that the pressure as a function of radius is given by $$\begin{aligned} p(r)=& \frac{4}{3} \pi \rho_{m} G b^{3}\left(\rho_{c}-\rho_{m}\right)\left(\frac{1}{r}-\frac{1}{a}\right) \\ &+\frac{2}{3} \pi G \rho_{m}^{2}\left(a^{2}-r^{2}\right) \quad b \leq r \leq a \\\ =& \frac{2}{3} \pi G \rho_{c}^{2}\left(b^{2}-r^{2}\right)+\frac{2}{3} \pi G \rho_{m}^{2}\left(a^{2}-b^{2}\right) \\ &+\frac{4}{3} \pi \rho_{m} G b^{3}\left(\rho_{c}-\rho_{m}\right)\left(\frac{1}{b}-\frac{1}{a}\right) \\ 0 \leq r \leq b \end{aligned}$$ Apply this model to the Earth. Assume \(\rho_{m}=\) \(4000 \mathrm{~kg} \mathrm{~m}^{-3}, b=3486 \mathrm{~km}, a=6371 \mathrm{~km}\) Calculate \(\rho_{c}\) given that the total mass of the Earth is \(5.97 \times 10^{24} \mathrm{~kg} .\) What are the pressures at the center of the Earth and at the core-mantle boundary? What is the acceleration of gravity at \(r=b ?\)
Problem 19
An overcoring stress measurement in a mine at a depth of \(1.5 \mathrm{~km}\) gives normal stresses of \(62 \mathrm{MPa}\) in the \(N-S\) direction, \(48 \mathrm{MPa}\) in the \(\mathrm{E}-\mathrm{W}\) direction, and 51 MPa in the NE- SW direction. Determine the magnitudes and directions of the principal stresses.
Problem 28
Triangulation measurements at monument 0 give the time rate of change of \(\theta_{1}, \dot{\theta}_{1}\) and the time rate of change of \(\theta_{2}, \dot{\theta}_{2}\) (Figure 2.32). Show that $$\dot{\varepsilon}_{x y}=\frac{1}{2} \frac{\left(\dot{\theta}_{2} \sec \theta_{2} \csc \theta_{2}-\dot{\theta}_{1} \sec \theta_{1} \csc \theta_{1}\right)}{\left(\tan \theta_{2}-\tan \theta_{1}\right)}$$ and $$\dot{\varepsilon}_{y y}-\dot{\varepsilon}_{x x}=\frac{\left(\dot{\theta}_{2} \csc ^{2} \theta_{2}-\dot{\theta}_{1} \csc ^{2} \theta_{1}\right)}{\left(\cot \theta_{1}-\cot \theta_{2}\right)}$$ where \(\dot{\varepsilon}_{x x}=d \varepsilon_{x x} / d t,\) and so on.
Problem 33
The displacement of the MOJA (Mojave) station is \(23.9 \mathrm{~mm} \mathrm{yr}^{-1}\) to the east and \(-26.6 \mathrm{~mm} \mathrm{yr}^{-1}\) to the north. Assuming the San Andreas fault to be pure strike-slip and that this displacement is associated only with motion on this fault, determine the mean slip velocity on the fault and its orientation. A MATLAB code for solving this problem is given in Appendix \(D\).
