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.
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 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.
Problem 10
Consider a block of rock with a height of \(1 \mathrm{~m}\) and horizontal dimensions of \(2 \mathrm{~m}\). The density of the rock is \(2750 \mathrm{~kg} \mathrm{~m}^{-3}\). If the coefficient of friction is \(0.8,\) what force is required to push the rock on a horizontal surface?
Problem 14
The state of stress at a point on a fault plane is \(\sigma_{y y}=150 \mathrm{MPa}, \sigma_{x x}=200 \mathrm{MPa}\), and \(\sigma_{x y}=0\) ( \(y\) is depth and the \(x\) axis points westward). What are the normal stress and the tangential stress on the fault plane if the fault strikes \(\mathrm{N}-\mathrm{S}\) and dips \(35^{\circ}\) to the west?
Problem 15
Show that the sum of the normal stresses on any two orthogonal planes is a constant. Evaluate the constant.
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 $$ \begin{aligned} g(r) &=\frac{4}{3} \pi \rho_{c} G r \quad 0 \leq r \leq b \\ &=\frac{4}{3} \pi G\left[r \rho_{m}+b^{3}\left(\rho_{c}-\rho_{m}\right) / r^{2}\right] \quad b \leq r \leq a \end{aligned} $$
Problem 24
Show that the principal strains are the minimum and the maximum fractional changes in length.
