Problem 51 - Fundamentals of Physics

Question

Rock faults are ruptures along which opposite faces of rock have slid past each other. In Fig. \(3-35\), points \(A\) and \(B\) coincided before the rock in the foreground slid down to the right. The net displacement \(\overrightarrow{A B}\) is along the plane of the fault. The horizontal component of \(\overrightarrow{A B}\) is the strike-slip \(A C\). The component of \(\overrightarrow{A B}\) that is directed down the plane of the fault is the dip-slip \(A D\). (a) What is the magnitude of the net displacement \(\overrightarrow{A B}\) if the strike-slip is \(22.0 \mathrm{~m}\) and the dip- slip is \(17.0 \mathrm{~m} ?\) (b) If the plane of the fault is inclined at angle \(\phi=52.0^{\circ}\) to the horizontal, what is the vertical component of \(\overrightarrow{A B}\) ?

Step-by-Step Solution

Answer

(a) 27.8 m, (b) 13.4 m

Step 1: Understanding the Problem

We are given two components of the net displacement \(\overrightarrow{A B}\): the strike-slip component \(A C = 22.0\, \text{m}\) and the dip-slip component \(A D = 17.0\, \text{m}\). We need to find (a) the magnitude of the net displacement \(\overrightarrow{A B}\) and (b) the vertical component of \(\overrightarrow{A B}\) given the angle \(\phi = 52.0^\circ\).

Step 1: Calculate the Magnitude of Net Displacement

The magnitude of the net displacement \(\overrightarrow{A B}\) can be found using the Pythagorean theorem since the strike-slip \(A C\) and dip-slip \(A D\) are perpendicular components:\[|\overrightarrow{A B}| = \sqrt{(A C)^2 + (A D)^2} = \sqrt{(22.0)^2 + (17.0)^2}\]Calculating this gives:\[|\overrightarrow{A B}| = \sqrt{484 + 289} = \sqrt{773} \approx 27.8 \, \text{m}\]

Step 2: Find the Vertical Component of Displacement

The vertical component of \(\overrightarrow{A B}\) can be found using the dip-slip \(A D\) and the inclination angle \(\phi\). The vertical component is the opposite side in the right triangle formed, thus calculated by:\[\text{Vertical Component} = A D \cdot \sin(\phi) = 17.0 \cdot \sin(52.0^\circ)\]Using the sine value of 52 degrees (\(\sin(52^\circ) \approx 0.788\)):\[\text{Vertical Component} = 17.0 \times 0.788 \approx 13.4 \, \text{m}\]

Key Concepts

Fault Rupture

In geology, a fault rupture represents a fracture or discontinuity in the Earth's crust that results from movement. These movements can range from several meters to just a fraction of a millimeter. Faults allow the crust to move, adjusting stress within the Earth. When a fault ruptures, rocks on either side of the fault slip past each other.
These slips can be calculated in various ways, resulting in different types of measurable components like the strike-slip and the dip-slip. The strike-slip refers to horizontal movements, while dip-slip pertains to vertical displacements. Understanding these components is crucial to measure the total displacement accurately.

Net Displacement

Net displacement in the context of faults and earthquakes refers to the overall movement between two points on separate blocks of crust, like points A and B in the exercise.
It considers both the strike-slip and dip-slip components, which are perpendicular to each other. To find out the net displacement vector, you combine these two components using vector addition.

• Net displacement directly relates to understanding the forces and energy released during an earthquake.
• It gives geologists insights into the dynamics of crustal movements.

In the exercise, calculating the net displacement helps us understand the spatial relationship between the faulted rock layers.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle used in geometry, and it is crucial for solving problems involving right triangles.
The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides: \[c^2 = a^2 + b^2\]
In the context of this problem, the theorem helps us find the magnitude of the net displacement \(|\overrightarrow{A B}|\) by treating the strike-slip and dip-slip as perpendicular sides of a right triangle.

• It simplifies calculating distances and movements in 2D and 3D spaces.
• Essential for converting component vectors into a single resultant vector.

Vertical Component

The vertical component of a vector is crucial when analyzing vectors in inclined planes. It helps in understanding how much a component contributes to vertical movement, especially relevant in fault analysis.
In this exercise, the vertical component is calculated using the dip-slip and the sine of the inclination angle. This calculation gives insight into how much of the slip is vertical as opposed to horizontal.

• Vertical component = Dip-slip \(\times \sin(\phi)\).
• Important for assessing the lifting or sinking of earth materials during faulting.

Calculations involving vertical components aid in predicting geological features like valleys and ridges after an earthquake.