Problem 10 - College Physics

Question

The earth's crust is broken up into a series of more-or-less rigid plates that slide around due to motion of material in the mantle below. Although the speeds of these plates vary somewhat, they are typically about \(5 \mathrm{~cm} / \mathrm{y}\). Assume that this rate remains constant over time. (a) If you and your neighbor live on opposite sides of a plate boundary at which one plate is moving northward at \(5.0 \mathrm{~cm} / \mathrm{y}\) with respect to the other plate, how far apart do your houses move in a century? (b) Los Angeles is presently \(550 \mathrm{~km}\) south of San Francisco but is on a plate moving northward relative to San Francisco. If the \(5.0 \mathrm{~cm} / \mathrm{y}\) velocity continues, how many years will it take before Los Angeles has moved up to San Francisco?

Step-by-Step Solution

Answer

(a) 5 meters apart in a century, (b) 11 million years for Los Angeles to reach San Francisco.

Step 1: Convert movement rate to compatible time units

The plate moves northward at a rate of \(5.0 \text{ cm/year}\). We need to calculate the distance moved in 100 years (a century). First, convert the rate from cm to meters for easier conversion later:\[\text{Rate} = 5.0 \text{ cm/year} = 0.05 \text{ m/year}\]

Step 2: Calculate distance moved in a century

Using the converted rate from Step 1, calculate how far the plates move in 100 years (a century):\[\text{Distance moved in 100 years} = 0.05 \text{ m/year} \times 100 \text{ years} = 5 \text{ meters}\]Thus, the houses move 5 meters apart in a century.

Step 3: Convert initial city distance to same units as movement

Los Angeles is \(550 \text{ km}\) south of San Francisco. Convert this distance to meters for convenience as the plate's movement rate is in meters:\[550 \text{ km} = 550,000 \text{ meters}\]

Step 4: Determine years required for Los Angeles to reach San Francisco

Knowing the rate of movement is \(0.05 \text{ m/year}\), calculate how long it takes for Los Angeles to cover the full \(550,000 \text{ meters}\):\[\text{Time} = \frac{\text{Distance}}{\text{Rate}} = \frac{550,000 \text{ meters}}{0.05 \text{ m/year}} = 11,000,000 \text{ years}\]Therefore, it will take Los Angeles 11 million years to reach San Francisco at the given rate.

Key Concepts

Earth's Crust

The Earth's crust is the outermost layer of our planet, marking the surface where we live. It includes the continents and ocean floors, composed primarily of rocks and minerals. The crust is relatively thin compared to other layers, averaging about 35 km under continents and around 7 km under oceans. However, it is not a continuous shell but is divided into pieces called tectonic plates. These plates fit together like a giant jigsaw puzzle, covering the entire Earth.

Tectonic plates are always moving, albeit very slowly, driven by the forces from deeper within the Earth. These movements are responsible for geological phenomena such as earthquakes and volcanic activity. Understanding the Earth's crust is essential for studying plate tectonics, as it sits atop the more dynamic layers beneath.

Mantle

The mantle is the thick layer of rock beneath the Earth's crust, constituting about 84% of Earth's total volume. It extends to a depth of about 2,900 kilometers until it meets the core. The mantle is primarily composed of silicate minerals which are dense and capable of flowing over geological time scales.

The mantle's heat comes from radioactive decay of elements and residual heat from Earth's formation. This heat creates convective currents, much like boiling water, that cause the material in the mantle to move. The slow, churning movement of the mantle's material drives the motion of tectonic plates on the surface. Thus, the mantle acts as both a conveyor belt and a heat engine for the planet, pushing the plates around and enabling plate tectonics.

Plate Movement

Plate movement refers to the shifting of Earth's tectonic plates, which float on the semi-fluid mantle below. These movements are typically measured in centimeters per year, as in the problem where a plate moves at 5 cm/year. Although this seems slow, it can lead to significant changes over long periods, affecting geological formations and landscapes.

Plate movements are powered by processes in the mantle, including slab pull, ridge push, and mantle convection. These processes can cause plates to slide past each other, collide, or move apart, resulting in different geological features such as mountains, earthquakes, and oceanic trenches. Understanding plate movements helps explain patterns of seismic and volcanic activity worldwide.

Distance Calculation

In the context of tectonic plates, calculating how far one plate has moved relative to another over time can be crucial for understanding geological changes. This is typically calculated using the formula: \[ \text{Distance moved} = \text{Rate} \times \text{Time} \]For example, if a plate moves at a constant rate of 5 cm per year, over 100 years, this equates to a movement of 5 meters, as shown in the solution example.

When dealing with large distances, it's important to convert units to ensure calculations are accurate. Converting between centimeters, meters, and kilometers is often required to ensure a precise understanding of plate movements over time.

Geological Time Scales

Geological time scales provide a framework for understanding the long-term dynamics and history of the Earth. They divide Earth's 4.5 billion-year history into different periods based on major geological and biological events.

Time scales range from eons to eras, periods, epochs, and ages. Plate tectonics plays a significant role in shaping the Earth over these vast spans of time. Even though plate movements appear slow, they accumulate significant changes over millions of years, leading to the formation and breakup of continents, mountain ranges, and ocean basins.

Understanding these scales helps geologists explain the past configurations of our continents and the processes that sculpted the Earth's surface over time.