Ch 11: Equilibrium & Elasticity

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 11: Equilibrium & Elasticity

Problem 34a

Chapter 11, Problem 34a

In the Challenger Deep of the Marianas Trench, the depth of seawater is 10.9 km and the pressure is 1.16×10⁸ Pa (about 1.15×10³ atm). If a cubic meter of water is taken from the surface to this depth, what is the change in its volume? (Normal atmospheric pressure is about 1.0×10⁵ Pa. Assume that k for seawater is the same as the freshwater value given in Table 11.2.)

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Identify the initial and final pressures. The initial pressure at the surface is the normal atmospheric pressure, \( P_0 = 1.0 \times 10^5 \) Pa. The final pressure at the depth of the Challenger Deep is \( P = 1.16 \times 10^8 \) Pa.

Determine the bulk modulus of seawater. The bulk modulus \( k \) is a measure of a substance's resistance to uniform compression. For freshwater, \( k \) is approximately \( 2.2 \times 10^9 \) Pa. Assume seawater has the same bulk modulus.

Use the formula for volume change due to pressure change: \( \Delta V = -\frac{\Delta P \cdot V_0}{k} \), where \( \Delta P = P - P_0 \) is the change in pressure, \( V_0 = 1 \) m³ is the initial volume, and \( k \) is the bulk modulus.

Calculate the change in pressure: \( \Delta P = 1.16 \times 10^8 \text{ Pa} - 1.0 \times 10^5 \text{ Pa} \).

Substitute the values into the volume change formula to find \( \Delta V \). This will give you the change in volume of the cubic meter of water when taken from the surface to the depth of the Challenger Deep.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Pressure

Pressure is the force exerted per unit area. In this context, it refers to the immense pressure exerted by the water column above a point in the ocean, which increases with depth. At the Challenger Deep, the pressure is significantly higher than at the surface, affecting the volume of submerged objects.

Bulk Modulus

The bulk modulus (k) is a measure of a substance's resistance to uniform compression. It is defined as the ratio of pressure change to the relative volume change. For seawater, it indicates how much the volume of water decreases under high pressure, such as at great ocean depths.

Volume Change Under Pressure

The change in volume of a substance under pressure can be calculated using the bulk modulus. The formula ΔV = -V₀(ΔP/k) relates the initial volume (V₀), pressure change (ΔP), and bulk modulus (k) to find the volume change (ΔV). This concept is crucial for determining how much a cubic meter of water compresses at the Challenger Deep.

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