Calculate the pressure due to the ocean at the bottom of the...

Calculate the pressure due to the ocean at the bottom of the Marianas Trench near the Philippines, given its depth is $11.0 \mathrm{km}$ and assuming the density of sea water is constant all the way down. (b) Calculate the percent decrease in volume of sea water due to such a pressure, assuming its bulk modulus is the same as water and is constant. (c) What would be the percent increase in its density? Is the assumption of constant density valid? Will the actual pressure be greater or smaller than that calculated under this assumption?

Key Concepts

Volume and Density Variation Under Pressure

This concept involves understanding that as pressure is applied to a fluid, it undergoes compression, leading to a reduction in volume and an increase in density. The degree of compression depends on the fluid’s bulk modulus. Awareness of these variations is crucial for assessing the limitations of the constant density assumption in high-pressure environments.

Bulk Modulus and Compressibility

The bulk modulus of a material is a measure of its resistance to uniform compression, defined as the ratio of applied pressure change to the relative change in volume (K = -V ?P/?V). In the context of fluids, it determines how much a fluid will compress under an increased pressure. This concept is essential for estimating the decrease in volume and corresponding increase in density when a fluid is subjected to high pressures.

Hydrostatic Pressure

This concept refers to the pressure exerted by a fluid at equilibrium due to the force of gravity. The pressure increases linearly with depth according to the formula P = ?gh, where ? is the density of the fluid, g is the acceleration due to gravity, and h is the depth below the surface. This relationship is the foundation for calculating the pressure at any depth in a fluid.

Constant Density Assumption

In many introductory problems, the assumption is made that the density of a fluid remains constant regardless of depth. This simplification makes calculations easier using the basic hydrostatic pressure formula; however, in reality, the density can change under high pressure. Understanding when this assumption is valid is important in accurately modeling real-world phenomena.

Transcript

00:01 So for part a, here we can just find the pressure below the sea level.

00:05 This would be equalling the density times the acceleration to the gravity times the depth.

00:10 And so the density here, this would be the density of sea water, 1 ,025 kilograms per cubic meter, multiplied by the acceleration due to gravity, 9 .80 meters per second squared, times the depth of 11 kilometers or we can say 11 ,000 meters.

00:39 And so we find that the pressure 11 kilometers underneath the surface of the ocean, this would be in mariana's trench.

00:48 This would be 1 .105 times 10 to the 8th newton's per square meter.

00:56 This would be the change, the pressure due to the ocean.

01:06 Again, the true pressure would actually be this value plus the atmospheric pressure, but we're just calculating the pressure due to the ocean.

01:17 For part b, we can find, we know that the pressure is simply a force over unit area, and we know that the change in volume would be equaling to 1 over b, the bulk modulus, multiplied by the force per unit area, times the initial volume.

01:37 And so the initial, the change in volume, rather, would be the pressure over the bulk modulus times the original volume.

01:47 And so we can say that then the change in volume over the original volume would be equaling the pressure over the bulk modulus.

01:55 And we can solve 1 .105 times 10 to the 8th newtons per square meter.

02:03 This would be divided by the bulk modulus of seawater 2 .2 times 10 to the 9th newtons per square meter.

02:14 And we find that this would be equal to 5 .0%.

02:20 So the decrease in volume is 5 .0%...

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