Problem 15 - College Physics

Question

By means of physiological adaptations that are still not very well understood, sperm whales are thought to be able to hunt for their food at depths of between 400 \(\mathrm{m}\) and 3000 \(\mathrm{m.}\) (a) What range of gauge pressures (in Pa and atm) do the whales withstand at these depths? (b) Estimate the total inward force of water pressure on the surface of a sperm whale at a depth of \(3000 \mathrm{m},\) modeling the whale as a cylinder 16 \(\mathrm{m}\) long and 4 \(\mathrm{m}\) in diameter.

Step-by-Step Solution

Answer

The gauge pressure ranges from 3,924,000 Pa (38.71 atm) to 29,430,000 Pa (290.41 atm), and the inward force at 3000 m is approximately 5,917,357,800 N.

Step 1: Understanding the Problem

We need to compute the gauge pressure the sperm whale experiences at depths of 400 m and 3000 m, and then determine the force exerted by water pressure on the whale at 3000 m depth, assuming the shape of a cylinder for the whale.

Step 1: Calculate Gauge Pressure at 400 m Depth

The gauge pressure is given by the formula: \( P = \rho \cdot g \cdot h \), where \( \rho \) is the density of water (approximately \( 1000 \text{ kg/m}^3 \)), \( g \) is the acceleration due to gravity (approximately \( 9.81 \text{ m/s}^2 \)), and \( h \) is the depth.For \( h = 400 \text{ m} \):\[ P = 1000 \cdot 9.81 \cdot 400 = 3,924,000 \text{ Pa} \]

Step 2: Calculate Gauge Pressure at 3000 m Depth

Using the gauge pressure formula for \( h = 3000 \text{ m} \):\[ P = 1000 \cdot 9.81 \cdot 3000 = 29,430,000 \text{ Pa} \]

Step 3: Convert Gauge Pressures to Atmospheres

To convert pressures from Pascals to atmospheres, use the conversion factor \( 1 \text{ atm} = 101,325 \text{ Pa} \).At 400 m:\[ P = \frac{3,924,000}{101,325} \approx 38.71 \text{ atm} \]At 3000 m:\[ P = \frac{29,430,000}{101,325} \approx 290.41 \text{ atm} \]

Step 4: Calculate the Surface Area of the Whale

Assuming the whale is a cylinder with length \( L = 16 \text{ m} \) and diameter \( d = 4 \text{ m} \), the radius \( r = 2 \text{ m} \). Calculate the surface area \( A \) using the formula for the lateral surface area of a cylinder: \[ A = 2 \pi r L \]\[ A = 2 \pi \cdot 2 \cdot 16 \approx 201.06 \text{ m}^2 \]

Step 5: Calculate Total Inward Force on the Whale at 3000 m

The total force \( F \) is given by the product of the pressure and the area: \[ F = P \cdot A \]Using \( P = 29,430,000 \text{ Pa} \) from 3000 m depth and \( A \approx 201.06 \text{ m}^2 \):\[ F = 29,430,000 \cdot 201.06 \approx 5,917,357,800 \text{ N} \]

Key Concepts

Cylinder Surface Area

When we explore the surface area of a cylinder, we're essentially interested in the part of the cylinder that wraps around its length, like a label around a soda can. For our purposes, the sperm whale is modeled as a perfect cylinder. This involves only the lateral or curved surface, excluding the top and bottom.

To calculate this lateral surface area, we use:

• Formula: \( A = 2 \pi r L \)
• Where: \( r \) is the radius of the cylinder, and \( L \) is its length.

For our sperm whale, which stretches 16 meters long and has a 4-meter diameter, its radius sits comfortably at 2 meters. By fitting these values into our formula,
we get:

• \( A = 2 \pi \times 2 \times 16 \)
• \( A \approx 201.06 \text{ m}^2 \)

This value denotes the total area over which water pressure will exert force, crucial for our understanding of the next calculations.

Sperm Whale Physics

The physics surrounding sperm whales are fascinating due to their ability to dive into extraordinary depths. Adaptations allow these creatures to withstand immense pressure as they hunt in the vast depths of our oceans.

When thinking about sperm whale physics:

• Think of buoyancy, which helps them maintain depth without exerting energy.
• Consider their specialized bodies which can manage the intense pressure from water at 3000 meters deep.

By modeling the whale as a cylinder, the calculations become manageable. Each aspect of these calculations plays into the larger picture of how these animals survive and thrive where the human physics begins to buckle under pressure.

Depth Pressure in Fluids

Pressure at various ocean depths is a vital part of fluid mechanics. From a whale's perspective, such knowledge is crucial to understand how they withstand deep-sea conditions.Gauge pressure at a certain depth is calculated by:

• \( P = \rho \cdot g \cdot h \)
• Where: \( \rho \) is water's density, \( g \) is gravitational acceleration, and \( h \) is the depth.

At 400 meters, the gauge pressure computed is 3,924,000 Pa, translating into approximately 38.71 atmospheres. Contrast this with at a depth of 3000 meters, where the calculation reaches 29,430,000 Pa, or about 290.41 atmospheres.

This immense pressure is crucial for understanding how sea creatures like whales endure and adapt to their conditions, and why diving to such depths challenges even the most advanced underwater technologies humans have developed.