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Chapter 18: Problem 64

\(.\) Penguin huddling. To withstand the harsh weather of the Antarctic, emperor penguins huddle in groups (Fig. \(18-49)\). Assume that a penguin is a circular cylinder with a top surface area \(a=0.34 \mathrm{~m}^{2}\) and height \(h=1.1 \mathrm{~m}\). Let \(P_{r}\) be the rate at which an individual penguin radiates energy to the environment (through the top and the sides); thus \(N P_{r}\) is the rate at which \(N\) identical, well-separated penguins radiate. If the penguins huddle closely to form a huddled cylinder with top surface area \(N a\) and height \(h\), the cylinder radiates at the rate \(P_{h}\). If \(N=1000,(\) a) what is the value of the fraction \(P_{h} / N P_{r}\) and (b) by what percentage does huddling reduce the total radiation loss?

Short Answer

Expert verified
The fraction \(P_h / N P_r\) is approximately 0.68, indicating a 32% reduction in radiation loss when penguins huddle.

Step by step solution

01

Understand Radiation from Individual Penguin

An individual penguin has a top surface area \(a = 0.34 \, \text{m}^2\) and height \(h = 1.1 \, \text{m}\). Each penguin radiates energy through the top and side surfaces. The lateral surface area of one penguin is \(2\pi rh\), where \(r\) is the radius. To find \(r\), we use \(a = \pi r^2\), hence \(r = \sqrt{\frac{a}{\pi}} = \sqrt{\frac{0.34}{\pi}}\). So, the total radiation area for one penguin is approximately \(a + 2\pi rh\).
02

Calculate Total Radiation for Separated Penguins

For \(N = 1000\) well-separated penguins, the total radiating area is \(N \times (a + 2\pi rh)\), which radiates energy at the rate \(N P_r\).
03

Calculate Radiation from Huddled Penguins

When the penguins huddle, they form a cylinder with a top surface area of \(N a\) and the same height \(h\). The lateral surface area of the huddled cylinder is \(2\pi R h\), where \(R\) is the radius of this large cylinder. Calculate \(R\) using \(N a = \pi R^2\), so \(R = \sqrt{\frac{N a}{\pi}}\), thus the huddled penguin cylinder radiates across an area of \(N a + 2\pi R h\) at rate \(P_h\).
04

Find Fraction of Radiated Energy in Huddle

Compare the radiated power of the huddled group vs. individual penguins: \(\frac{P_h}{N P_r} = \frac{(N a + 2\pi R h)}{N \times (a + 2\pi rh)}\). Simplify this fraction to find the value.
05

Calculate Percentage Reduction Due to Huddling

Calculate the percentage reduction in radiation loss due to huddling: \(\left(1 - \frac{P_h}{N P_r}\right) \times 100\%\). This represents the percentage of energy savings when penguins huddle together.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radiation
Radiation refers to the process through which energy, particularly in the form of electromagnetic waves, is emitted from a body. In the context of penguins huddling, it is crucial to understand how these birds lose heat through radiation. Penguins, like all warm-blooded animals, release thermal energy into the environment to regulate their body temperature. This transfer happens through their top and lateral surfaces.
The given problem models each penguin as a circular cylinder to simplify the radiation calculations. For one penguin, the radiation occurs from the top surface and the sides. The amount of energy radiated depends on the surface area through which heat is lost. In physics, this is usually calculated using the Stefan-Boltzmann law, which states that the power radiated by a surface is proportional to the fourth power of its temperature and its surface area. Therefore, understanding radiation is at the core of calculating how different arrangements of penguins affect thermal energy loss.
Energy Conservation
Energy conservation, in this scenario, involves limiting the energy loss through radiation. Penguins huddle together as a natural strategy to conserve energy. By coming together, they minimize the amount of surface area exposed to the cold temperatures of the Antarctic environment.
When penguins stand separately, the radiation occurs through a greater individual surface area. When penguins huddle, this reduces the total surface area exposed to the cold, decreasing energy loss. This is achieved by reducing the lateral surface area that emits heat. Huddling effectively insulates the group, as the sides of each penguin are less exposed.
In mathematical terms, this conservation is calculated by comparing the energy radiated when penguins are apart versus when they are huddled. Huddling leads to a lesser rate of energy radiation per penguin, demonstrating effective energy conservation.
Mathematical Modeling
Mathematical modeling is a powerful tool used here to represent and solve real-world problems. In the case of penguin huddling, the mathematical model involves simplifying penguins as circular cylinders.
To solve this problem, we must consider both individual and group configurations of penguins and calculate the rates of energy radiation. Initially, for one penguin, the top surface area and lateral surface dimensions are used to calculate the overall radiating surface. Using basic geometry, like the formula for the surface area of a cylinder, helps clarify the problem.
For the huddled penguins, the model calculates the radiating surface of the entire group as one large cylinder. This approach helps demonstrate the energy savings. By using equations to describe how surface areas translate into energy loss, the model provides a clear picture of the physical principles in action. This makes mathematical modeling a key component in understanding how penguins manage their energy effectively.

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Most popular questions from this chapter

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