4.7: Summary and further problems

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Proportional reasoning focuses our attention on how one quantity determines another. (A wonderful collection of pointers to further reading is the American Journal of Physics’s “Resource Letter” on scaling laws [49].) By guiding us toward what is often the most important characteristic of a problem, the scaling exponent, it helps us discard spurious complexity.

Exercise \(\PageIndex{1}\): Cruising speed versus mass

For geometrically similar animals (the same shape and composition but different sizes) in forward flight, how does the animal’s minimum-energy flight speed v depend on its mass m? In other words, what is the scaling exponent \(\beta\) in \(v \: \propto \: m \beta\)?

Exercise \(\PageIndex{2}\): Hovering power versus size

In Section 3.6.1, we derived the power required to hover. For geometrically similar birds, how does the power per mass depend on the animal’s size L? In other words, what is the scaling exponent \(\gamma\) in \(P_{hover}/m \: \propto \: L^{ \gamma}\)? Why are there no large hummingbirds?

Exercise \(\PageIndex{3}\): Cruising speed versus air density

How does a plane’s (or a bird’s) minimum-energy speed v depend on \(\rho_{air}\)? In other words, what is the scaling exponent \(\gamma\) in \(v \: \propto \: \rho_{air}^{\gamma}\)?

Exercise \(\PageIndex{4}\): Speed of a bar-tailed godwit

Use the results of Problem \(\PageIndex{1}\) and Problem \(\PageIndex{3}\) to write the ratio v₇₄₇/v_godwit as a product of dimensionless factors, where v₇₄₇ is the minimum-energy (cruising) speed of a 747, and v_godwit is the minimum-energy (cruising) speed of a bar-tailed godwit. Using m_godwit ≈ 400 grams, estimate the cruising speed of a bar-tailed godwit. Compare your result to the average speed of the record-setting bar-tailed godwit that was studied by Robert Gill and his colleagues [19], which made its 11 680-kilometer journey in 8.1 days.

Exercise \(\PageIndex{5}\): Thermal resistance of a house versus a tea mug

When we developed the analogy between low-pass electrical and thermal filters (Section 2.4.5)—whether RC circuits, tea mugs, or houses—we introduced the abstraction of thermal resistance R_thermal. In this problem, you estimate the ratio of thermal resistances \(R_{thermal}^{house}/R_{thermal}^{tea \: mug}.\).

House walls are thicker than teacup walls. Because thermal resistance, like electrical resistance, is proportional to the length of the resistor, the house’s thicker walls raise its thermal resistance. However, the house’s larger surface area, like having many resistors in parallel, lowers the house’s thermal resistance. Estimate the size of these two effects and thus the ratio of the two thermal resistances.

Exercise \(\PageIndex{6}\): General birthday problem

Extend the analysis of Problem 4.4.2 to k people sharing a birthday. Then compare your predictions to the exact results given by Diaconis and Mosteller in [10].

Exercise \(\PageIndex{7}\): Flight of the housefly

Estimate the mechanical power required for a common housefly Musca domestica (m ≈ 12 milligrams) to hover. From everyday experience, estimate its typical flight speed. At this flight speed, compare the power requirements for forward flight and for hovering.