 D103. Thinking about Scaling
Introduction We've seen in a previous assignment that the relative strength of an ant is much greater than that of a human. Since weight increases as the cube of the scale factor while muscle and bone strength increase as the square of the scale factor, weight increases much faster than muscle and bone strength. For this reason, larger animals generally have thicker legs in relation to the size of their bodies. Consider the thin legs of an insect (relative to their body size) compared to the thick legs of an elephant or rhinoceros. By the way, the largest mammals--whales--don't have legs at all. They use the water to help with support. Similar arguments apply to the stresses that are applied to the body in other situations. Humans can jump from a height of 2 m onto their feet without injury. If an elephant were dropped that distance, its legs would be crushed. Here's a related argument. Some insects can walk on water. Humans can't. The force that supports an insect is surface tension, and this force scales as the surface area that is in contact with the water. Thus, surface tension scales in the same way as bone strength. For this reason, water can more easily support insects than people. If, however, the insect falls into the water, it will most likely die. In that case, the surface tension becomes a formidable force trapping the insect in a layer of water. A human, on the other hand, could just shake it off. Let's now consider elephant ears. The purpose of these giant appendages is to help cool the animal. The rate of cooling through evaporation of water from the surface is proportional to the surface area of the ears. Why don't smaller animals require such oversized ears? In order to answer this, consider a sphere. The volume of a sphere scales as the cube of the diameter, while the surface area scales as the square of the diameter. The ratio of volume to surface area (V/A) is therefore proportional to the diameter. A spherical elephant a thousand times more massive than a spherical human would have a V/A ratio 10 times greater. Most of the mass of the animal produces thermal energy through normal biological processes, but there's only one-tenth the surface area, in relation to that of a human, to provide evaporative cooling. Now stick 2 large, thin disks on the sphere. The disks, due to their shape, maximize the amount of surface area for their weight. They add very little to the overall weight of the sphere while adding greatly to the surface area. Thus, the efficiency of cooling is greatly increased. Polar bears demonstrate the opposite extreme. Their problem is to keep warm. The roly-poly shape of their bodies minimizes the surface area in relation to weight, thus reducing heat loss by evaporation. The Assignment In the assignment to follow, you'll attempt to answer a question using scaling arguments. Here's an example question with sample responses of increasing quality. Question: Giant spiders in horror movies generally look like scaled-up versions of real spiders. Even if super-sized spiders existed, why isn't the movie variety realistic?
The Questions You'll be assigned one of the following questions to which to respond.
In responding to your question, explicitly use your knowledge of scaling relationships and provide an explanatory response. Here's some additional information that may be helpful in discussing some of the questions.
Here is your assigned question:
The above assignments allot 4 students per question. Posting your response Do the following:
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