## Dinosaur Models and the Estimation of Dinosaur Weights

An example of how science, art, and popular culture can be combined is through information derived from models of dinosaurs. Dinosaurs are often associated with huge sizes, but how can the question "How big were dinosaurs?" be answered? This book refers to the kilograms or metric tonnage (1000 kg, which equals 2200 pounds) of a particular dinosaur, even though no one has actually weighed a living (or even recently dead) one. Arriving at such figures requires a few simple principles of physics, a little bit of math in the form of biometry, and some help from the dinosaur models.

Dinosaur models, usually encountered in toy stores or gift shops of natural history museums, are a form of mass-produced "artwork" for which the artists are usually not credited. Nonetheless, many of the models are based on at least some scientifically-derived estimates for dinosaur morphology. Moreover, they are sometimes scaled to a standard size in relation to a full-sized species of dinosaur. Armed with these models, a vessel containing water, some measuring tools, and a little bit of knowledge, the approximate weight of a dinosaur can be calculated.

Weight is a measurement of the amount of force exerted by gravity, which is caused by the attraction of the matter for matter. In the case of the Earth, the force of gravity is expressed by the following equation:

where G is the gravitational constant (9.8 meters/second2); m1 and m2 are the masses of the objects attracted to one another (one of them being the Earth, the other being any other object); and d is the distance separating the two objects. The force is measured in newtons (N), expressed as kg/m/s2. This shows that weight, in this case, is a force expressed by the mass of an object multiplied by the acceleration that is imparted to it from its attraction to the Earth. As a force, a person's weight will vary very slightly on the Earth's surface. This variation depends on whether a person is directly over an area of the Earth with slightly more or less mass interacting with their mass, as well as the distance between those two masses. For dinosaurs that had much mass, which we have interpreted on the basis of the large size of their skeletal parts and inferred musculature, a logical conclusion is that they correspondingly had much weight.

If a scale was not to hand to measure someone's weight, it could still be estimated on the basis of two parameters:

1 volume, which is the three-dimensional space occupied by a certain amount of matter and normally expressed in cubic centimeters (cm3); and

2 density, which is the mass of that matter divided by volume and expressed in grams per cubic centimeter (g/cm3).

Dipping someone into a bathtub and measuring the volume of water displaced could measure the volume. For example, once immersed, the person might displace 72.0 liters of water, which converts to 72,000 cm3 (because 1.0 ml = 1.0 cm3 = 1.0 g, with pure water as a standard). Because the human body is mostly composed of water, its density is also close to that of water, about 0.9 g!cm3. To find out the weight, simply multiply mass by volume, where W is weight, d is density, and v is volume:

(converting to kilograms) Step 2. = 64,800 g - 1000 g!kg = 64.8 kg

The present mass of the Earth is assumed to be identical to that in the Mesozoic Era. A model of a tyrannosaur, scaled at 0.033 (3.3%) of the original size of the dinosaur, would displace 235 ml (235 cm3) of water, if fully immersed. However, the assumed density for the tyrannosaur is 0.8 g/cc, which is less dense than a person because of the degree of "hollowness" in some dinosaur bones (Chapter 8). Is 0.8 g/cm3 then multiplied by 235 cm3? No, because the tyrannosaur must be made "larger" by scaling it to life-size. This means recognizing that 3.3% is about equal to 1/30 and that it had three dimensions (length, width, height), which corresponds approximately to its original volume. Thus, scaling involves making the tyrannosaur 30 times longer, wider, and higher than the model, which results in the following volume change, where V is volume, l is length, w is width, and h is height:

Step 1. V = 30 x 30 x 30 = 27,000 times the volume of the model

Using this volume increase and multiplying it by the density and the measured volume yields the following results for the tyrannosaur:

Step 1. W = 0.8 g/cm3 x 235 cm3 x 27,000 = 5,076,000

(Converting to kilograms) Step 2. = 5,076,000 g + 1000 g/kg = 5076 kg

(Converting to metric tons) Step 3. = 5076 kg + 1000 kg/ton = 5.076 metric tons where W is weight, d is density, v1 is measured volume, v2 is the volume increase.

Hence an initial estimate of how much a particular dinosaur weighed can be calculated. This is probably not accurate, because the first assumption is that the model is an accurate representation of the dinosaur. This assumption is made despite the fact that many species of dinosaurs are known from less than 90% complete skeletons. As a result, their reconstruction is sometimes sketchy (Chapters 6 and 7). Furthermore, not all model-makers are concerned with constructing scientifically accurate figures. Another assumption is that the density was 0.8 g/cm3, whereas other researchers have made estimates of 0.9-1.1 g/cm3.

Alternative methods have been used for estimating dinosaur weight. One method uses measurements of leg-bone circumferences of extant mammal species and correlates these data with animal weights. This results in different values for dinosaurs, suggesting that either method might work, or not.

The important point here is that some artistic interpretations of dinosaurs, which are based on at least some available scientific information, can be tested in a scientific manner for their feasibility. Such tests can demonstrate that any supposed gap between science and popular art is not as wide as we sometimes think. These weight estimates derived from models also help us to better appreciate the possible weights of some dinosaurs relative to living animals. For perspective, an adult African elephant can weigh 5 metric tons, which is about the same weight as our hypothetical tyrannosaur. Realizing that a carnivore, such as T. rex, may have weighed as much as an African elephant adds a sense of realism to it that transcends models, paintings, or photographs of its remains, and brings it more to life.

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