Dynamics of dinosaurs

Transcription

1 MATH0011 Numbers and Patterns in Nature and Life Lecture 1 Dynamics of Dinosaurs Questions: Dynamics of dinosaurs How heavy are dinosaurs? Would sand support a big dinosaur just as well as a small one? How about clay? 1 Basic Facts about Dinosaurs Basic Facts about Dinosaurs (cont.) Dinosaurs were a particular subclass of reptiles, lived in the Mesozoic geological era (230 million 65 million years ago). (Human appeared only about 1 million years ago.) Although they are reptile-like, dinosaurs are not ancestors of modern reptiles (such as crocodiles and lizards). In fact, dinosaurs are ancestors of birds. Not all dinosaur species lived at the same time. 2 3

2 Basic Facts about Dinosaurs (cont.) The largest known dinosaur is Argentinosaurus (35-45 m long). Another well-known big one is Diplodocus (25 m long). Basic Facts about Dinosaurs (cont.) The smallest dinosaur fossils found so far are 23mm and 25mm in length, found in Guiyang of China in The most famous is the Tyrannosaurus (T-rex, > 13 m long). It is the largest known land predators. Diplodocus Argentinosaurus 4 5 Basic Facts about Dinosaurs (cont.) Some dinosaurs walked on two legs (bipedal), some walked on four legs (qradrupedal). stegosaurus Iguanodon Basic Facts about Dinosaurs (cont.) Dinosaurs are studied mainly through their fossils, which are mainly bone (including teeth) fossils. Over the millions of years, organic matters in those bones were replaced by minerals, forming complex compounds (fossilization). 6 7

3 Basic Facts about Dinosaurs (cont.) Therefore, the remains of dinosaurs that we see in museums which resemble bones are actually some kind of stones. How Heavy Are Dinosaurs? Difficulty: No live specimens available, nor even dead ones with their whole body (including blood and body fluids, etc.) intact. Remark: One may use a weighing bridge to weigh large mammals such as elephants. Weighing an elephant 8 9 Weighing Dinosaurs Using Scale Models Weighing Dinosaurs Using Scale Models (cont.) Method 1: Using Archimedes Principle with scale models. Apparatus: anatomically accurate dinosaur model, beaker, beam balance, water. Example: Suppose a model of Triceratops, of scale 1:40, is used. Suspend it from one arm of a beam balance over a beaker. (If the model is not as dense as water then hang a metal weight from its tail.) Setup of apparatus 10 11

4 Weighing Dinosaurs Using Scale Models (cont.) Make sure that the model is not touching the bottom or the side of the beaker. Pour water into the beaker so that the tail weight (if any) is submerged in water but the model is above the water. Enough weights are put on the pan to balance the system. Record the weight as W a (kg). Add water until the model is completely submerged. The upthrust of water on the model will put the system unbalanced. Remove enough weight on the pan until the system is balanced again. Record the weight as W b (kg). Weighing Dinosaurs Using Scale Models (cont.) By Archimedes principle, upthrust is equal to the weight of water displaced. Therefore: W a W b = upthrust on model = mass of volume of water occupied by the model. Hence volume of model = (W a W b )/(density of water). Suppose W a W b = kg. Since density of water is 1,000 kg / m 3, volume of the model is m 3. Since the (linear) scale of the model is 1:40, the dinosaur had 40x40x40=64,000 times the volume of the model Weighing Dinosaurs Using Scale Models (cont.) Hence volume of the Triceratops was x 64,000 = 6.08 m 3. To get the mass of the dinosaur from its volume, we must estimate its density. Since most animal have about the same density as water (because they either just float in water, with very little body parts above water surface, or they just sink), we may assume dinosaurs had a density equal to that of water, which is 1,000 kg/m 3. Thus, in our case, the mass of the Triceratops is estimated as 6.08 m 3 x (1,000kg/m 3 ) = 6,080 kg. Weighing Dinosaurs Using Scale Models (cont.) Theorem 1 Suppose a model of scale 1: α is used. If an upthrust of c kg (= W a W b ) is observed, and a body density of d kg/m 3 is assumed, then this method will give an estimate of the dinosaur s body mass (in kg) as α 3 cd 1000 Remark: Accuracy of this method relies on: (1) Accuracy of the model, and (2) Accuracy of the estimate on the body density of dinosaurs

5 Weighing Dinosaurs with Leg Bone Measurement Method 2: This method requires leg bone measurements. Fortunately, major leg bones are often well preserved in fossils. Dinosaurs stood and moved with legs underneath their bodies, not splayed out to each side like modern reptiles such as crocodiles and lizards. Footprints of (a) a lizard, (b) a bird, and (c) a mammal. Weighing Dinosaurs with Leg Bone Measurement (cont.) Anderson et al. (1985) measured the circumferences of the humerus and femur (upper bones in the fore leg, and hind leg, respectively) of several dinosaur fossils and some modern mammals. They added the two circumferences together, and plotted these against the body mass of modern mammals Weighing Dinosaurs with Leg Bone Measurement (cont.) Weighing Dinosaurs with Leg Bone Measurement (cont.) They observed that an equation of Log-log plot of data (quadrupedal mammals) obtained by Anderson et al. body mass in kg = a (total circumferences in mm) b where a = , b = 2.73, would fit the data of modern quadrupedal mammals very well. They then used the above equation to estimate the body mass of quadrupedal dinosaurs. For bipeds, they used the formula body mass in kg = a (circumference of femur in mm) b where a = , b =

6 Weighing Dinosaurs with Leg Bone Measurement (cont.) This formula underestimates the masses of kangaroos and overestimates the masses of ostriches. They used this formula to estimate body masses of bipedal dinosaurs. Estimated body masses of Dinosaurs (in tonnes, 1 tonne = 1000 kg) Dinosaur species 1 st method (Alexander, 1985) 2 nd method (Anderson et al., 1985) Apatosaurus 37.5 Brachiosurus Diplodocus Tyrannosaurus Anatosaurus 4.0 Triceratops 6.1 Iguanodon 5.4 Stegosaurus Masses of Some Modern Mammals (in tonnes) Here are some data on modern mammals for comparison. Males Females Blue whale African elephant Hippopotamus Black rhinoceros African buffalo 0.75 Lion Human Weighing Dinosaurs Using Scale Models (cont.) Note that the second method consistently gives smaller estimates than the first method, and in the case of Diplodocus the discrepancy is large. Which of these two methods give more accurate estimates is debatable. Although there are uncertainties about the masses of dinosaurs, from these estimates it is quite clear that the largest dinosaurs were exceedingly heavy

7 Trace Fossils of Dinosaurs Apart from body fossils, there are also trace fossils of dinosaurs, which include footprints, trackways, or even teeth marks, nests, and droppings. Trace Fossils of Dinosaurs Footprints of dinosaurs show that dinosaurs had walked with their feet directly under their body, like modern birds and mammals, and not splayed out to the sides like crocodiles. Tracks of dinosaur tails are not usually found. This suggests that dinosaur tails may not have dragged along the ground. Hence the whole body weight of the dinosaur was supported by only its legs. As large dinosaurs are heavy, one may wonder whether they got bogged down, or would sunk in soggy wetland Trace Fossils of Dinosaurs As we noted in the previous lecture, the body weight and the body mass are two different concepts. The body weight is a force W, exerted towards the center of the earth, due to earth s gravity g acting on the body mass M. The equation is W (newton) = M(kg) x g (m/s 2 ) The value of g is customarily taken as 9.8 m/s 2. For estimation purposes, one may simply take g = 10 m/s 2. We define stress s as force per unit area. Thus, when a bipedal dinosaur whose weight is W N (N stands for newton) is standing symmetrically, each foot will support ½W. The compression stress s acting over a foot or a cross-section of leg whose area is ½A cm 2 is s = (½ W) (½ A) = W/A (N/cm 2 ) 26 27

8 Theorem 2 A big dinosaur, which was α times the linear size of a similar shaped small one with the same body density, would exert α times the stress on the ground. Proof. Let W, A, s denote the body weight, the total cross-section area of the feet touching the ground, and the amount of stress exerted on the ground, respectively, of the small dinosaur; let W, A, s be those of the big dinosaur. Now (volume of big dinosaur) = α 3 (volume of small dinosaur) Proof (cont.) Since body density kept unchanged, body weight is proportional to body volume. Hence W = α 3 W On the other hand, A = α 2 A. Therefore s = W A = (α 3 W) (α 2 A) = αw A = α s The strength of ground is measured by its yield stress, which is the particular value of stress which causes it to collapse. In other words, the ground will support an animal (or a building, etc.) standing on it as long as s < y, where s is the stress the animal exerted on the ground, and y is the yield stress of the ground. Let A be the area over which weight is being applied. It is found that the yield stress follows an equation Y = a A b, where a and b are constants whose values depend on the type of the ground. For clay, its strength depends on cohesion between particles, and its yield stress follows the formula y = a A b, where b = 0, i.e. y = a, where the constant a depends on the water content. For dry sand, its supporting strength depends on friction between particles, and its yield stress follows formula y = a A 1/

9 In general, the stress exerted by an animal of weight W standing on an area A is W A, so the ground is safe if its yield stress y > W A, i.e., if Ay > W. On the other hand, the ground is ready to yield if W = Ay = A (a A b ) = aa b+1 = W max, where W max is the maximum weight of animal which can be supported on an area A of ground. 32 Theorem 3 Sand would support a big dinosaur as well as a small one. Proof. Using b = ½, the maximum weight of dinosaur which can be supported on an area A of sand is W max = aa 3/2 ( ) If this dinosaur is scaled up by a length factor of α, then its body weight is scaled up α 3 times, i.e., LHS of equation ( ) is scaled up α 3 times. On the other hand, its feet area A will be scaled up α 2 times; thus A 3/2 will be scaled up α 3 times, which means RHS of equation ( ) is scaled up α 3 times, and the equality in ( ) remains unchanged. This means that the larger dinosaur is still safe. 33 The table below shows stress exerted by different animals on the ground. Theorem 4 Clay would not support a big dinosaur as well as a small one. Proof. Using b = 0, the maximum weight of dinosaur which can be supported on an area A of sand is W max = aa ( ) If this dinosaur is scaled up by a length factor of α, then its body weight is scaled up α 3 times, i.e., LHS of equation ( ) is scaled up α 3 times. On the other hand, its feet area A will be scaled up α 2 times; thus RHS of equation ( ) is scaled up α 2 times, and the equality in ( ) will become >, which means that the larger dinosaur is not safe. 34 Mass (kg) Total foot area A (m 2 ) Ground stress W/A (N/m 2 ) Apatosaurus 35, ,000 Tyrannosaurus 7, ,000 Iguanodon 5, ,000 African elephant 4, ,000 Human ,000 35

10 Reference Mathematics Masterclasses, M. Sewell (ed.), Oxford University Press, Dynamics of Dinosaurs & Other Extinct Giants, R.M. Alexander, Columbia University Press, WWW Resources 36