What is the qualitative relationship between the bone diameter and size of the a

Question

What is the qualitative relationship between the bone diameter and size of the animal from where it comes from? For example, the an elephant's femur is 50 cm long and diameter is 10 cm. What is the relationship here for elephant? And what is the relationship for other animals? What is the qualitative relationship between the bone diameter and size of the animal from where it comes from? For example, the an elephant's femur is 50 cm long and diameter is 10 cm. What is the relationship here for elephant? And what is the relationship for other animals?

Explanation / Answer

Scaling relationships between skeletal measurements and body mass in birds and mammals is vital in predicting the body mass of animals.

Let us consider an animal with single bone.

The force Mg due to the animal's mass M acting on a bone of diameter D and cross-sectional area A loads it in axial compression s

s = (Mg/A) is proportional to (M/D2) (1)

(g is the gravitational acceleration). Suppose our animal grows while keeping its shape unaltered. Constant shape means M is directly proportional to D3, because D grows just like any other linear dimension of the animal. From equation (1), a steadily growing stress s proportional to D results, and the part fails when s exceeds some critical value. To achieve a constant stress, the part should scale as M directly proportional to D. In the long run this will not do – the part would soon be too thick and heavy for any use.

Even more serious a limitation to animal size is the relative weakening of muscle strength. Stress in part's muscles obeys equation (1), too, but the relative size of muscles is more restricted, for the following reason: muscle mass Mm, like most morphological and physiological variables in biology, has so-called allometric dependence on body mass M

Mm = aMb (2)

where a, b are parameters to be determined from theory or measurements. Now, the ratio Mm/M is about 0.4 irrespective of body size , so a 10 kg animal has 4 kg as Mm . According to the D is directly proportional to M1/2 law (b = 1.5) a 60 kg animal of the same shape would be nothing but muscle! A much weaker allometric dependence, b = 1.1, produces a large elephant (weighing 10 tons) of 80% muscle-bone would fill the rest of the body.

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