Question

1.) The femur of an elephant is about 90 cm long and 15 cm in diameter. This is a scaling problem. The largest dinosaur probably weighed about 10 times as much as a large elephant. In this problem we will be discussing scaling. For reference, areas scale as length squared(A=L^2 for a square and A=3.14*r^2 for a circle) and volume scales as length cubed. To describe the size of a dinosaur's femur compared to that of an elephant's, the dinosaur's femur appears to be 3 times as wide and one and a half times as tall.

a.) How do you expect the mass of an animal to scale with the length of typical bone? Use this to estimate the length of the dinosaur's femur.

b.) How do you expect the strength of a bone to scale with its width?(We want a uniform strain (delta L/L) for all animals.) Use this to estimate the width of the dinosaur's femur.

c.) Are your answers to a and b consistent with the description of the size of a dinosaur's femur stated above? Why or why not?

d.) Based on this, briefly argue why whales (2.5 times larger than the largest dinosaurs) are not land animals.

Answer 1

a)

Mass of dino/mass of elephant = 10

size of femur of dino = 1.5 * 90 = 135 cm

hence (135/90)^n = 10

n = 5.67

hence mass of animal scales as length of bone raised to the power of 5.67 (approx.)

b)

as the strain is proportional to the stress hence and strength is scales as area of the bone or it can be represented as strength scales as width squared.

c)

Answers in a and b are not consistent exactly with real data because as the size increases, due to internal structure of the body and bones strength can't be scaled accordingly hence the results are marginally inconsistent.

d)
whale are very large sized animals hence have very large amount of mass which makes it difficult for their body (bones ) to support the entire structure against the gravity while standing on and, whereas under water due to buyont force the apparent wieght is less than actual wieght and the bone structure of whale is able to support that huge amount of mass in reduced wieght condition