Larger animals have sturdier bones than smaller animals

Question:

Larger animals have sturdier bones than smaller animals. A mouse’s skeleton is only a few percent of its body weight, compared to 16% for an elephant. To see why this must be so, recall, that the stress on the femur for a man standing on one leg is 1.4% of the bone’s tensile strength. Suppose we scale this man up by a factor of 10 in all dimensions, keeping the same body proportions. (Assume that a 70 person has a femur with a cross-section area (of the cortical bone) of, a typical value.)

A. Both the inside and outside diameter of the femur, the region of cortical bone, will increase by a factor of 10. What will be the new cross-section area?

B. The man’s body will increase by a factor of 10 in each dimension. What will be his new mass?

C. If the scaled-up man now stands on one leg, what fraction of the tensile strength is the stress on the femur?

Answer:

A. The geometry of scaling is:
when the Length (L) factor increases by 10.
the Area (L²) increases by 10² = 100

B. The volume (L³) increases by 10³ = 1000
The mass is proportional to Volume so it increases by 1000

C. Stress = Force/Area
Force in the “scaled up man” is his weight which is proportional to his mass
The area in the case of the “scaled up man” is the cross-sectional area of his femur.
so in the 10X “scaled up” man his:
Weight is 1000 times that of the original man
The area is 100 times that of the original man
Stress in femur = 1000/100 = 10 x that of the original man = 14%

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