**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%