Depending on how you fall, you can break a bone easily. The severity of the brea

Question

Depending on how you fall, you can break a bone easily. The severity of the break depends on how much energy the bone absorbs in the accident, and to evaluate this let us treat the bone as an ideal spring. The maximum applied force of compression that one man’s thighbone can endure without breaking is 6.8 x104 N. The minimum effective cross-sectional area of the bone is 5 x10-4 m2, its length is 0.54 m, and Young’s modulus is Y=9.4x109 N/m2. The mass of the man is 69 kg. He falls straight down without rotating, strikes the ground stiff-legged on one foot, and comes to a halt without rotating. To see that it is easy to break a thighbone when falling in this fashion, find the maximum distance through which his center of gravity can fall without his breaking a bone.

Explanation / Answer

Young's modulus, E = Fl/(xA), so F = (EA/l)x
For a spring F = kx ,where k is the spring constant. So by comparing, (EA/l) corresponds to the spring constant k.

k = EA/l = (9.4*109 * 5*10-4) / 0.54 = 8.704*106 N/m

The amount by which the bone is compressed, x, when the breaking force is reached.
F = kx
x = F/k = (6.8*104)/(8.704*106) = 7.813*10-3 m

The energy stored in the bone when compressed to its breaking point.
Energy = (1/2)kx2 = (1/2)(8.704*106)(7.813*10-3)2 = 265.64 J

The energy will come from the gravitational potential energy (mgh) turned to kinetic energy which is then transferred to the bone.
mgh = 265.64
h = 265.64/(mg) = 265.64/(69*9.81) = 0.39 m

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