Question

Materials fracture when the force per unit area, called STRESS, exceeds a critical value. In general, the critical stress depends on the direction as well as the magnitude of the force. For example, bones break differently when subjected to torsional (twisting) stress than when subjected to compressional (or squeezing) stress. Let us consider compressional stress. Wet human bone for 20-39 year olds has an ultimate compressive strength of 1.6x104N/cm2 [8]. Consider a person who falls from a height to the ground. Upon landing, the person's kinetic energy is converted to work. If the person decelerates over a distance Δ h, we can estimate the average force during the collision by Kinetic Energy- -m v'- Work Fdx FAh If a person lands stiff-legged on a hard surface and doesn't bounce, the deceleration occurs over a very short distance, Δ h ~ 1 cm Since the force is transmitted up the leg, the stress is greatest where the cross-sectional area is least, i.e., the tibia just above the ankle The following lists some different conditions that may occur during landing and physical explanations for why they make bone fracture from a fall more or less likely [8] (a) bounce [more likely. The impulse is twice as large for an elastic collision as for an inelastic collision.] (b) bent knees less likely. Bending the knees increases the time and distance over which deceleration occurs, which reduces the force.] (c) landing in loose earth or sand [less likely. This increases A h which reduces F] (d) turning the body to land on the side [less likely. This increases Δ h. It also distributes the force widely over the body (increases A) thereby reducing the stress.] So, why can a properly executed karate kick break a concrete block without fracturing bones [16]? First, bone is a very strong material Its ultimate compressive strength is approximately 40 times larger than concrete. Second, contact is made with the edge of the foot. This concentrates the force into a small area of the target and reduces the likelihood of bending a bone to the point of fracture. Third the collision with the target is essentially inelastic and extends over several milliseconds, so the peak force, though large, does not exceed the strength of the bone. [8] George B. Benedek and Felix M. H. Villars, PHYSICS WITH ILLUSTRATIVE EXAMPLES FROM MEDICINE AND BIOLOGY, Vol. 1. (Menlo Park: Addison-Wesley Publishing Co., 1974). [16] S. R. Wilk, R. E. McNair, and M. S. Feld, AM J. PHYS. 51, 783 (1983). If A (cross-sectional area of the tibia) ~ 2.7 cm, compute how far a 79 kg person can fall and land stiff-legged on both legs without breaking a bone. Assume F is split evenly between two legs. (Hint: F < 2ơA) cm

Answer 1

Suppose person can fall maximum height h

Then K.E. = mgh

And

FΔh=mgh

Or

F = mgh/Δh

Here m=79kg g=9.8 m/s² Δh=1cm

Hence F = 774.2h

Now F should be less than Or equal to 2σA

F<=2σA

Or

774.2h <= 2σΑ

Or

h <= 2σΑ/774.2

here σ=1.6×10⁴ N/cm^{​​​​​​2} A=2.7 cm²

Hence

h <= 111.6 cm

Person can fall a maximum height of 111.6 cm