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 7.6 x104 N. The minimum effective cross-sectional area of the bone is 4 x10-4 m2, its length is 0.52 m, and Young’s modulus is Y=9.4x109 N/m2. The mass of the man is 66 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.

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 7.6 x104 N. The minimum effective cross-sectional area of the bone is 4 x10-4 m2, its length is 0.52 m, and Young’s modulus is Y=9.4x109 N/m2. The mass of the man is 66 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.

Answer 1

To find the maximum distance through which the center of gravity can fall without breaking a bone, we need to calculate the maximum amount of potential energy that can be absorbed by the bone without exceeding its maximum applied force of compression.

The potential energy of an object is given by the equation:

PE = mgh

where PE is the potential energy, m is the mass, g is the acceleration due to gravity, and h is the height through which the object falls.

In this case, we want to find the maximum value of h without exceeding the maximum force that the bone can handle. The force applied to the bone can be calculated using Hooke's law:

F = kx

where F is the force, k is the spring constant (which can be calculated using Young's modulus), and x is the displacement or compression of the bone.

First, let's calculate the maximum force that the bone can handle:

F = 7.6 x 10^4 N

Next, let's calculate the spring constant (k) using Young's modulus:

k = YA / L

where Y is Young's modulus, A is the cross-sectional area of the bone, and L is the length of the bone.

k = (9.4 x 10^9 N/m^2) * (4 x 10^-4 m^2) / 0.52 m

Now, let's rearrange Hooke's law to solve for x:

x = F / k

Finally, we can calculate the maximum value of h by equating the potential energy to the work done by the bone:

PE = F * x = mg * h

h = (F * x) / (m * g)

Substituting the given values:

h = (7.6 x 10^4 N) * (F / k) / (66 kg * 9.8 m/s^2)

After performing the calculations, the maximum distance through which the center of gravity can fall without breaking a bone will be obtained.