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 3 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 61 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.
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To determine the maximum distance through which the man's center of gravity can fall without breaking a bone, we need to find the maximum amount of potential energy that can be absorbed by the thighbone without exceeding its breaking point.
First, we find the total potential energy when the person falls:
Potential Energy (PE) = mass * gravity * height
The mass of the person is given as 61 kg, and the acceleration due to gravity is approximately 9.8 m/s^2. The height can be calculated using the maximum distance the center of gravity can fall without breaking the bone. Let's denote it as "h."
PE = 61 kg * 9.8 m/s^2 * h
Now, let's calculate the maximum amount of potential energy that can be absorbed by the thighbone without breaking it:
Maximum Potential Energy (MPE) = maximum force * maximum distance
The maximum force that can be applied to the thighbone without breaking is given as 7.6 x 10^4 N. We need to find the maximum distance "d" that can be traveled without exceeding this force.
MPE = 7.6 x 10^4 N * d
Since potential energy is equivalent to work done, we can equate the potential energy to the maximum potential energy:
PE = MPE
61 kg * 9.8 m/s^2 * h = 7.6 x 10^4 N * d
Simplifying the equation:
h = (7.6 x 10^4 N * d) / (61 kg * 9.8 m/s^2)
Now, let's substitute the given values for the minimum effective cross-sectional area of the bone (A), the length of the bone (L), and Young’s modulus (Y) into the formula for the force experienced by the bone (F).
F = YAΔL / L
Where:
- Y is Young's modulus (9.4 x 10^9 N/m^2)
- A is the minimum effective cross-sectional area of the bone (3 x 10^-4 m^2)
- ΔL is the change in length of the bone (which we can assume to be equal to the distance traveled "d")
- L is the length of the bone (0.54 m)
Substituting these values in:
7.6 x 10^4 N = (9.4 x 10^9 N/m^2) * (3 x 10^-4 m^2) * d / (0.54 m)
Now, solve the equation for "d":
d = (7.6 x 10^4 N * 0.54 m) / ((9.4 x 10^9 N/m^2) * (3 x 10^-4 m^2))
Calculate the value of "d" to find the maximum distance through which the center of gravity can fall without breaking a bone.