Consider a human femur of length L = 469 mm. Model the bone as a hollow cylinder, with inner radius Ra = 4.17 mm and outer radius Rb = 12.51 mm. If a 74 kg person's weight is supported by one leg, calculate the stress on the femur; express as percentage of the UCS. Ignore the weight of the leg.
To calculate the stress on the femur, we need to first determine the force exerted on the bone by the person's weight. We can do this using Newton's second law of motion, which states that force (F) is equal to mass (m) multiplied by acceleration due to gravity (g).
Given that the mass of the person is 74 kg and acceleration due to gravity is approximately 9.8 m/s², we can calculate the force as follows:
F = m * g
F = 74 kg * 9.8 m/s²
F = 725.2 N
Now, let's consider the cross-sectional area of the femur. Since it is modeled as a hollow cylinder, we can calculate the difference in the areas between the outer (A_outer) and inner (A_inner) radii as follows:
A_outer = π * Rb²
A_inner = π * Ra²
To find the cross-sectional area of the bone (A_bone), we subtract the inner area from the outer area:
A_bone = A_outer - A_inner
To calculate the stress on the femur, we divide the force by the cross-sectional area:
Stress = F / A_bone
Now, let's calculate the stress as a percentage of the ultimate compressive strength (UCS). The UCS is a measure of the maximum stress that a material can withstand before failure.
Finally, we can express the stress as a percentage of the UCS using the following formula:
Stress_percentage = (Stress / UCS) * 100
Please provide the value of the ultimate compressive strength (UCS) to complete the calculation.