To solve this, let's analyze the forces and energies involved. We're interested in determining the maximum height from which the woman can jump without her femur fracturing. Given that her gravitational potential energy on landing converts entirely into strain energy in the femur, we need to equate these two energies to find the maximum height.

Given dataTo solve this, let's analyze the forces and energies involved. We're interested in determining the maximum height from which the woman can jump without her femur fracturing. Given that her gravitational potential energy on landing converts entirely into strain energy in the femur, we need to equate these two energies to find the maximum height.

Given

• Mass of the woman, $m = 73.0 \, \text{kg}$
• Length of the femur, $l = 77.0 \, \text{cm} = 0.77 \, \text{m}$
• Cross-sectional area of the femur, $A = 4.40 \, \text{cm}^2 = 4.40 \times 10^{-4} \, \text{m}^2$
• Young’s modulus of bone, $Y = 1.35 \times 10^{10} \, \text{N/m}^2$
• Maximum stress (rupture stress) of bone, $\sigma_{\text{max}} = 8.15 \times 10^7 \, \text{N/m}^2$
• Gravitational acceleration, $g = 9.81 \, \text{m/s}^2$

Step-by-Step Solution:

1. Calculate the maximum force the femur can withstand without breaking:

   Using the definition of stress: $\sigma_{\text{max}} = \frac{F_{\text{max}}}{A}$ Solving for $F_{\text{max}}$: $F_{\text{max}} = \sigma_{\text{max}} \times A$ Substituting the values: $F_{\text{max}} = (8.15 \times 10^7 \, \text{N/m}^2) \times (4.40 \times 10^{-4} \, \text{m}^2)$ $F_{\text{max}} = 35860 \, \text{N}$

2. Calculate the maximum strain energy stored in the femur:

   The strain energy $U$ stored in the femur is given by: $U = \frac{F_{\text{max}}^2 \cdot l}{2 \cdot A \cdot Y}$ Substituting in the values: $U = \frac{(35860 \, \text{N})^2 \times 0.77 \, \text{m}}{2 \times (4.40 \times 10^{-4} \, \text{m}^2) \times (1.35 \times 10^{10} \, \text{N/m}^2)}$ $U = \frac{1.285 \times 10^9}{1.188 \times 10^7}$ $U \approx 108.2 \, \text{J}$

3. Calculate the maximum height:

   The gravitational potential energy at the height $h$ is given by: $U = m \cdot g \cdot h$ Solving for $h$: $h = \frac{U}{m \cdot g}$ Substituting in the values: $h = \frac{108.2 \, \text{J}}{73.0 \, \text{kg} \times 9.81 \, \text{m/s}^2}$ $h \approx 0.151 \, \text{m} = 15.1 \, \text{cm}$

Answer:

The maximum height from which the woman can jump onto one heel without fracturing her femur is 15.1 cm.

Do you want further clarification on any step? Here are some additional questions you might explore:

1. What factors affect the Young's modulus of bone and how does it vary among individuals?
2. How would a decrease in femur cross-sectional area impact the maximum height?
3. How does Young's modulus affect the energy absorption capacity of bones in general?
4. If the woman bent her knees during landing, how would this alter the results?
5. How do different types of materials compare to bone in terms of stress and strain capacity?

Tip: When solving for maximum allowable stresses in structures or biological materials, always double-check units to ensure that force, area, and modulus are consistently used.