Problem 92 - Physics

Question

The maximum applied force of compression that one man's thighbone can endure without breaking is \(7.0 \times 10^{4} \mathrm{~N}\). The minimum effective cross-sectional area of the bone is \(4.0 \times 10^{-4} \mathrm{~m}^{2}\), and its length is \(0.55 \mathrm{~m}\). The mass of the man is \(65 \mathrm{~kg}\). 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 left of gravity can fall without his breaking a bone.

Step-by-Step Solution

Answer

The maximum fall distance without breaking the bone is approximately 1.82 cm.

Step 1: Identify the Key Values

Start by identifying the critical values from the problem: maximum compressive force (\( F_{max} = 7.0 \times 10^4 \, \mathrm{N} \)), cross-sectional area (\( A = 4.0 \times 10^{-4} \, \mathrm{m^2} \)), length of the bone (\( L = 0.55 \, \mathrm{m} \)), and the man's mass (\( m = 65 \, \mathrm{kg} \)).

Step 2: Calculate the Force Exerted by Gravity During Fall

The force exerted by gravity on the man is given by \( F_g = m \cdot g \), where \( g = 9.8 \, \mathrm{m/s^2} \) is the acceleration due to gravity. Therefore, \( F_g = 65 \, \mathrm{kg} \times 9.8 \, \mathrm{m/s^2} = 637 \, \mathrm{N} \).

Step 3: Determine the Work Done to Stop the Fall

The work done by the bone to bring the man to a stop is equal to the work done by the force due to gravity over the distance \( d \). The work done, when he falls and comes to a stop with a compressive force, is \( W = F_{max} \times d \).

Step 4: Equate Work Done to Change in Potential Energy

The potential energy lost by the man when he falls is \( \, \mathrm{PE} = m \cdot g \cdot h \). If the maximum strain energy occurs when \( h = d \), equate it with the work done: \( m \cdot g \cdot d = \frac{1}{2} F_{max} \times d \).

Step 5: Solve for Maximum Safe Falling Distance

Isolate \( d \) from the equation in Step 4: \( m \cdot g \cdot d = \frac{1}{2} F_{max} \times d \) simplifies to \( d = \frac{2m\cdot g}{F_{max}} \). Substituting, \( d = \frac{2 \times 65 \, \mathrm{kg} \times 9.8 \, \mathrm{m/s^2}}{7.0 \times 10^4 \, \mathrm{N}} \approx 0.0182 \, \mathrm{m} \).

Step 6: Conclusion

The maximum distance through which the man's center of gravity can fall without breaking a bone is approximately \(0.0182\) meters, or \(1.82\) centimeters.

Key Concepts

Compressive Force

When we talk about compressive force, we're referring to the force applied to compress or shorten something. In this problem, we're dealing with the human thighbone, a crucial part of our skeletal structure.

Compressive force is essentially pushing forces coming from different sides towards the center. Imagine squeezing a sponge; that's a simple form of compression.

In physics, especially biomechanics, understanding compressive force helps us see how much stress bones or materials can handle before breaking. In the context of our problem, the thighbone has a maximum compressive force it can withstand, given as \(7.0 \times 10^4 \; \mathrm{N}\). This means any force above this threshold will cause the bone to break.

Every material, including human bones, comes with its limits, so engineers and scientists measure these to ensure safety. Factors like material type, area, and conditions affect how compressive force is experienced. That's why the bone's cross-sectional area and the force due to gravity are significant here.

Potential Energy

Imagine holding a book high above a table. At that height, the book has potential energy because of its position. Potential energy is all about the "potential" to do work due to position or height.

In our problem, the man's potential energy depends on how high his center of gravity is above the ground before he falls. Mathematically, potential energy is expressed as \( \, \mathrm{PE} = m \cdot g \cdot h \), where \(m\) is mass, \(g\) is gravitational acceleration, and \(h\) is height.

When he falls, this potential energy converts into kinetic energy until he stops. The energy needs to be absorbed, in this case, by the thighbone, which can only handle so much due to its compressive force limit.

This conversion explains why high falls are risky: the greater the height, the more potential energy, and thus more strain on objects or bones absorbing the fall.

Work-Energy Principle

The work-energy principle connects the concepts of work and energy, stating that the work done on an object equals the change in its kinetic energy. In layman's terms, it describes how work and energy interact.

For the problem at hand, when the man falls and comes to a stop, his thighbone must do the work to exert a force that negates his kinetic energy built up during the fall. If the force exceeds the bone's compressive capacity, it results in breaking.

We applied the principle to find the maximum distance he could fall without breaking his bone. By equating the work done (force times distance) to the change in potential energy (mass times gravity times height), we see how much energy the bone has to absorb when he lands.

This is why minimizing "\(d\)", the distance of the fall, ensures his bones remain intact, as the energy and therefore work required to stop the fall remain within safe limits.