Problem 63 - College Physics

Question

\(\bullet\) \(\bullet\) Stress on the shinbone. The compressive strength of our bones is important in everyday life. Young's modulus for bone is approximately 14 GPa. Bone can take only about a 1.0\(\%\) change in its length before fracturing. If Hooke's law were to hold up to fracture: (a) What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 3.0 \(\mathrm{cm}^{2} .\) . This is approximately the cross-sectional area of a tibia, or shinbone, at its narrowest point.) (b) Estimate the maximum height from which a 70 \(\mathrm{kg}\) man can jump and not fracture the tibia. Take the time between when he first touches the floor and when he has stopped to be \(0.030 \mathrm{s},\) and assume that the stress is distributed equally between his legs.

Step-by-Step Solution

Answer

(a) Maximum force is 42,000 N. (b) Maximum height is approximately 0.36 m.

Step 1: Identify the formula

To find the maximum force that can be applied to the bone, we use the formula derived from Hooke's Law and stress equation, which is \( E = \frac{\sigma}{\epsilon} \), where \( E \) is Young's modulus, \( \sigma \) is stress, and \( \epsilon \) is strain (change in length/original length). Given \( E = 14 \text{ GPa} = 14 \times 10^9 \text{ N/m}^2 \) and \( \epsilon = 1.0\% = 0.01 \).

Step 2: Calculate maximum stress

Using the formula for stress \( \sigma = E \times \epsilon \), we find the maximum stress the bone can endure before fracturing: \( \sigma = 14 \times 10^9 \text{ N/m}^2 \times 0.01 = 1.4 \times 10^8 \text{ N/m}^2 \).

Step 3: Calculate maximum force

The force can be calculated using the formula \( F = \sigma \times A \), where \( A \) is the cross-sectional area: \( 3.0 \text{ cm}^2 = 3.0 \times 10^{-4} \text{ m}^2 \). Thus, \( F = 1.4 \times 10^8 \text{ N/m}^2 \times 3.0 \times 10^{-4} \text{ m}^2 = 4.2 \times 10^4 \text{ N} \).

Step 4: Determine deceleration during impact

The average force during deceleration can be calculated using the impulse-momentum theorem. Given the man's mass \( m = 70 \text{ kg} \), initial velocity \( v = 0 \text{ m/s} \) and time of impact \( \Delta t = 0.030 \text{ s} \). The change in momentum equals force time time: \( F = m \times a \), and \( a = \Delta v / \Delta t \).

Step 5: Relate force to jump height

Considering the force required to stop the man, set \( F_{max} = F_{impact} \), and find \( v \) using energy conservation: \( \frac{1}{2} m v^2 = mgh \). Solving for height \( h = \frac{v^2}{2g} \). Substituting \( F = 70\times a \) and \( F = F_{max}/2 \) (force distributed over two legs), and solving for \( a \), then use \( v = a \times \Delta t \) to find maximum jumping height \( h \).

Key Concepts

Young's Modulus

Young's modulus is a measure of the stiffness of a material. It describes how much a material will deform under stress. It is defined as the ratio of stress to strain in the linear elastic region of a material. In simple terms, Young's modulus tells us how much a material will stretch or compress when force is applied, and it is a critical factor when analyzing bone mechanics. For bones, this measure helps us understand their rigidity and resistance to being deformed.
Young's modulus for bone is around 14 GPa (gigapascals), which indicates that bone is quite strong and resilient. The higher the Young’s modulus, the stiffer the material. For example, steel has a much higher Young's modulus than bone, meaning it is much less flexible under similar stress conditions.
**Key Points:**

• Young's modulus defines stiffness.
• It helps predict deformation under stress.
• For bone, Young's modulus is about 14 GPa.

Hooke's Law

Hooke's Law is a fundamental principle that defines the relationship between force and deformation in elastic materials. It states that the force needed to extend or compress a spring by a distance is proportional to that distance. In mathematical terms, it can be expressed as:$$ F = k imes x $$where \( F \) is the force applied, \( k \) is the spring constant (or stiffness), and \( x \) is the change in length.
In the context of bones, Hooke's Law helps us calculate how much a bone will flex under a certain force, assuming the deformation is within the elastic limit. This law is applicable up to the point where materials no longer respond elastically and experience permanent deformation.
**Key Points:**

• Hooke's Law describes force and deformation relation.
• It applies to the elastic region of materials.
• Useful for predicting bone flex under force.

Compressive Strength

Compressive strength refers to the ability of a material to withstand loads that tend to reduce size. In bones, compressive strength is crucial as it relates to how much force they can endure without cracking or shattering. It determines the maximum force that bone structures like the tibia can withstand.
Bones are designed by nature to endure a significant amount of compressive force, which is critical for supporting body weight during activities such as standing, walking, and jumping. The compressive strength of bones is inherently linked to their mineral content and structural properties.
**Key Points:**

• Compressive strength withstands force reducing size.
• Essential for load-bearing capacity in bones.
• Linked to bone structure and mineral composition.

Strain and Stress

Strain and stress are concepts used to describe how materials react to forces. Stress is the force applied to a material divided by the area over which it is applied, typically measured in pascals (Pa). Strain, on the other hand, is the deformation experienced by the material as a result of applied stress, measured as a percentage or a dimensionless ratio.
In the context of bones, stress and strain help quantify how much pressure they experience during activities like jumping or lifting weights, and how this pressure affects their shape and structure. Understanding the relationship between stress and strain is essential to ensuring that the force does not surpass the bone's ability to withstand it without fracturing.
**Key Points:**

• Stress is force per unit area.
• Strain is deformation due to stress.
• Key for analyzing bone pressure and deformation.

Impulse-Momentum Theorem

The impulse-momentum theorem relates the change in momentum of an object to the impulse applied to it. This is represented mathematically by the formula:$$ ext{Impulse} = ext{Change in Momentum} = F \times \Delta t $$where \( F \) is the force applied and \( \Delta t \) is the time period over which the force is applied.
In practical situations, this theorem helps in calculating the forces involved during collisions or impacts, such as determining the forces on a bone when a person lands after a jump. The time period over which the force acts is critical, as longer contact times can reduce the maximum force experienced by the tibia, reducing the risk of fracture.
**Key Points:**

• Impulse equals change in momentum.
• Crucial for analyzing forces during impacts.
• Helps assess risk of bone fracture.