Problem 11 - College Physics

Question

\(\bullet\) \(\bullet\) The effect of jogging on the knees. High-impact activities such as jogging can cause considerable damage to the cartilage at the knee joints. Peak loads on each knee can be eight times body weight during jogging. The bones at the knee are separated by cartilage called the medial and lateral meniscus. Although it varies considerably, the force at impact acts over approximately 10 \(\mathrm{cm}^{2}\) of this cartilage. Human cartilage has a Young's modulus of about 24 MPa (although that also varies). (a) By what percent does the peak load impact of jogging compress the knee cartilage of a 75 kg person? (b) What would be the percentage for a lower-impact activity, such as power walking, for which the peak load is about four times body weight?

Step-by-Step Solution

Answer

24.5% compression for jogging; 12.25% for walking.

Step 1: Understand the Problem and Formulate the Plan

To solve this problem, we'll use the formula for stress, which is force per unit area, and the formula for strain, which is stress divided by Young's modulus. Then, we'll find how much the cartilage compresses by multiplying the original thickness of the cartilage by the strain. Finally, we'll convert this compression into a percentage.

Step 2: Calculate Force Due to Jogging and Walking

During jogging, the peak load on each knee is 8 times the body weight, and during power walking, it is 4 times the body weight. Given a body weight of 75 kg, calculate these forces using the gravitational constant, \[\text{Force}_{\text{jogging}} = 8 \times 75 \times 9.8 = 5880 \, \text{N} \]\[\text{Force}_{\text{walking}} = 4 \times 75 \times 9.8 = 2940 \, \text{N} \]

Step 3: Determine the Stress on Knee Cartilage

Stress is defined as force divided by the area over which it acts. The area of impact is approximately 10 \, \text{cm}^2 or 0.001 \, \text{m}^2.For jogging:\[\text{Stress}_{\text{jogging}} = \frac{5880 \, \text{N}}{0.001 \, \text{m}^2} = 5.88 \times 10^6 \, \text{Pa}\]For walking:\[\text{Stress}_{\text{walking}} = \frac{2940 \, \text{N}}{0.001 \, \text{m}^2} = 2.94 \times 10^6 \, \text{Pa}\]

Step 4: Calculate the Strain on Knee Cartilage

Strain is the stress divided by Young's modulus (24 MPa = 24 \times 10^6 \, \text{Pa}).For jogging:\[\text{Strain}_{\text{jogging}} = \frac{5.88 \times 10^6 \, \text{Pa}}{24 \times 10^6 \, \text{Pa}} = 0.245\]For walking:\[\text{Strain}_{\text{walking}} = \frac{2.94 \times 10^6 \, \text{Pa}}{24 \times 10^6 \, \text{Pa}} = 0.1225\]

Step 5: Convert Strain to Percentage Compression

Strain as a decimal value can be converted into a percentage by multiplying by 100.For jogging:\[\text{Compression Percentage}_{\text{jogging}} = 0.245 \times 100 = 24.5\%\]For walking:\[\text{Compression Percentage}_{\text{walking}} = 0.1225 \times 100 = 12.25\%\]

Key Concepts

Stress-Strain Relationship

The stress-strain relationship is a fundamental concept in understanding material behavior under force. Stress is defined as the force applied per unit area on a material, calculated using the formula, \( \text{Stress} = \frac{\text{Force}}{\text{Area}} \). Strain, on the other hand, is a measure of deformation and is obtained by dividing stress by the material's Young's modulus, \( \text{Strain} = \frac{\text{Stress}}{\text{Young's Modulus}} \). This relationship helps to predict how materials like cartilage will deform under various loads.
In the context of knee cartilage, when a person jogs or engages in other activities, forces exerted on the knee joint cause stress to build up in the cartilage. The higher the impact, such as in jogging, the greater the stress compared to lower-impact activities like walking. This in turn results in a corresponding strain or change in the shape or volume of the cartilage.
Understanding this relationship is crucial for predicting how different physical activities might affect the health and longevity of knee joints, allowing individuals to adjust their activity level to avoid potential damage.

Biomechanics

Biomechanics is the study of movement in biological systems, applying principles from mechanics. It helps us understand how forces affect body movements and the structural integrity of tissues like muscles and cartilage.
Using biomechanics, we can analyze how jogging affects knee joint health. When a person jogs, each step places the knee cartilage under significant stress due to the forces involved. This understanding helps us gauge the limits cartilage can withstand before damage and pain might occur. Biomechanics thus not only aids in injury prevention but also in the development of better rehabilitation strategies and athletic training regimens.
In our problem, the biomechanics concept is applied by calculating the forces involved in jogging and walking. By doing so, we can differentiate the impact level between activities and better understand safe thresholds for various movements and exercises.

Knee Cartilage

Knee cartilage plays a pivotal role in joint health and movement. It acts as a cushion between the bones, absorbing impact and ensuring smooth, pain-free movement.
In high-impact activities like jogging, knee cartilage is subjected to intense forces. Each step can exert peak loads up to eight times the body weight on the knees, which is significant stress for the cartilage to manage. Over time, this could lead to wear and tear if not managed appropriately.
Understanding the role of knee cartilage and how it responds to forces is crucial. Knowing the Young's modulus of cartilage helps predict its response to stress, aiding individuals to make informed decisions about activity levels to maintain joint health and functionality.