Problem 10 - College Physics

Question

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 \(75 \mathrm{~kg}\) 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

Knee cartilage compresses by 24.5% during jogging and 12.25% during power walking.

Step 1: Calculate Body Weight Force

First, calculate the body weight force using the gravitational constant. The force due to gravity on a person of mass 75 kg is calculated as:\[ F = mg = 75 \times 9.8 = 735 \text{ N} \]

Step 2: Calculate Peak Load During Jogging

For jogging, the peak load on each knee can be eight times the body weight. Thus, the peak load is:\[ F_j = 8 \times 735 = 5880 \text{ N} \]

Step 3: Calculate Stress on Cartilage

Calculate the stress by dividing the force by the area. The area over which this force acts is \(10 \text{ cm}^2 = 0.001 \text{ m}^2\).So, the stress \( \sigma \) is:\[ \sigma = \frac{F_j}{A} = \frac{5880}{0.001} = 5.88 \times 10^6 \text{ N/m}^2 = 5.88 \text{ MPa} \]

Step 4: Calculate Strain Using Young's Modulus

Use Young's Modulus to find the strain. Young's Modulus \( E \) is given as 24 MPa. Relate stress and strain using:\[ E = \frac{\sigma}{\varepsilon} \rightarrow \varepsilon = \frac{\sigma}{E} \]Substitute the values:\[ \varepsilon = \frac{5.88 \text{ MPa}}{24 \text{ MPa}} = 0.245 \]

Step 5: Determine Percentage Compression for Jogging

Convert the strain into percentage by multiplying by 100:\[ \text{Percent Compression} = \varepsilon \times 100 = 0.245 \times 100 = 24.5\% \]

Step 6: Calculate Peak Load for Power Walking

For power walking, the peak load is four times the body weight. Thus, the peak load is:\[ F_w = 4 \times 735 = 2940 \text{ N} \]

Step 7: Calculate Stress for Power Walking

Use the peak load for power walking to find stress:\[ \sigma = \frac{F_w}{A} = \frac{2940}{0.001} = 2.94 \times 10^6 \text{ N/m}^2 = 2.94 \text{ MPa} \]

Step 8: Calculate Strain for Power Walking

Use Young's Modulus to calculate strain as before:\[ \varepsilon = \frac{\sigma}{E} = \frac{2.94 \text{ MPa}}{24 \text{ MPa}} = 0.1225 \]

Step 9: Determine Percentage Compression for Power Walking

Convert strain into percentage:\[ \text{Percent Compression} = \varepsilon \times 100 = 0.1225 \times 100 = 12.25\% \]

Key Concepts

Knee Cartilage Compression

Knee cartilage plays a crucial role in facilitating smooth joint movement and absorbing impact. When we jog or engage in high-impact activities, our knee cartilage must endure significant loads. Imagine it as a cushion between the bones, helping to reduce friction and wear. This cushion gets compressed under pressure. The percent by which knee cartilage compresses during these activities can be an essential factor in understanding joint health. Compression is calculated using the strain in the cartilage which can indicate how much stress the cartilage can bear without deteriorating.

Stress and Strain Calculations

Stress and strain are pivotal in analyzing knee cartilage compression. **Stress** is the force applied on an area, and it is measured in Pascals (Pa). In terms of our example, it is the force exerted by body weight during jogging, which acts on the knee cartilage area. The formula is: \[ \sigma = \frac{F}{A} \] where \( F \) is the force in newtons, and \( A \) is the area in square meters. On the other hand, **strain** is the deformation caused due to stress, indicating the extent to which the cartilage stretches or compresses. This is: \[ \varepsilon = \frac{\sigma}{E} \] Here, \( \varepsilon \) is the strain, \( E \) is Young's Modulus, and \( \sigma \) is stress. Calculating these values allows us to compute the percent change in cartilage length, which shows how much the cartilage compresses under stress.

Impact of Jogging on Joints

Jogging is a popular activity for its cardiovascular benefits, yet it comes with considerable impact on the joints, particularly the knees. When jogging, the impact on knee joints can be as much as eight times the body weight. This substantial force impacts the cartilage, increasing wear over time. Prolonged exposure to these forces can potentially lead to cartilage degradation and conditions such as osteoarthritis. However, understanding these impacts can help us choose suitable footwear and running surfaces, or adapt exercises to lessen the load on joints.

Biomechanics of Human Motion

Biomechanics helps to understand human movement intricately, focusing on forces and their effects on our bodies. By examining biomechanics, especially in activities like jogging, we gain insights into how muscles, bones, tendons, and ligaments work together in harmony. Knowledge of biomechanics aids in optimizing performance and safety in physical activities, allowing adjustments to movement techniques to minimize injury risks and improve efficiency. For instance, recognizing how knee joints respond to various forces can guide the development of effective physical therapy and injury prevention programs, enhancing overall joint health.