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\(\bullet\) \(\bullet\) 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?

Short Answer

Expert verified
24.5% compression for jogging; 12.25% for walking.

Step by step solution

01

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.
02

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, \[\text{Force}_{\text{jogging}} = 8 \times 75 \times 9.8 = 5880 \, \text{N} \]\[\text{Force}_{\text{walking}} = 4 \times 75 \times 9.8 = 2940 \, \text{N} \]
03

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:\[\text{Stress}_{\text{walking}} = \frac{2940 \, \text{N}}{0.001 \, \text{m}^2} = 2.94 \times 10^6 \, \text{Pa}\]
04

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:\[\text{Strain}_{\text{walking}} = \frac{2.94 \times 10^6 \, \text{Pa}}{24 \times 10^6 \, \text{Pa}} = 0.1225\]
05

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:\[\text{Compression Percentage}_{\text{walking}} = 0.1225 \times 100 = 12.25\%\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

\(\bullet\) A mass of 0.20 \(\mathrm{kg}\) on the end of a spring oscillates with a period of 0.45 s and an amplitude of 0.15 \(\mathrm{m}\) . Find (a) the velocity when it passes the equilibrium point, (b) the total energy of the system, and (c) the equation describing the motion of the mass, assuming that \(x\) was a maximum at time \(t=0\) .

\(\bullet\) \(\bullet\) An object suspended from a spring vibrates with simple harmonic motion. At an instant when the displacement of the object is equal to one-half the amplitude, what fraction of the total energy of the system is kinetic and what fraction is potential?

\(\bullet\) Find the period, frequency, and angular frequency of (a) the second hand and (b) the minute hand of a wall clock.

\(\bullet\) You've made a simple pendulum with a length of 1.55 \(\mathrm{m}\) , and you also have a (very light) spring with force constant 2.45 \(\mathrm{N} / \mathrm{m} .\) What mass should you add to the spring so that its period will be the same as that of your pendulum?

\(\bullet\) Effect of diving on blood. It is reasonable to assume that the bulk modulus of blood is about the same as that of water \((2.2 \mathrm{GPa}) .\) As one goes deeper and deeper in the ocean, the pressure increases by \(1.0 \times 10^{4}\) Pa for every meter below the surface. (a) If a diver goes down 33 \(\mathrm{m}\) (a bit over 100 \(\mathrm{ft} )\) in the ocean, by how much does each cubic centimeter of her blood change in volume? (b) How deep must a diver go so that each drop of blood compresses to half its volume at the surface? Is the ocean deep enough to have this effect on the diver?

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