Problem 7 - College Physics

Question

Muscles are attached to bones by means of tendons. The maximum force that a muscle can exert is directly proportional to its cross-sectional area \(A\) at the widest point. We can express this relationship mathematically as \(F_{\max }=\sigma A\) where \(\sigma\) (sigma) is a proportionality constant. Surprisingly, \(\sigma\) is about the same for the muscles of all animals and has the numerical value of \(3.0 \times 10^{5}\) in SI units. The gastrocnemius muscle, in the back of the leg, has two portions, known as the medial and lateral heads. Assume that they attach to the Achilles tendon as shown in Figure \(5.45 .\) The cross sectional area of each of these two muscles is typically \(30 \mathrm{~cm}^{2}\) for many adults. What is the maximum tension they can produce in the Achilles tendon?

Step-by-Step Solution

Answer

The maximum tension in the Achilles tendon is 1800 N.

Step 1: Understand the Given Information

We are given that the maximum force a muscle can exert is directly proportional to its cross-sectional area, described by the equation \( F_{\max} = \sigma A \), where \( \sigma = 3.0 \times 10^{5} \) N/m². Each muscle head has an area of \( 30 \text{ cm}^2 \).

Step 2: Convert Area to SI Units

The formula requires area in square meters (\( \text{m}^2 \)). Convert \( 30 \text{ cm}^2 \) to \( \text{m}^2 \) by using the conversion factor \( 1 \text{ cm}^2 = 0.0001 \text{ m}^2 \). Therefore, \( 30 \text{ cm}^2 = 30 \times 0.0001 = 0.003 \text{ m}^2 \).

Step 3: Calculate Maximum Force for One Muscle Head

Using the formula \( F_{\max} = \sigma A \), substitute \( \sigma = 3.0 \times 10^{5} \) N/m² and \( A = 0.003 \text{ m}^2 \), to calculate \( F_{\max} \) for one muscle head: \( F_{\max} = 3.0 \times 10^{5} \times 0.003 \).

Step 4: Simplify the Calculation

Compute \( F_{\max} = 3.0 \times 10^{5} \times 0.003 = 900 \) N for one muscle head.

Step 5: Calculate Total Maximum Tension

Since there are two muscle heads, the total force they can exert together is \( 2 \times 900 \) N. Therefore, the total maximum tension is \( 1800 \) N.

Key Concepts

Biomechanics

Biomechanics is the study of the movement of living organisms, combining principles of mechanics with biology. Essentially, it looks at how biological systems like muscles and tendons work. These systems follow mechanical rules similar to those governing machines. For instance, the force exerted by muscles is crucial to understanding movement and stability.
Biomechanics examines:

• The relationship between force and motion in the body.
• How different structures within the body can withstand forces.
• Ways to enhance movement efficiency and prevent injury.

Understanding biomechanics is essential for fields like sports science and physical therapy, where improving physical function or maximizing performance without injuring is crucial.
In the given problem, biomechanics helps us study how the gastrocnemius muscles apply force through the Achilles tendon, impacting both stability and movement.

Force Calculation

Force calculation involves using known values and physics formulas to find out the force a muscle can exert. In the context of muscle movement, force determines how much effort a muscle can deliver. Calculating this force is fundamental to understanding various physical activities and biological processes.In our exercise, the force calculation follows the relationship:\[ F_{\text{max}} = \sigma A \]Where:

• \( F_{\text{max}} \) is the maximum force.
• \( \sigma \) represents the proportionality constant (muscle tensile strength).
• \( A \) stands for the cross-sectional area of the muscle.

We calculated the maximum force per muscle using these values, showing that multiplying the muscle's cross-sectional area by the constant gives the maximum force each muscle head can exert. This force is then combined to understand the total tension in the Achilles tendon.

Physics of Muscles

The physics of muscles involves understanding how muscles generate force and produce movement. Muscles work like biological machines, converting chemical energy into mechanical energy. This energy allows muscles to contract, exert force, and perform work.
Key aspects of muscle physics include:

• Cross-sectional area: Larger muscle area often means greater force, as muscles with larger areas have more fibers to exert force.
• Proportionality constant (\( \sigma \)): A universal constant indicating the tensile strength per unit area, approximately \(3.0 \times 10^{5}\) for all animals.
• Contraction mechanism: Interaction of proteins inside muscle fibers fuels contraction and force.

Through understanding these elements, we gain insight into how muscles maintain posture, generate movement, and contribute to body stabilization. In this case, muscles attached to the Achilles tendon convert energy into movement during walking or running.

SI Units Conversion

SI units, or the International System of Units, are the standard units of measurement in science. Converting measurements into SI units is critical for consistency and accuracy in calculations across different scientific fields.For our problem, the area given in square centimeters needs to be converted into square meters. The area conversion follows the rule:\[ 1 \text{ cm}^2 = 0.0001 \text{ m}^2 \]Therefore, we converted an area of \( 30 \text{ cm}^2 \) to \( 0.003 \text{ m}^2 \) by multiplying the given value by 0.0001. This conversion ensures the correct application of the formula for calculating force.

• Accurate unit conversion avoids errors in calculation and ensures meaningful results.
• It simplifies communication and allows scientists worldwide to understand each other's work.

Conversions often appear in physics problems, as they align measurements with the required standards for precise scientific inquiry and engineering applications.