Problem 7 - College Physics

Question

\(\bullet$$\bullet\) Tension in a muscle. 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 \(F_{\text { max }}=\sigma A,\) 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. (a) What are the SI units of \(\sigma\) in terms of newtons and meters and also in terms of the fundamental quantities \((\mathrm{kg}, \mathrm{m}, \mathrm{s})\) (b) In one set of experiments, the average maximum force that the gastrocne- mius muscle in the back of the lower leg could exert was meas- ured to be 755 \(\mathrm{N}\) for healthy males in their midtwenties. What does this result tell us was the average cross-sectional area, in \(\mathrm{cm}^{2},\) of that muscle for the people in the study?

Step-by-Step Solution

Answer

The SI unit of \(\sigma\) is \(\text{Pa} = \text{N/m}^2\); the average area is 25.2 \(\text{cm}^2\).

Step 1: Determine SI Units of \(\sigma\)

The equation given is \( F_{\text{max}} = \sigma A \). Here, \( F_{\text{max}} \) is measured in newtons (N) and \( A \) is measured in square meters \( \text{m}^2 \). Therefore, the units of \( \sigma \) can be determined by rearranging to \( \sigma = \frac{F_{\text{max}}}{A} \). So, \( \sigma \) has units \( \frac{N}{\text{m}^2} \) (newtons per square meter). This is equivalent to pascals (Pa). Also, in terms of fundamental quantities: \( N = \text{kg}\cdot\text{m/s}^2 \). So, \( \sigma \) is \( \frac{\text{kg} \cdot \text{m/s}^2}{\text{m}^2} = \text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2} \).

Step 2: Calculate the Cross-Sectional Area A

Given the maximum force \( F_{\text{max}} = 755 \) N and the proportionality constant \( \sigma = 3.0 \times 10^5 \) Pa, we can find \( A \) using the formula \( F_{\text{max}} = \sigma A \). Solving for \( A \) gives: \( A = \frac{F_{\text{max}}}{\sigma} = \frac{755}{3.0 \times 10^5} \text{ m}^2 \).

Step 3: Convert A from Square Meters to Square Centimeters

First, calculate \( A \) in square meters: \( A = \frac{755}{3.0 \times 10^5} = 2.52 \times 10^{-3} \text{ m}^2 \). To convert this to square centimeters, note that \( 1 \text{ m}^2 = 10^4 \text{ cm}^2 \). So, \( A = 2.52 \times 10^{-3} \times 10^4 \text{ cm}^2 = 25.2 \text{ cm}^2 \).

Key Concepts

Tendons and Bones: The Bridge Connecting Forces

Muscles play a crucial role in movement by generating force, which is transmitted to bones through tendons. Tendons, acting as robust connective tissues, are vital in attaching muscles to their respective bones.
The interaction between muscles, tendons, and bones is integral in achieving efficient movement and stability.

• Tendons: These are the resilient bands of connective tissue that connect muscle to bone. Their elasticity allows them to absorb some of the shock and stress placed on muscles during movement.
• Bones: Acting as levers, bones facilitate movement when muscles exert tension upon them through tendons.

This harmonious collaboration allows our bodies to perform various activities, from walking and running to lifting and pushing objects. Tendons help in maintaining posture and play a regenerative role since some can repair themselves over time. However, the health and functionality of these components must remain optimal to prevent injuries such as tendonitis, which is the inflammation of a tendon.

Cross-Sectional Area and Its Importance in Muscle Function

The cross-sectional area of a muscle is a fundamental property that correlates with the muscle's ability to generate force. In terms of mechanics, a larger cross-sectional area means a muscle can exert a greater force. This is due to the increased number of muscle fibers that can contract simultaneously.
When considering muscle strength or the study of biomechanics, looking at the muscle's cross-sectional area provides critical insights.

• Mathematical Relationship: The maximum force a muscle exerts, denoted as \( F_{\text{max}} \), is directly proportional to its cross-sectional area \( A \). This is expressed by the equation \( F_{\text{max}} = \sigma A \), where \( \sigma \) is the proportionality constant.
• Measuring: Typically, the cross-sectional area is measured in units of square meters (\( \text{m}^2 \)), or converted to square centimeters (\( \text{cm}^2 \)) for more granular analyses.

Understanding this relationship helps in areas such as diagnosing muscle atrophy or monitoring muscle development. In physical training or rehabilitation, increasing a muscle's cross-sectional area through resistance training can enhance its force-generating capacity, thereby improving overall muscle strength and function.

Proportionality Constant: Bridging the Muscle-to-Force Equation

In the equation \( F_{\text{max}} = \sigma A \), \( \sigma \) symbolizes the proportionality constant. This constant is pivotal because it allows for the conversion of cross-sectional area into a quantifiable force.
For muscles, this constant remains surprisingly consistent across different species, including humans, and provides a means of comparison.

• SI Units: The units of \( \sigma \) are derived from rearranging the equation into \( \sigma = \frac{F_{\text{max}}}{A} \). This results in units of newtons per square meter (\( \text{N/m}^2 \)), also known as pascals (Pa).
• Fundamental Quantities: In terms of basic units: \( \sigma \) is expressed as \( \text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2} \), showcasing its roots in physics fundamentals like mass, distance, and time.

The consistency in \( \sigma \) implies that no matter the size or species, muscles produce a similar force per unit area, making it an essential tool in comparing muscle physiology across different organisms. This characteristic allows biomechanists and physiologists to study muscles under varying conditions and design appropriate training or treatment regimes.