Problem 5 - College Physics

Question

\(\bullet\) Biceps muscle. A relaxed biceps muscle requires a force of 25.0 \(\mathrm{N}\) for an elongation of \(3.0 \mathrm{cm} ;\) under maximum tension, the same muscle requires a force of 500 \(\mathrm{N}\) for the same elongation. Find Young's modulus for the muscle tissue under each of these conditions if the muscle can be modeled as a uniform cylinder with an initial length of 0.200 \(\mathrm{m}\) and a cross-sectional area of 50.0 \(\mathrm{cm}^{2}.\)

Step-by-Step Solution

Answer

Young's modulus is 33333.3 Pa when relaxed and 666666.7 Pa under max tension.

Step 1: Understand Young's Modulus Formula

Young's modulus \( E \) is a measure of the stiffness of a solid material. It is defined by the formula: \( E = \frac{F \cdot L_0}{A \cdot \Delta L} \), where \( F \) is the force applied, \( L_0 \) is the original length of the material, \( A \) is the cross-sectional area, and \( \Delta L \) is the elongation.

Step 2: Convert Units

Convert the given elongation and cross-sectional area into meters and square meters, respectively. The elongation \( \Delta L = 3.0 \; \text{cm} = 0.03 \; \text{m} \) and the cross-sectional area \( A = 50.0 \; \text{cm}^2 = 0.005 \; \text{m}^2 \).

Step 3: Calculate Young's Modulus for Relaxed Muscle

For the relaxed muscle, plug in the values: \( F = 25.0 \; \text{N}, \; L_0 = 0.200 \; \text{m}, \; A = 0.005 \; \text{m}^2, \; \Delta L = 0.03 \; \text{m} \) into the Young's modulus formula. The calculation becomes: \[ E_{\text{relaxed}} = \frac{25.0 \cdot 0.200}{0.005 \cdot 0.03} \approx 33333.3 \; \text{Pa} \]

Step 4: Calculate Young's Modulus for Muscle Under Max Tension

For the muscle under maximum tension, use: \( F = 500.0 \; \text{N}, \; L_0 = 0.200 \; \text{m}, \; A = 0.005 \; \text{m}^2, \; \Delta L = 0.03 \; \text{m} \). Plug these into the formula: \[ E_{\text{max tension}} = \frac{500.0 \cdot 0.200}{0.005 \cdot 0.03} \approx 666666.7 \; \text{Pa} \]

Step 5: Interpret the Results

The Young's modulus for the relaxed muscle is approximately 33333.3 Pa, whereas for the muscle under maximum tension, it is approximately 666666.7 Pa, indicating that the muscle tissue becomes significantly stiffer under high tension.

Key Concepts

Elasticity in Physics

Elasticity is a key concept in physics, describing how materials deform and return to their original shape when a force is applied and then removed. The ability of materials to resist deformation and recover is what makes them elastic. In physics, elasticity is quantified using a parameter called Young's modulus.
Young's modulus, denoted as \( E \), is a measure of stiffness or resistance to elastic deformation. It is determined by the formula:

• \( E = \frac{F \cdot L_0}{A \cdot \Delta L} \)

here \( F \) is the force applied, \( L_0 \) is the original length, \( A \) is the cross-sectional area, and \( \Delta L \) is the change in length.
Higher values of Young's modulus indicate stiffer materials. This fundamental measure is crucial not only for understanding basic physics but also for engineering and biology applications, such as analyzing tissues and constructing materials that mimic these properties.

Muscle Tissue Properties

Muscle tissues have unique properties that allow them to perform complex and demanding functions. They must combine elasticity and strength to endure stretching and contracting repeatedly without damage.
Consider the example of a biceps muscle, which extends and contracts during arm movement. This muscle's ability to elongate under tension is characterized by a distinct range of Young's modulus depending on its state.
When the muscle is relaxed, it has a lower Young's modulus, indicating it is less stiff and more flexible. However, under maximum tension, the muscle becomes significantly stiffer, with a much higher Young's modulus. This adaptability is crucial for muscular function, allowing muscles to handle heavy loads or forceful contractions while maintaining structural integrity. Understanding these properties helps in fields like sports science and rehabilitation, where muscle performance is key.

Force and Deformation

Force and deformation are interrelated in the study of material properties, particularly in understanding elasticity. When a force is exerted on a material, it may change shape. This change in shape or deformation can be temporary or permanent.
In the context of elasticity, our focus is on temporary deformation, where the material returns to its original shape once the force is removed. Take the biceps muscle as an example:

• Applying a force of 25.0 N causes the muscle to elongate by 3 cm when relaxed.
• Under maximum tension, a much larger force of 500 N is needed to achieve the same elongation.

The relationship between the applied force and the resulting deformation for a particular state of the muscle gives insights into its stiffness and flexibility. This relationship is again expressed through Young's Modulus, which captures how resistant the muscle is to changes in length under various forces.