Problem 16 - University Physics with Modern Physics

Question

Tendons are strong elastic fibers that attach muscles to bones. To a reasonable approximation, they obey Hooke's law. In laboratory tests on a particular tendon, it was found that, when a 250-g object was hung from it, the tendon stretched 1.23 cm. (a) Find the force constant of this tendon in N/m. (b) Because of its thickness, the maximum tension this tendon can support without rupturing is 138 N. By how much can the tendon stretch without rupturing, and how much energy is stored in it at that point?

Step-by-Step Solution

Answer

(a) Force constant is approximately 199.19 N/m. (b) Maximum stretch is 0.6926 m, storing 47.86 J of energy.

Step 1: Understand Hooke's Law

According to Hooke's Law, the force exerted by a spring (or tendon in this case) is proportional to its extension or stretch i.e. \( F = kx \), where \( F \) is the force applied, \( k \) is the force constant, and \( x \) is the displacement or stretch.

Step 2: Calculate the Force Applied to the Tendon

The weight of the 250-g object provides the force causing the stretch. First convert 250 g to kilograms (\( 0.250 \) kg) and use \( F = mg \) to find the force: \( F = 0.250 \times 9.8 = 2.45 \) N.

Step 3: Solve for the Force Constant

From the information \( F = 2.45 \) N and \( x = 1.23 \) cm (converted to meters as 0.0123 m), use Hooke's Law: \( F = kx \) or \( k = \frac{F}{x} = \frac{2.45}{0.0123} \approx 199.19 \) N/m.

Step 4: Calculate Maximum Stretch Before Rupture

Use the maximum force (138 N) and force constant to find the maximum stretch: \( x = \frac{F}{k} = \frac{138}{199.19} \approx 0.6926 \) m.

Step 5: Calculate Energy Stored in the Tendon

The energy stored in a stretched tendon is given by elastic potential energy: \( U = \frac{1}{2} k x^2 \). Substituting the maximum stretch \( x = 0.6926 \) m: \( U = \frac{1}{2} \times 199.19 \times (0.6926)^2 \approx 47.86 \) J.

Key Concepts

Force Constant

When studying materials like tendons, which behave (approximately) according to Hooke's Law, the term "force constant" plays a crucial role. The force constant, represented by the symbol \( k \), measures how rigid or stiff a spring-like object is. In simple terms, it's a measure of how difficult it is to stretch or compress the object. The higher the force constant, the stiffer the material.
To determine the force constant of a tendon, as in the given exercise, you would use Hooke's Law, which can be mathematically expressed as \( F = kx \). Here, \( F \) is the force applied to stretch or compress the tendon, \( k \) is the force constant, and \( x \) is the extension or stretch caused by the force.
To calculate \( k \) for our tendon example:

• First, we know the force (\( F \)) is the weight of the object causing the stretch. For a 250-g object, that's \( 2.45 \) N.
• The displacement (\( x \)) is the stretch of \( 0.0123 \) meters.
• Substitute into the equation: \( k = \frac{F}{x} = \frac{2.45}{0.0123} \approx 199.19 \text{ N/m} \).

This tells us the tendon is relatively stiff, as it resists stretching until subjected to significant force.

Elastic Potential Energy

Elastic potential energy is a form of stored energy due to the deformation of an object, such as the stretching or compressing of a tendon. This energy can be calculated when a material obeys Hooke's Law. When a tendon is stretched, it stores energy that can be released when the stretch is removed, much like a rubber band snapping back to its original shape.
The amount of stored elastic potential energy \( U \) can be calculated using the formula:
\[ U = \frac{1}{2} k x^2 \]
where \( k \) is the force constant and \( x \) is the displacement from the natural length of the tendon.
In our example, to find out how much energy is stored when the tendon is stretched to its limit:

• The force constant \( k \) is \( 199.19 \text{ N/m} \).
• The maximum stretch \( x \) that the tendon can handle before rupturing is \( 0.6926 \) meters.
• Plug these into the equation: \( U = \frac{1}{2} \times 199.19 \times (0.6926)^2 \approx 47.86 \text{ J} \).

Understanding elastic potential energy is key in biomechanics as it denotes how energy is stored and utilized during physical activities.

Tendon Mechanics

Tendon mechanics involves studying how tendons respond to forces, including how they stretch, support loads, and store energy. Tendons, which are robust and slightly flexible, connect muscles to bones. The mechanical properties of tendons are essential for understanding bodily movements and preventing injuries.
A tendon acting according to Hooke's Law will stretch linearly with applied force, depicted in a relationship akin to a spring.
Key factors in tendon mechanics include:

• **Stiffness:** Represented by the force constant, which quantifies tendon rigidity.
• **Stretching Limit:** Determined by its ability to endure force without tearing. In our problem, it bears up to 138 N before snapping.
• **Energy Absorption:** It encompasses how much energy a tendon can store elastically when stretched, calculated via elastic potential energy.
• **Biological Adaptation:** Tendons can adapt over time to resist forces, crucial for athletes who subject their tendons to high stress.

These properties highlight the importance of tendons in daily movements as well as specialized activities, helping to distribute force efficiently and enabling dynamic action without injury.