For small stretches, the Achilles tendon can be modeled as an ideal spring. Experiments using a particular tendon showed that it stretched \(2.66 \mathrm{mm}\) when a \(125-\mathrm{kg}\) mass was hung from it. (a) Find the spring constant of this tendon. (b) How much would it have to stretch to store \(50.0 \mathrm{J}\) of energy?

One of the fundamental principles to comprehend when dealing with elastic materials, such as springs, is Hooke's Law. This law states that the force exerted by a spring is directly proportional to the displacement or stretch from its original length, as long as the material is not stretched beyond its elastic limit.

The Achilles tendon in the exercise behaves similarly to an ideal spring as described by Hooke's Law. More formally, Hooke's Law is represented by the equation: \[ F = -kx \]where \( F \) represents the force applied to the spring, \( k \) is the spring constant indicating the stiffness of the spring, and \( x \) indicates the displacement or stretch of the spring from its equilibrium position. The negative sign indicates that the force exerted by the spring is in the opposite direction to the stretch.

Through these relationships, we can understand motion and predict behavior of elastic materials. The calculation of the spring constant in the given exercise is a direct application of Hooke's Law.

When a spring is stretched or compressed, it stores potential energy. The energy stored in a spring is known as elastic potential energy since it arises from the elastic properties of the spring. Mathematically, the potential energy (P.E.) in springs can be represented using the formula:\[ P.E. = \frac{1}{2} k x^2 \]In this equation, \( k \) refers to the spring constant, and \( x \) is the displacement or stretch beyond the spring's natural length.

This equation implies that the energy is proportional to the square of the displacement, which means small stretches can store a considerable amount of energy if the spring constant is high. The example of the Achilles tendon in the exercise demonstrates this concept: a small stretch can store enough energy to perform significant work, such as supporting the weight of the body in motion.

The relationship between force and motion is a fundamental aspect of physics that describes how an object interacts with the forces applied to it, leading to changes in motion. According to Newton's second law of motion, the force acting on an object is equal to the mass of the object multiplied by its acceleration.

The calculation of the force applied to the Achilles tendon in the textbook exercise (\( F = m \cdot g \)) is a direct application of this principle. Here, \( m \) represents the mass of the object, and \( g \) is the acceleration due to gravity. Force, a vector quantity, not only has magnitude but also direction. As demonstrated in our solution, when the force acts downwards due to the mass of an object, the Achilles tendon stretches, subsequently storing potential energy according to Hooke's Law and the formulas for potential energy in springs.

Understanding this relationship allows physicists and engineers to design systems that can efficiently manipulate force and motion, such as in suspension systems or even in the human body, as our exercise showcases with the Achilles tendon functioning similarly to a spring.