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The Work-Energy Theorem is a fundamental principle in physics that relates the work done on an object to the change in its kinetic energy. When we apply a force to stretch or compress a spring, we're doing work on the spring. According to the theorem, the work done by the forces acting on an object is equal to the change in kinetic energy of that object.

To put it into context with springs, when you stretch a spring and then release it, the energy you used to pull the spring gets converted into kinetic energy, causing the spring to move back to its original position. However, if the spring doesn't move (as in the case of our exercise), the energy is stored as potential energy, known as elastic potential energy, in the spring. That's what's happening in the exercise provided; the work done to stretch the spring is stored as potential energy.

Potential energy is the energy stored within a physical system as a result of the position of the objects within that system. For springs, this is known as elastic potential energy. This kind of energy is directly related to the displacement of the spring from its equilibrium position, that is, how much it has been stretched or compressed.

The potential energy (\(U\text{elastic}\)) stored in a spring is given by \( U_{\text{elastic}} = \frac{1}{2} k x^2 \), where \(k\) is the spring constant and \(x\) is the displacement from equilibrium. This formula comes into play when we want to calculate the spring constant from a given amount of work, as seen in the original exercise. By understanding this concept, students can better understand why the work done to stretch the spring is equivalent to the potential energy stored in it.

Hooke's Law is a principle of physics that states the force needed to extend or compress a spring by some distance scales linearly with respect to that distance. This can be represented by the formula \( F = -kx \), where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement from the spring's original length (equilibrium position).

The negative sign indicates that the force exerted by the spring is in the opposite direction to the force causing the displacement. The spring constant, \( k \), represents the stiffness of the spring—the higher the value of \( k \) the stiffer the spring. Using Hooke's Law, one can understand how the constant plays a crucial role in the force-displacement relationship and provides insights into the physical properties of the spring itself. By calculating the spring constant, we can predict how a spring will respond to varying forces.