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Elastic potential energy is the stored energy in objects that can be stretched or compressed, like springs. In the context of springs, this energy arises when you apply a force to change the spring's shape. When you release the force, the spring can return to its original form, releasing the stored energy.

The elastic potential energy of a spring is dependent on how much the spring is stretched or compressed from its natural length. It's calculated with the formula:

• \(PE = \frac{1}{2}kx^2\)

Here, \(PE\) represents the potential energy, \(k\) is the spring constant, and \(x\) denotes the displacement from the spring's equilibrium position.

In our exercise, the work done by the force on the spring, which is 200 J, becomes the elastic potential energy, since it's stored in the stretched spring. Understanding this concept helps in figuring out how energy transformations occur in spring systems.

Hooke's Law describes the behavior of springs and is fundamental to understanding spring mechanics. It states that the force needed to extend or compress a spring by some distance \(x\) is proportional to that distance. The equation is:

• \(F = kx\)

Here, \(F\) is the force applied to the spring, \(k\) is known as the spring constant, and \(x\) is the displacement.

This principle is key to deducing how much force is needed during stretching or compressing. It also helps determine the force at maximum elongation or compression. For the spring in our problem, applying Hooke’s Law means at maximum stretch of 2 meters, the force is 200 N.

The spring constant, denoted as \(k\), is a measure of a spring's stiffness. It quantifies how resistant a spring is to being compressed or stretched. The larger the spring constant, the stiffer the spring, and a greater force will be required to displace it by one unit of length.

In the exercise given, the spring constant is 100 N/m, illustrating a moderate stiffness. Knowing this value is crucial for any calculations involving Hooke's Law and elastic potential energy, as it directly affects the force and energy calculations.

For example, with a known spring constant, you can accurately determine the force on the spring when it is displaced, as well as the amount of potential energy stored at different displacements.

Displacement in the context of springs refers to the change in length of a spring from its unstretched or relaxed position. This change can be either an extension or a compression.

In our initial exercise, we calculated a displacement of 2 meters. This was derived from the work-energy principle by equating the work done on the spring to the change in elastic potential energy. The displacement is calculated by rearranging the elastic potential energy formula \(PE = \frac{1}{2}kx^2\) to solve for \(x\).

Understanding displacement is important as it tells us how much a spring has been stretched or compressed from its natural state, which is useful for calculating forces and stored energy within the spring system. This understanding is crucial in various applications, from simple gadgets to complex mechanical systems.