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Hooke's Law is a principle of physics that relates the force needed to extend or compress a spring to the distance it extends or compresses. In simpler terms, it tells us that the displacement of the spring is directly proportional to the force applied. The law is named after the 17th-century British scientist Robert Hooke. He first stated the law in 1678 as a Latin anagram, and then translated it to the now-familiar form:
\[ F = k \cdot \Delta x \] where \(\text{F}\) is the force applied to the spring in Newtons (N), \(\text{k}\) is the spring constant that measures the stiffness of the spring in Newtons per meter (N/m), and \(\text{\Delta x}\) is the displacement of the spring from its equilibrium position in meters (m).
This law is the cornerstone for understanding elastic behavior in physics. For an object that can be modeled as a spring, such as a wire as in the above problem, Hooke's law is crucial for calculating how much it will stretch when a force is applied.

The spring constant \(\text{k}\) is a measure of the stiffness of a spring or elastic material. It is a central concept in Hooke's law, indicating how resistant a spring is to being compressed or stretched. The higher the spring constant, the more force is required for the same amount of displacement. Mathematically, it's represented by the formula:
\[ k = \frac{F}{\Delta x} \] from the original Hooke's law equation. The unit of the spring constant is Newtons per meter (N/m).
In the context of the problem provided, knowing the force applied and the resulting displacement allowed for the calculation of the spring constant. It's essential to ensure that the units are consistent—forces in Newtons and displacement in meters—to avoid calculation errors.

Elastic potential energy is the energy stored in an object when it is stretched or compressed. To calculate this type of energy, one must understand that it directly depends on both the spring constant \(\text{k}\) and the displacement \(\text{\Delta x}\) squared. The formula used for the calculation of elastic potential energy is:
\[ PE_{elastic} = \frac{1}{2} k \cdot \Delta x^2 \] The result will give you the potential energy in Joules (J), which is the standard unit of energy. In the exercise under consideration, this formula was used after first determining the spring constant using Hooke's law. The calculated elastic potential energy was set equal to the work done by the force to stretch the spring, giving the amount of energy stored in the wire.

Solving physics problems like the one about the stretched wire involves a systematic approach. First, as with the problem given, understanding the question and identifying the relevant physics principles—namely Hooke's law in this case—is critical. Afterward, converting units for consistency is often necessary, as calculations must be in standard units like meters for displacement.
The next step involves algebraically manipulating the equations to find the desired quantities. For this problem, isolating \(\text{k}\) from Hooke's law led to the spring constant, which was then used to find the elastic potential energy. Lastly, substituting the correct values into the energy formula and solving the equation gives the final result. Careful and methodical work, checking units at each step, is key to successful physics problem solving.