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Hooke's Law is a fundamental principle in mechanics that describes the linear relationship between the force applied to a spring and the displacement caused by that force. It is expressed mathematically 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 equilibrium position. This law is crucial for understanding how springs and elastic materials behave under force. It essentially tells us that the further a spring is stretched or compressed, the more force is required to do so, provided the material remains within its elastic limit. This concept lays the groundwork for calculating other properties like potential energy in springs.

The spring constant, denoted as \( k \), is a measure of a spring's stiffness. It is a key part of Hooke's Law. When you know the force applied to a spring and the displacement it caused, you can calculate the spring constant using the formula \( k = \frac{F}{x} \). This calculation is crucial because the spring constant allows us to predict how a spring will respond to different forces. A higher spring constant means a stiffer spring that requires more force for a given displacement. To find \( k \) for our spring, we took the force of \( 4.1 \ \text{N} \) and the displacement of \( 0.014 \ \text{m} \), resulting in a spring constant \( k = \frac{4.1}{0.014} \ \text{N/m} \). Understanding \( k \) helps us further explore the behavior of the spring under various conditions.

Calculating the displacement is a central part of understanding how energy is stored in a spring. We use the potential energy formula for a spring: \( PE = \frac{1}{2} k x^2 \). To find how far a spring must be stretched for a certain potential energy, we rearrange this formula to solve for \( x \). For example, if the desired potential energy is \( 0.020 \ \text{J} \), we set up the equation: \( 0.020 = \frac{1}{2} \times \frac{4.1}{0.014} \times x^2 \). Solving this equation will yield the displacement \( x \) needed to achieve the specified energy. Similarly, for a potential energy of \( 0.080 \ \text{J} \), we use the same approach to calculate \( x \). This helps us understand the precise adjustments needed to attain different energy states in the spring.

The concepts of force and energy are intricately linked when it comes to springs. The force exerted by, or required to move, a spring is a function of its displacement, dictated by Hooke's Law. This force is what allows a spring to store energy. The energy stored in a spring is called elastic or potential energy, captured by the formula \( PE = \frac{1}{2} k x^2 \). This formula indicates that the energy is proportional to the square of the displacement. So, doubling the displacement increases the energy by a factor of four, showcasing the non-linear relationship between displacement and energy. Understanding this relationship helps us calculate either one, knowing the values of other parameters, which is key in applications involving mechanical systems like shock absorbers or load stabilizers.