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The spring constant, often denoted as \( k \), is a measure of a spring's stiffness. According to Hooke's Law, the force \( F \) exerted by a spring is directly proportional to the displacement \( x \) it experiences from its equilibrium position. This relationship is expressed by the equation \( F = kx \). If the spring is not very stiff, \( k \) will be smaller, indicating that a given force will cause a larger displacement.
The spring constant is crucial in determining how a spring behaves when a force is applied to it. In the context of the exercise, we calculated \( k \) by considering the weight of the fish, which acts as the force stretching the spring. By rearranging Hooke's Law, we found that \( k = \frac{F}{x} \), where \( F \) is the gravitational force (\( mg \)) and \( x \) is the displacement caused by this force. This gives us information about how much force is required to compress or extend a spring by a specific distance. In this example, the spring constant was determined to be approximately \( 5313.75 \text{ N/m} \), indicating a fairly stiff spring suitable for measuring larger weights.

Simple Harmonic Motion (SHM) describes the oscillatory motion that repeats itself in a regular cycle. This type of motion is characterized by the restoring force being directly proportional to the displacement and acting in the opposite direction.
Consider a mass-spring system, like in the exercise. When the fish is pulled slightly from its equilibrium position and released, it oscillates up and down. The springs have this tendency because their force is always directed towards the equilibrium position, making the motion periodic. SHM can be described mathematically with the equation for the period \( T \), the time for one complete cycle. It is given by \[ T = 2\pi \sqrt{\frac{m}{k}} \] where \( m \) is the mass and \( k \) is the spring constant.
In our exercise, we found the period of the oscillation to be approximately \( 0.694 \) seconds. This tells us how long it takes for one full up-and-down oscillation of the fish, which is due to the natural properties of the mass-spring system.

Angular frequency, often symbolized by \( \omega \), relates to how rapidly something oscillates in simple harmonic motion. It is measured in radians per second and offers a way to understand the rate of motion in a circular or oscillatory system.
The angular frequency is related to the spring constant \( k \) and mass \( m \) by the equation \( \omega = \sqrt{\frac{k}{m}} \). It essentially captures the frequency of oscillations in terms of radians, rather than cycles, which is particularly useful in various physics and engineering contexts.
In the given problem, the fish's oscillation frequency helps predict how often the maximum speed occurs at the equilibrium point. We calculated \( \omega \) to be about \( 9.04 \text{ rad/s} \). Knowing this allows us to find the maximum speed of the fish using \( v_{max} = \omega A \), where \( A \) is the amplitude of motion. Understanding \( \omega \) gives insights into not just how fast but also how the motion is orchestrated in periodic systems like springs.