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Hooke's Law is a fundamental principle in physics that describes the behavior of springs under load. It states that the force exerted by a spring is directly proportional to the displacement it undergoes. This relationship can be expressed mathematically as:

• \( F = kx \)

Here, \( F \) is the force applied to the spring, \( k \) is the spring constant, and \( x \) is the displacement from its original position. The spring constant \( k \) is a vital part of this equation, representing the stiffness of the spring. A higher spring constant indicates a stiffer spring that doesn't stretch easily.
In practical terms, Hooke's Law helps understand how much force is needed to stretch or compress a spring by a given amount. This principle is not only crucial in physics but also in engineering applications, where precise control of mechanical systems is required.

The spring constant, denoted by \( k \), measures a spring's stiffness. It determines how much force is needed to extend or compress a spring by a certain distance. In the formula \( F = kx \), the spring constant \( k \) tells you how much force (\( F \)) is needed for each unit of displacement (\( x \)).
The unit of the spring constant is newtons per meter (N/m). A larger \( k \) means a stiffer spring, requiring more force for the same displacement. In our exercise, using the data with a 2.8 kg mass stretching the spring by 0.018 m, the spring constant calculated was approximately 1523.33 N/m.
Understanding the spring constant is essential for tasks like designing suspension systems or calculating oscillations in mechanical setups. It provides insights into how a spring will behave when forces are applied, which is crucial for safety and performance in applications ranging from vehicle suspensions to measuring devices.

Frequency of vibration refers to how often an oscillating system completes a cycle per unit time. In the context of a spring-mass system, it indicates how quickly the mass goes back and forth. It is measured in hertz (Hz), where one hertz is one complete cycle per second.
The frequency \( f \) of a simple harmonic oscillator like a spring-mass system is given by the formula:

• \( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \)

Here, \( k \) is the spring constant, and \( m \) is the mass attached to the spring. This equation shows that the frequency is influenced directly by the spring constant and inversely by the mass. Thus, increasing the mass decreases the frequency, while a stronger spring (higher \( k \)) increases the frequency.
In our problem, given a desired frequency of 3.0 Hz, we can use this relationship to calculate the appropriate mass that needs to be attached to achieve such vibrations. This involves solving the equation by rearranging it to find \( m \), showcasing how changes in \( m \) or \( k \) affect the system's behavior.