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Hooke's Law is a fundamental principle that defines the relationship between the force exerted by a spring and its displacement. When you apply a load to a spring, it stretches or compresses depending on the direction of the force.
The mathematical representation of Hooke's Law is:

• \( F = kx \)

Here, \( F \) is the force exerted by the spring, \( k \) is the spring constant (which indicates how stiff the spring is), and \( x \) is the displacement of the spring from its original position. The spring constant \( k \) is unique for each spring and tells you how much force is required to move the spring a certain distance. A higher \( k \) means a stiffer spring.
In our exercise, the spring balance can handle a maximum load of \(15.0 \) kg, which translates to a gravitational force \( F_{max} = 147.15 \, \text{N}\). The spring stretches by \(0.12 \, \text{m}\), giving a spring constant \( k \approx 1226.25 \, \text{N/m}\). This constant represents the spring's resistance to being stretched or compressed.

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium. It's the same kind of motion that a mass attached to a spring undergoes as it moves back and forth about a central point.
In the realm of springs, when a mass is attached, the system oscillates in a regular pattern if displaced. Important quantities in these systems include frequency \( f \) and period \( T \). In our exercise, the frequency is given as \(2.00 \, \text{Hz} \), meaning two complete oscillations occur each second.
The frequency of oscillation for a mass-spring system is defined by:

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

This formula shows that the frequency depends on the spring constant \( k \) and mass \( m \) attached to the spring. If you know the frequency and spring constant, you can determine the mass. For our scenario, rearranging the formula gives us an approximate mass of \(3.1 \, \text{kg} \). Simple Harmonic Motion is vital in understanding oscillatory systems in physics and engineering.

A Mass-Spring System is a classic model in physics that consists of a mass attached to a spring. This system is used to study oscillatory motion and principles of dynamics.
In our example, the Mass-Spring System consists of a package tied to a spring balance, oscillating due to gravity. The spring constant \( k \) is vital in such systems as it determines the behavior of the oscillation. The system's frequency (2.00 Hz in our case) aids in calculating other components like the mass.
The equation for the frequency as mentioned earlier is

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

By knowing useful values like the spring constant and frequency, you can re-engineer to find missing quantities such as mass. In our exercise, after determining \( m \) as \(3.1 \, \text{kg} \), we can calculate the system’s weight, which is equivalent to the force due to gravity, \( W = mg \). This results in approximately \(30.4 \, \text{N}\). Mass-Spring Systems form the bedrock for understanding more complex oscillations and are extensively used to model real-world applications.