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The spring constant, often denoted by the symbol \( k \), is a fundamental concept in harmonic motion. It measures the stiffness of a spring. Think of \( k \) as a sort of "springiness" factor. More robust springs have higher values of \( k \), meaning they require more force to be stretched or compressed. In our problem, the spring constant is given as 315 N/m. This means that for every meter the spring is stretched or compressed, 315 newtons of force are needed. Understanding the spring constant is crucial because it directly affects the behavior of the spring in motion. A higher spring constant results in a stiffer spring, affecting the oscillation's frequency and potential energy stored.

Hooke's Law is a simple yet powerful principle that relates the force exerted by a spring to its displacement from equilibrium. Mathematically, it's expressed as:\[ F = -kx \]where

• \( F \) is the force exerted by the spring,
• \( k \) is the spring constant,
• \( x \) is the displacement from the spring's natural length.

The negative sign indicates that the force exerted by the spring is restorative, meaning it acts in the opposite direction of displacement, trying to bring the spring back to its equilibrium position. In the context of the problem, Hooke's Law helps us determine the maximum force when the spring is at its maximum displacement (amplitude). This force is crucial for finding how the spring affects the block's motion and its acceleration.

Understanding kinetic and potential energy is essential for solving problems involving harmonic motion. These two types of energy continually transform into each other in a spring-system. - **Kinetic Energy (KE):** This is the energy due to motion. For an object of mass \( m \) moving at velocity \( v \), kinetic energy is given by:\[ KE = \frac{1}{2}mv^2 \]In our scenario, the block initially has complete kinetic energy since it moves at 12.0 m/s while the spring is at equilibrium.- **Potential Energy (PE):** In springs, potential energy is stored as elastic potential energy. When the spring is compressed or stretched, potential energy is at its maximum. The equation for potential energy in a spring is:\[ PE = \frac{1}{2}kx^2 \]At the amplitude, the spring's potential energy equals the system's total initial mechanical energy.The total mechanical energy in a spring-block system remains constant, allowing us to use this relationship to find unknown quantities like amplitude, as energy transitions between kinetic and potential forms during the oscillation.