91Ó°ÊÓ

When studying spring systems, one fundamental principle is Hooke's law. It states that the force required to compress or extend a spring by a certain distance is proportional to that distance. The formula is expressed as

\[ F = -kx \]

where \( F \) is the force exerted by the spring, \( k \) is the spring constant, a measure of the spring's stiffness, and \( x \) is the displacement from the spring's equilibrium position. This law applies as long as the spring is not stretched beyond its elastic limit, meaning the spring can return to its initial shape after being deformed. In our textbook exercise, we used Hooke's law to relate the spring constant and the compression of the spring to the force on the block.

Elastic potential energy is the form of potential energy that is stored in elastic materials when they are stretched or compressed. It's significant in the study of conservation of energy in spring systems. The formula to calculate the elastic potential energy in a spring is

\[ E_{\text{elastic}} = \frac{1}{2} kx^2 \]

The term \( \frac{1}{2} \) appears because the energy is proportional to the square of the displacement, and \( k \) represents the spring constant. \( x \) stands for the distance the spring has been deformed from its equilibrium position. In the exercise's context, we calculated the initial elastic potential energy of the spring when it was compressed, and we also considered changes to this energy as the spring extended or compressed more at different points.

Kinetic energy is the energy that an object possesses due to its motion. It is calculated by the equation

\[ E_{\text{kinetic}} = \frac{1}{2} mv^2 \]

with \( m \) being the mass of the object and \( v \) its velocity. For our block attached to the spring, we analyzed how the kinetic energy varied as the spring went from being compressed to uncompressed. At any point in time, the sum of the block's kinetic energy and the spring's elastic potential energy must be constant, assuming no external forces like friction are at work. This concept demonstrates the conservation of energy, which is pivotal in understanding the problem at hand.

The spring constant, denoted by the symbol \( k \), quantifies the stiffness of a spring. It can be found from Hooke's law and is measured in Newtons per meter (N/m). The stiffer the spring, the greater its spring constant. In our exercise, we were given a spring constant of \( 100 \mathrm{N/m} \), which was used to differentiate the force applied by the spring at various levels of compression or extension. The spring constant's role in the equations of elastic potential energy and Hooke's law helped us determine the relationship between the displacement of the spring and the energies involved in the system.

Harmonic motion, or simple harmonic motion (SHM), refers to repetitive oscillation back and forth through an equilibrium position. A system where a mass is attached to a spring exhibits SHM if the spring follows Hooke’s law and there are no external forces like friction affecting the motion. Such a system's motion can be described by sinusoidal functions of time.

In the context of the exercise, when the block is released from a compressed state, it oscillates around the equilibrium position where the spring is at its natural length. This motion includes cycles of transferring energy back and forth between kinetic and elastic potential forms, perfectly illustrating the concept of conservation of energy in physics.