91Ó°ÊÓ

The spring constant, often denoted as \( k \), is a fundamental concept in understanding harmonic motion. It essentially measures a spring's stiffness. The larger the spring constant, the stiffer the spring. In our context, two springs are involved, each having different constants \( k_1 = 2500 \text{ N/m} \) and \( k_2 = 2000 \text{ N/m} \). This exercise demonstrates a scenario where both springs are in parallel, meaning their combined effect must be considered when determining the system's behavior.
The total effective spring constant in such a parallel setup is the sum of the individual constants. Thus,

• \( k_{total} = k_1 + k_2 = 4500 \text{ N/m} \)

This combined spring constant provides a comprehensive measure of the system's overall stiffness. Understanding this is crucial for analyzing the system's potential energy and how it'll behave when displaced.

Conservation of energy is a pivotal principle in physics, asserting that energy cannot be created or destroyed, only transformed from one form to another. In the setup with our block and springs, this principle helps us explore how energy transitions between potential and kinetic states.
The system initially shows all energy stored as potential due to the displacement of the springs. As the block is released and moves towards equilibrium, this potential energy (\( U \)) is converted to kinetic energy (\( E_k \))

• Initial potential energy: \( U = \frac{1}{2}k_{total}x^2 \)
• When the block reaches its maximum speed, \( E_k = U \)

This conversion indicates the maximum kinetic energy is equal to the potential energy when the block is at its farthest from the equilibrium position.

Kinetic energy is the energy of motion. When the block moves after being released, it transitions from potential to kinetic energy as it passes through equilibrium.
At this point, all the stored energy in the springs has converted to kinetic energy, which is calculated as:

• \( E_k = \frac{1}{2}mv^2 \)

For the block, kinetic energy becomes maximum at equilibrium because it possesses the highest velocity at that point. Using the derived kinetic energy formula:

• \( v = \sqrt{\frac{2E_k}{m}} \)

Inserting our values reveals that at maximum speed, \( v \approx 5.81 \text{ m/s} \). This reflects the speed transitions from potential energy at maximum displacement.

Potential energy in springs is stored due to the deformation, either compression or stretching from equilibrium. It's described by the formula \( U = \frac{1}{2}kx^2 \), where \( x \) is the displacement from equilibrium.
This exercise highlights how potential energy is initially stored when the block is pushed 0.15 meters. It converts entirely to kinetic energy as the block rushes back to equilibrium.
During parts of motion where the springs compress or extend, this stored potential energy reaches its peak. Specifically, the maximum compression of spring 1 is found using:

• \( x_1 = \sqrt{\frac{2U}{k_1}} \)

Calculating gives \( x_1 \approx 20.1 \text{ cm} \). This figure represents the point where spring 1 is fully storing the system's mechanical energy—ensuring a full grasp of potential energy's role.