91Ó°ÊÓ

In spring mechanics, we understand the behavior of springs subjected to forces. A spring's fundamental characteristic is its ability to store mechanical energy, exhibited as potential energy when the spring is either compressed or extended. The spring constant, denoted by \( k \), measures the stiffness of a spring. In our example, each spring attached to the 10-kg block has a spring constant of 2 kN/m, indicating how resistant it is to deformation.
Springs follow Hooke’s Law, which states that the force needed to extend or compress a spring by some distance \( x \) is proportional to that distance. This law is mathematically represented by \( F = kx \). When a spring returns to its original shape, it releases stored potential energy. This energy transformation is central to spring mechanics and is crucial when analyzing systems with multiple springs, like the one in the exercise.
Some key aspects of spring mechanics include:

• Understanding spring constant \( k \): Determines energy storage capacity.
• Hooke’s Law \( F = kx \): Describes the proportional relationship between force and displacement.
• Potential energy stored \( \frac{1}{2} k x^2 \): Calculated when a spring is displaced by \( x \).

Kinetic energy is the energy an object possesses due to its motion. When the 10-kg block moves after being released, it gains kinetic energy. The kinetic energy \( KE \) of the moving block is given by the formula \( KE = \frac{1}{2} mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the block.
In this problem, when the block is first released and moves 50 mm downward, the accumulated potential energy in the springs is converted into kinetic energy, making the block accelerate. Initially, when the block is at rest, its kinetic energy is zero. Once it starts moving, the equation \( \frac{1}{2} k x^2 + \frac{1}{2} k x^2 = \frac{1}{2} m v^2 \) helps us find the velocity by calculating the amount of potential energy transformed into kinetic energy.
Some important points about kinetic energy:

• Depends on mass (10 kg in this case) and velocity (\( v \)).
• Calculated using \( \frac{1}{2} mv^2 \).
• Essential in analyzing movement resulting from potential energy release.

Potential energy is the stored energy of an object due to its position or configuration. In our exercise, potential energy is stored in the springs when they are compressed or extended. This energy can be calculated using the formula \( PE = \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the displacement from its equilibrium position, which in this case is 0.05 m or 50 mm.
When the block is released, the potential energy stored in the springs is gradually converted into kinetic energy as the block moves. Initially, when both springs are unstressed, the potential energy is zero. As the springs stretch or compress, potential energy increases, peaking as they reach their maximum displacement before converting fully to kinetic energy.
Here's a summary of potential energy in spring systems:

• Calculated using \( \frac{1}{2} k x^2 \) — significant in energy transformations.
• Reflects the capability to convert into kinetic energy.
• Peaks when springs are fully compressed or stretched, then transforms into kinetic energy.