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Hooke's Law is a fundamental principle for understanding how springs behave under force. Imagine a spring like a rubber band; it can stretch and compress. This law states that the force needed to stretch or compress a spring by a distance is directly proportional to that distance. It can be mathematically represented as \[ F = kx \]where:

• \( F \) is the force applied to the spring,
• \( k \) is the spring constant, which tells us how stiff the spring is, and
• \( x \) is the displacement or change in length of the spring.

The spring constant \( k \) is a measure of the force required to stretch or compress the spring by one unit of distance. In other words, a bigger \( k \) means a stiffer spring. For the given exercise, knowing the spring stretches to a certain length gives us a chance to calculate this constant, revealing the spring's stiffness. Understanding Hooke’s Law is vital as it lays the groundwork for further calculations involving the spring's potential energy when it stretches.

Energy conservation is a key concept in physics, suggesting that energy cannot be created or destroyed, only transformed from one form to another. In our spring problem, energy conservation is handy to find how much energy shifts between two forms as the block moves.

Initially, the block has potential energy due to its position. As it falls, this potential energy turns into kinetic energy and eventually into elastic potential energy in the spring. Earth's gravitational force (weight of the block) converts to the elastic potential energy stored in the spring. The equation used for this energy transformation is:\[ mgx = \frac{1}{2}kx^2 \]where:

• \( m \) is the mass of the block
• \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \))
• \( x \) is the displacement
• \( k \) is the spring constant

This equation helps us find the spring constant \( k \) by showing how gravitational energy translates into energy stored in the spring, when the block momentarily rests at its lowest point.

Angular frequency is crucial to understanding how quickly something oscillates. It tells us how fast an object moves through its cycle in oscillatory motions like those of the block on the spring. It's directly tied to the spring's stiffness and the object's mass attached to it.

The formula for angular frequency \( \omega \) is:\[ \omega = \sqrt{\frac{k}{m}} \]where:

• \( \omega \) is the angular frequency measured in radians per second,
• \( k \) is the spring constant, and
• \( m \) is the mass of the oscillating block.

This relationship reveals that a stiffer spring or a lighter mass results in a higher angular frequency, meaning the block vibrates faster. For problems involving frequency and vibrations, grasping this connection helps in predicting how an object behaves in oscillatory motion. The given exercise uses this formula to calculate how rapidly the block moves up and down, showing the practical applications of these theoretical principles.