91Ó°ÊÓ

Hooke's Law describes the relationship between the force exerted by a spring and the amount it is stretched or compressed. This law states that the force needed to extend or compress a spring by a distance \( x \) is proportional to that distance. Mathematically, it is expressed as:

• \( F = kx \)

where \( F \) is the force applied to the spring (in newtons), \( k \) is the spring constant (a measure of the spring's stiffness), and \( x \) is the displacement of the spring from its equilibrium position (in meters).
The spring constant \( k \) tells us how stiff the spring is: larger \( k \) values indicate stiffer springs. In the exercise, the spring constant was calculated using Hooke's Law in combination with the work-energy principle. This provides insight into how much force is required to compress or extend the spring by a specific distance.

The work-energy principle connects the work done on an object with its energy changes. For springs, this principle is often applied to calculate how much work is done during compression or extension, linking it to the potential energy stored in the spring. The formula used here is:

• \( W = \frac{1}{2}kx^2 \)

where \( W \) is the work done on the spring (in joules), \( k \) is the spring constant, and \( x \) is the displacement.
In practical terms, when a spring is compressed or extended, it stores energy. This stored energy can be calculated, and in our exercise, it allowed us to determine the spring constant by rearranging the formula to solve for \( k \). This demonstrates how the work-energy principle can be a powerful tool in both problem-solving and understanding the transfer and storage of energy in physical systems.

Newton's second law is fundamental in understanding mechanical motion and can be expressed as:

• \( F = ma \)

where \( F \) is the force applied to an object (in newtons), \( m \) is the object's mass (in kilograms), and \( a \) is the acceleration produced (in meters per second squared).
In the context of our exercise, Newton's second law works hand-in-hand with Hooke's Law to find the mass attached to the spring. Once we know the force exerted by the spring at maximum compression (using Hooke's Law), the mass can be derived using the relationship between force and acceleration provided by Newton's second law.
Understanding this law is crucial because it not only helps calculate mass but also provides insight into how various forces and mass interact to affect an object's motion. It forms the basis for analyzing mechanical systems in equilibrium and motion, as shown in the exercise when determining the mass with a given acceleration.