Uncompressed, the spring for an automobile suspension is \(45 \mathrm{cm}\) long. It needs to be fitted into a space \(32 \mathrm{cm}\) long. If the spring constant is \(3.8 \mathrm{kN} / \mathrm{m},\) how much work does a mechanic have to do to fit the spring?

Understanding Hooke's Law is crucial when it comes to dealing with springs and their behavior. This fundamental principle of physics states that the force required to compress or extend a spring linearly varies with respect to its displacement from the original position.

Imagine holding a simple coiled spring between your fingers. If you pull it apart or compress it, you’re exerting a force on the spring. Hooke’s Law captures this relationship with the formula:
\[ F = -kx \]
where \( F \) is the force applied to the spring, \( k \) represents the spring constant, and \( x \) is the displacement of the spring from its equilibrium position. The negative sign indicates that the force is restorative - trying to bring the spring back to its original length. Therefore, when you compress a spring like the one in a vehicle's suspension system, you are witnessing Hooke's Law in action. The more you compress it, the more force you need to apply. It's this straight-forward yet foundational concept that predicts how much force a mechanic needs to exert to compress a spring a certain distance.

The spring constant, often denoted by \( k \) in physics, is a measure of a spring's stiffness. It's essentially a conversion factor that tells you how much force is needed to change the spring's length by a certain amount.

Think of it as a numerical rating for a spring's resistance to being deformed. A larger spring constant means a stiffer spring that’s harder to compress or extend. Conversely, a smaller spring constant indicates a more elastic spring that you can stretch or compress quite easily. In formal terms,
\[ k = \frac{F}{x} \]
where \( F \) is the force exerted on the spring and \( x \) is the deformation experienced by the spring (either compression or extension).

In the context of the exercise involving the automobile suspension spring, the spring constant of \(3.8 kN/m\) or \(3800 N/m\) meant that for every meter you compress the spring, you would need to apply a force of 3800 Newtons. This constant is fundamental in determining the force exerted by a spring given a certain amount of compression and is essential for calculating work done on the system, as seen in the given problem.

The work-energy theorem is a pivotal concept in physics that relates to how the movement of an object is affected by work. This theorem tells us that the work done on an object results in a change in its kinetic energy.

In mathematical terms, work done (\( W \)) is the product of the force applied (\( F \)) and the distance over which the force is applied (\( d \)):
\[ W = Fd \].
It’s important to note that the direction of the force and the direction of motion must be the same to use this simplified equation.

In the case of compressing a spring, this theorem helps to quantify the work needed to compress the spring from its uncompressed state to a specific length. Even though the spring is not moving and thus has no kinetic energy once compressed, the work energy theorem helps here in a slightly modified conceptual framework - the work done on the spring is stored as potential energy within the spring's deformation. This work is also the energy you'd need to supply, say, as a mechanic applying force to compress the spring in an automobile's suspension system, as described in the problem.