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Understanding the spring constant is essential when studying Hooke's law, which describes how a spring stretches or compresses in response to force. The spring constant, denoted by the symbol 'k', is a measure of a spring’s stiffness. It's calculated by dividing the force applied to the spring by the displacement caused by that force, as expressed in the formula \( k = \frac{F}{x} \) where \( F \) is the force in newtons (N) and \( x \) is the displacement in meters (m).

In the exercise, the calculation of the spring constant uses the weight of the mass hanging from the spring as the applied force. Since the spring constant is a ratio that remains constant for a given spring, knowing it allows us to predict how the spring will behave under different forces. This is why after calculating it initially, we can apply it to figure out the displacement for different masses, as the spring constant doesn’t change.

The relation between force and displacement in the context of Hooke's law is direct and proportional. This implies that the amount by which a spring stretches (displacement) is directly proportional to the force applied to it. If you double the force, the displacement doubles, provided the spring has not reached its elastic limit.

When materials like springs are deformed, the internal restoring forces try to bring them back to their original shape. In our exercise, the elongation of the spring from its original 10 cm to 15 cm is due to the force exerted by the mass due to gravity, calculated as \( F = mg \) where \( m \) is the mass and \( g \) is the acceleration due to gravity. The change in length (displacement) is then used alongside this force to compute the spring constant using Hooke’s law.

A mass-spring system is a simple harmonic oscillator comprising of a mass attached to a spring that can stretch or compress. When the mass hangs stationary, the system is in equilibrium. However, if the mass is pulled down and released, it will oscillate about the equilibrium point.

Calculating the behavior of a mass-spring system often begins by identifying the spring constant and examining how the system responds to different masses. The application of a mass leads to a force due to gravity which stretches the spring, and the spring exerts an equal and opposite force to support the mass. In our example, when we change the mass from 2.0 kg to 3.0 kg, using the constant 'k' we previously computed, we determine the new displacement and consequently the new length of the spring. The system's predictable nature, governed by Hooke’s law, allows us quantify such scenarios accurately.