Hey for physics 20 unit 4 SHM and waves explain to me about how the vertical spring system works when a mass is hung and when a mass is not hung.

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Vertical spring systems are fundamental in physics. When no mass is attached, the spring hangs at its natural length. When we add mass, gravity stretches the spring to a new equilibrium position. Hooke's Law, F equals negative k x, describes the restoring force where k is the spring constant. When no mass is attached, the spring hangs freely at its natural length L zero. The only force acting is the spring's own weight, which is usually negligible. This creates zero net force and establishes our reference equilibrium position. This unloaded state serves as our baseline for comparing loaded spring behavior. When we attach a mass to the spring, gravity immediately pulls it downward with force mg. This stretches the spring, creating an upward restoring force kx zero. The system reaches a new equilibrium when these forces balance: mg equals kx zero. The stretch distance x zero depends on both the mass and spring constant. At the new equilibrium position, forces balance perfectly. The gravitational force mg equals the spring restoring force kx zero. Solving for displacement gives x zero equals mg over k. For example, a 2 kilogram mass on a spring with constant 100 newtons per meter stretches 0.196 meters. Heavier masses create proportionally larger equilibrium displacements, while stiffer springs resist stretching more. When the mass is displaced from equilibrium, it experiences a net restoring force proportional to displacement. This creates simple harmonic motion with acceleration a equals negative k over m times A. The angular frequency omega equals square root of k over m, and the period T equals 2 pi times square root of m over k. The system oscillates, converting between kinetic and potential energy.