the mass-spring system

The mass-spring system is the simplest oscillator you can build and the textbook model for simple harmonic motion: a block attached to a spring. Picture a mass hanging from a spring, or resting on a frictionless table and tied to a wall by a spring. Pull it aside and let go, and it bobs back and forth. Almost every idea about oscillation is first learned here.

Precisely, combine Hooke's law F = -k x with Newton's second law F = m a to get m a = -k x, which rearranges to a = -(k/m) x. This is exactly the SHM condition, and its solution is x(t) = A cos(omega t + phi) with angular frequency omega = sqrt(k/m) and period T = 2 pi sqrt(m/k). So a heavier mass oscillates more slowly, and a stiffer spring (larger k) oscillates faster.

Notice that the period depends on the mass m and stiffness k, but not on the amplitude, and not on gravity for a horizontal spring. For a spring hung vertically, gravity simply shifts the equilibrium point down to a new resting spot; the oscillation about that new point behaves identically. As always, this clean result assumes an idealized, massless spring and no friction.