Harmonic Motion and Springs

Harmonic motion occurs when an object oscillates with a restoring force proportional to its displacement. When this motion gradually decreases over time, we call it damped harmonic motion or attenuation, where the oscillation amplitude steadily diminishes as energy dissipates.

Springs follow Hooke's Law, expressed as F = kx, where F is the force applied, k represents the spring constant, and x measures the extension. Every spring has an "elastic limit" beyond which it becomes permanently deformed. When compressed or stretched within this limit, springs store elastic potential energy $Us = ½kx²$.

The total energy in a spring system combines kinetic energy, spring potential energy, and gravitational potential energy: Etot = ½mv² + ½kx² + mgh. This energy remains constant in ideal systems, demonstrating the principle of energy conservation.

💡 The position of an oscillating object follows a cosine function: x = Acos(ωt), where A represents the amplitude and ω is the angular velocity. This is why harmonic motion creates such predictable, repeating patterns.

The period (T, measured in seconds) represents the time for one complete oscillation, while frequency (f, measured in Hertz) indicates how many oscillations occur per second. For an oscillating spring, the period depends only on the mass and spring constant: T = 2π√$m/k$. Remarkably, this period doesn't depend on gravity or the oscillation amplitude, which explains why springs oscillate consistently regardless of how far you stretch them initially.