Oscillations
Anything that oscillates follows a periodic and repetitive pattern of motion. Typically, measured with time, oscillatory motion usually revolves around springs or pendulums. However, there are examples of oscillations in almost every branch of science; the beating of a heart, predator-prey population cycles, geothermal geysers, vibration in string instruments, and even business cycles in economics are all examples that oscillate.
Simple harmonic oscillation refers to a mass of some sort attached to something that oscillates, such as a ball on a string acting as a pendulum. For it to follow simple harmonic motion, it must only be under the forces of gravity and tension. When the system is not static (a.k.a. not at rest), there is a net restoring force that brings the system back into equilibrium. However, whilst in motion back to equilibrium, its acquired momentum pushes the mass into yet another displaced position. The time that the mass takes to leave its original, displaced position and eventually return is known as the period of oscillation.
The frequency, f, of the system, is a simple calculation: f = 1/T, where T is the period. Whilst at its furthest position, a stretched spring has its highest spring potential energy, given by the equation: U = 1/2 k x. All of the spring’s potential energy is converted to kinetic energy at its equilibrium point, which is where its speed is highest.
The restoring force in a spring is calculated using Hooke’s Law: F = -kx, where k is the spring constant and x is the distance from equilibrium. The spring constant represents the stiffness of any elastic material. This is why a heavier spring is harder to stretch out because its restoring force is much greater than a light spring.
When multiple strings are connected, only two configurations can still stretch or compress as a normal string would. In these cases, the only change is the spring constant. When springs are in series, or attached end to end, the resulting spring constant can be calculated with 1/k = 1/k₁ + 1/k₂. When springs are in parallel, or attached side by side, you can simply add each individual spring constant together, and the sum is the total spring constant.
