In that branch of physics called mechanics, a simple pendulum is a mental construct or idealized theoretical model in which a point-sized mass is suspended from a rod or string, itself of negligible mass, used in a friction-free and otherwise perfect environment. If the string is lengthy and the mass travels in an arc of only a few degrees under the influence of gravity, resultant motion is both linear and harmonic. The point mass acts as if a spring is pulling it repetitively back-and-forth along a line through a central point. A simple pendulum’s line of motion serves as an axis with the point being its origin. This system is mathematically described by a number of equations directly related to real-world processes.
The period or swing time of a simple pendulum, operating under the restrictions mentioned above is T=2π(g/L)-1/2 — in this equation gravitational force is represented by "g" and "L" represents the length of the string. If the arc of motion is much more than a few degrees, the simple equation listed above — an approximation only — no longer suffices, and must then include one or more added terms from an infinite virial equation. That equation is written T=2π(g/L)-1/2[1+(1/16)θ2+(11/3072)θ4+…]. Theta (θ) is the angle of arc in radians. In practical application, the larger the arc, the less a real pendulum resembles a simple pendulum.
As for many mechanical systems, it is of interest to consider both kinetic and potential energies. A simple pendulum must stop and reverse direction at both ends of its swing. Kinetic energy reaches a minimum — zero — at these points, so in accordance with the conservation of energy, potential energy reaches its maximum. Conversely, potential energy minimizes at the center of the swing, whereas kinetic energy achieves its maximum. Velocity zeroes out at both ends, but reaches a peak at the center point.
Mathematical consideration validates the pendulum’s use in timepieces. As recently as 1929, the Riefler pendulum clock was still in use as the United States' time standard. Even after that, it was replaced by another pendulum clock, the Shortt clock. Although no longer the standard or among the world’s most accurate clocks, the Shortt variety did achieve the amazingly accurate resolution of one second per year. As technology advanced, it was inevitable that the basic design idealized in a simple pendulum would be replaced by an electronic, and later, an atomic clock.