The periodic motion has its basis in the simple movement produced by a pendulum (a hanging mass) when is set free in a gravitational field (air effects neglected)
Let me draw the pendulum
Now, let's identify the right triangle that is embedded in this pendulum
There is a right triangle hidden in the system!
By solving it for each one of its sides we can know the coordinates of the hanging object.
This kind of movement is pretty useful in physics since it has this solution, in general:
Where w is called the 'period' of the moment
The particular characteristic of this kind of movement is that it repeats itself after every period because of the properties of the sine function. (It behaves like a wave)
This kind of 'wave' behavior can be found in many places in nature. The life and death of generations of a certain species can be modeled with this kind of motion. A notable use of periodic motion is in quantum mechanics, the wave function of a particle is a periodic function. Some even argue that the financial system is a periodic motion, a series of abundance and crises though we don't know by far the function otherwise we would be able to anticipate every financial crises.