Why is it important to be able to calculate the periodic motion for objects?

Answer 1

Because it is related to vibration and natural frequencies which affect how suspensions on cars work and such things as failures of bridges and buildings under wind stress. Any machine or vehicle which moves in a rhythmic way from a hammer press to a car bouncing off a chuckhole to a tall crane moving a load from one side to another and back is going to be affected if it hits a resonate frequency of part of it.