Final answer:
Periodic motion is defined by its regular repetition and is quantified by its period and frequency. Oscillating motion is a form of periodic motion characterized by a back-and-forth movement from a central position, with Simple Harmonic Motion being a special case where the restoring force is proportional to displacement. To differentiate between them in equations, periodic functions with constant coefficients may indicate SHM, a type of oscillating motion.
Step-by-step explanation:
Periodic motion refers to a repetitious oscillation where an object moves in a regular cycle. Its defining characteristic is that it repeats after equal intervals of time, known as the period (T). The frequency (f) of periodic motion is the number of times the motion repeats per unit time, and these two quantities are inversely related (f = 1/T).
Oscillating motion, on the other hand, is a specific type of periodic motion where an object moves back and forth through a central neutral position due to a restoring force. Simple Harmonic Motion (SHM) is a refined type of oscillating motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that displacement.
For a simple pendulum, which consists of a mass m suspended by a string or wire of length L, the motion can be approximated as SHM for small amplitudes (less than about 15 degrees). The frequency of the simple pendulum in SHM is independent of its amplitude. This is not true for all types of oscillating motions, as some may have a frequency dependent on amplitude.
To determine from the equation of motion whether it represents periodic or oscillating motion, you can look for the presence of periodic functions (like sine or cosine) with constant coefficients related to frequency and period, which suggest SHM and therefore an oscillating motion.