From Periodic Motion to SHM
Any motion that repeats itself after equal interval of time is called periodic motion.
If a particle, undergoing periodic motion, covers the same path back and forth about a mean position, it is said to be executing an oscillation (or vibration).
Using advanced mathematics, it can be shown that the displacement of a particle in an oscillation can always be expressed in terms of sines and cosines.
This, coupled with the fact that the term harmonic is generally applied to expressions containing sine and cosine functions actuates us to use the term harmonic motion for the oscillation.
Every harmonic motion is characterised by:
(i) time period
(iii) amplitude and
The Time Period (T) of a harmonic motion is the time required to complete one oscillation (or cycle).
The frequency (ν or n or f) of a harmonic motion is the number of oscillations per unit time.
From these definitions, it follows;
Our topic under discission is simple harmonic motion (SHM) and at this stage we just state it is a special type of harmonic motion.
Recalling Equilibrium :
Before investigating what exactly an SHM is, let us have a brief review of equilibrium of a particle (or body).
Without disturbing the continuity of discussion, it may be stated here that when a body is slightly disturbed from its stable equilibrium it may execute SHM.
This statement shows that equilibrium plays an important role in SHM and that’s why we are doing this review.
Meaning of Equilibrium :
For a particle or body,
If ΣF→ = 0 it is said to be in translational equilibrium.
If Στ→ = 0 it is said to be in rotational equilibrium.
If ΣF→ is zero for a particle then Στ→ is also zero (or Στ→ is zero then ΣF→ is also zero). Therefore for a particle to be in equilibrium either we have to show ΣF→ = 0 or Στ→ = 0
A rigid body is said to be in equilibrium only when it simultaneously confirms translational as well as rotational equilibrium. So, for showing a rigid body in equilibrium we have to show ΣF→ = 0 as well as Στ→ = 0