Simple Harmonic Motion (SHM) is a type of periodic motion where an object oscillates back and forth around a central point, and the acceleration of the object is directly proportional to its displacement from that central point, and is always directed towards the central point.

1. Period(T): This is the total time taken by a vibrating body to complete one oscillation or revolution about a central point. The S.I. unit of period is seconds(s). $$ period (T) = \frac{\text{time taken}}{\text{no of revolutions}} $$ $$ T = \frac{t}{n} $$
2. Frequency(f) : This is number of oscillations or revolutions per second made by a vibrating body. The S.I. unit of frequency is Hertz (one cycle per second). $$ Frequency (f) = \frac{\text{no of rev.}}{\text{time taken}} $$ $$ f = \frac{n}{t} $$ $$ f = \frac{1}{T} $$
3. Amplitude(A): This is the maximum displacement of the body performing S.H.M from the central point.

The angular velocity \( (\omega) \) of a body undergoing S.H.M can be defined as the angle turned by the body per time. $$ \omega = \frac{\theta}{t} $$ $$ \text{where, }{\theta} = 2π \text{ radians or 360°} $$ $$ \therefore \omega = \frac{2π}{T} = 2πf $$ $$ \text{Where, T} = \text{period of revolution} $$ $$ Also, $$ $$ \text{linear speed, v} = \frac{s}{t} $$ $$ \text{where, s} = displacement $$ $$ \text{But, one radian, } \theta= \frac{s}{r} $$ $$ \text{Hence, s} = r\theta $$ $$ \text{Also, } \omega = \frac{\theta}{t} $$ $$ \text{Substituting, } \theta = \frac{s}{r} $$ $$ \omega = \frac{s}{r}.\frac{1}{t} $$ $$ \text{Since, v} = \frac{s}{t} $$ $$ \omega = v.\frac{1}{r} $$ $$ v = \omega r = \omega A $$ $$ \text{where, A} = Amplitude $$ Note: The important formulas are:

1. \( v = \omega r = \omega A \)
2. \( \omega = 2πf = \frac{2π}{T} \)
3. \( \omega = \frac{\theta}{t} \)

Example 1: The period of oscillation of a particle executing Simple Harmonic motion is 4π seconds. If the amplitude of oscillation is 3m, calculate the maximum speed of the particle. (WAEC)

From the parameters given, we can solve for \( \omega \) first $$ \text{Since, } \omega = \frac{2π}{T} $$ $$ \omega = \frac{2π}{4π} $$ $$ \omega = 0.5 \text{ rad/s} $$ We can then find the speed (v) $$ v = \omega A $$ $$ v = 0.5 × 3 $$ $$ v = 1.5 \text{ m/s} $$

Example 2: A body executing a simple harmonic motion has an angular velocity of 40 radians per second. If it has a maximum displacement of 6cm, calculate the linear speed. (NECO)

Example 3: A particle in circular motion performs 30 oscillations in 6 seconds. Its angular velocity is? (JAMB)