(i) Periodic Motion: When a moving object returns back to its original position after a particular interval of time and repeats its motion then this motion is called the periodic motion. A motion that repeats itself after regular intervals of time is known as the periodic motion. Example: Motion of planets around the Sun, motion of hands of a clock motion of the blades of a fan, Moon revolving around the Earth. To complete a full cycle the body takes a definite time which is called time period and is denoted by T
(ii) Non-periodic Motion: When an object does not repeat its motion after a definite time interval then its motion is called non-periodic motion. Example: Throwing a ball, shooting an arrow from a bow, etc.
(iii) Oscillatory Motion: If a body moves to and fro repeatedly about a mean position; then the motion of the body is called vibratory motion or oscillatory motion. It is a special form of periodic motion.
Example:
(i) When guitar strings are vibrated then they oscillate around its mean position.
(ii) A piece of wood which is floating on the surface of water, if slightly pushed and left then it does up and down vibration motion.
(iii) An iron plate clamped at one end if pushed on one side and left then it does vibrational motion around the equilibrium position.
(iv) A weight tied (hanged) from a spring if pulled and left, then it does up and down periodic motion.
Every oscillatory motion is periodic motion, but all periodic motions are not oscillatory motions.
(v) Time Period: The time taken to complete one oscillation by a body is called its time period. It is generally represented by T. Its SI unit is second (s). Time period is that minimum time interval after which the body again starts repeating its vibration motion.
(vi) Frequency: The number of periodic motions in one second(s) i.e. the number of completed oscillations is called frequency. It is generally represented as n or y. Its SI unit is Hertz (Hz).
Relationship between Frequency and Time Period
1f the time period of a body executing periodic motion is T and frequency is n.
Number of oscillations completed in T second = 1.
Hence, number of oscillations completed in 1 second
= \(\frac{1}{T}\)
But, the number of oscillations completed in 1 second is called frequency (n).
\(\therefore\ n = \frac{1}{T}\ ....(1)\)
(vii) Displacement and Amplitude: We consider a physical quantity in normal oscillatory motion which changes with time and whose measure is possible. The change in this physical quantity relative to its mean position is called displacement and in oscillatory motion, the displacement at any instant relative to the mean position is called instantaneous displacement. The value of instantaneous displacement changes with time.
The maximum displacement of a particle from its mean position when a particle does oscillatory or vibratory motion is called amplitude. In linear oscillatory motion linear amplitude is represented as ‘a’ and in angular oscillatory motion angular amplitude is represented as θ0.
(viii) Phase Angle: The physical quantity representing the nature (state) of vibrating particle’s motion at any instant is called phase angle.
From instantaneous displacement
y= a sin(ω0t + Φ);
(ω0t + Φ) is known as phase angle or phase. By knowing the phase the instantaneous position of the oscillating system can be calculated. In phase angle keeping the value of t = 0, then the value of phase angle Φ is called the First (Initial) Phase Angle in simple recurring (periodic) motion or is also known as phase constant.
The difference in the phase angles of two particles executing simple periodic motion is called “phase difference”. It is represented as ∆Φ.
When phase difference is ∆Φ = π then, both simple periodic motions are called in opposite phases. Whereas, when ∆Φ = 0 or 2π then the two particles are said to be in same phase. In same phase position both the particles do motion in the same direction.
(ix) Periodic Function: Periodic function is that function whose repetition is after a definite time interval. Periodic function is that function which is used to represent periodic motion.
If there is any function F(θ);
F(θ + T) = F(θ) .......(2)
Then, F(θ) is called periodic function.
Here, T is time period and θ is the argument of a function.
It is clear from Eq.(2) that if the value of the argument changes from θ to θ + T then the value of the function does not change. Similarly, if θ is increasesd by one more period then also the value of the function does not change.
F(θ + 2T) = F(θ)
F(θ + kT) = F(θ)
where, k = 0, ±1, ±2, .....................
A simple periodic function is a sine or cosine function.
f(t + T) = f(t)
g(t + T) = g(t)
where, f(t) = A sin ωt and g(t) = A cos ωt can be taken as the periodic functions.
