PERIODIC AND OSCILLATORY MOTIONS -

PERIODIC MOTION

The motion which repeats itself after a regular time interval is called periodic motion, and the time taken is called its Time Period.

Examples of some periodic motions are

An insect climbs up a ramp and falls down it comes back to the initial point and repeats the process identically. Its graph looks like as

Similar, a child climbs up a step, come down, and repeats the process.

Similar, When we play the game of bouncing a ball off the ground, between our palm and the ground. ( The curve is parabolic because of the equation of motion $\fn_cm s=ut+\frac{1}{2}gt^2$ )

OSCILLATORY MOTION

If a body in periodic motion moves along the same path but to and fro nature about an equilibrium position is called oscillatory motion (Oscillation) or vibratory motion (Vibration.

Example. The motion of bob in simple pendulum.

NOTE

All Oscillatory motions are periodic motions but all periodic motions are not necessary oscillations.
The difference between oscillation and vibration is, In oscillation generally frequency is small but in vibration generally frequency is large.
In oscillatory motions, a restoring force is one that tries to pull or push an object back to its equilibrium position.

SIMPLE HARMONIC MOTION

SHM is the simplest form of oscillatory motion. In SHM particle moves to and fro manner about the mean position but in a straight path.

In SHM force is always acts along with the mean position and it is proportional to the displacement from the mean position.

i.e $\fn_cm \large \vec{F}\propto \vec{X}$

$\fn_cm \large \left [ \vec{F}=-k \vec{X} \right ]$

where k is force constant and F is restoring force.

Now, $\fn_cm \large \frac{\vec{F}}{m}=-\frac{k}{m}\vec{X}$

$\fn_cm \large \left [ \vec{a}=-\omega ^2 \vec{x} \right ]\;\;\;\;\;\left [ \omega=\sqrt{\frac{k}{m}} \right ]$

where $\fn_cm \large \omega^2$ is positive constant ( $\fn_cm \large \omega$ is known as angular frequency) [ This will be discussed later]

NOTE

All simple harmonic motions are oscillatory motion but all oscillatory motion is not necessary SHM.

DISPLACEMENT

In SHM, it refers to change with time of any physical property under consideration. For example, In simple pendulum angle from vertical refer to displacement, the voltage across a capacitor in A.C, pressure variation in the propagation of the sound wave, the variation of the electric and magnetic field in electromagnetic wave etc are the common example of displacement variable.

AMPLITUDE

It is the maximum displacement variable from the mean position. It is generally denoted by A.

TIME PERIOD

It is the time taken to complete one repetition. It is denoted by T

$\fn_cm \large T=\frac{2\pi}{\omega}\;\;\;\;\; or \;\;\;\;\; T=2\pi \sqrt{\frac{m}{k}}$

FREQUENCY

It is the number of repetitions per second. It is denoted by $\fn_cm \large f$ or $\fn_cm \large \nu$ .

i.e $\fn_cm \large f=\frac{1}{T}\;\;\;\;\; or \;\;\;\;\; f=\frac{\omega}{2 \pi}$

Its S.I unit is cycle/second or Hertz (Hz)

NOTE:- Frequency is not necessarily an integer.

Angular frequency is the complete revolution of a circle i.e 2π per second. It is denoted by ω.

$\fn_cm \large \left [ \omega=2\pi f \right ]$

1. On average a human heart is found to beat 75 times in a minute. Calculate its frequency and period.

2. Which of the following examples represent periodic motion?
(a) A swimmer completing one (return) trip from one bank of a river to the other and back.
(b) A freely suspended bar magnet displaced from its N-S direction and released.
(c) A hydrogen molecule rotating about its center of mass.
(d) An arrow released from a bow.

3. Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
(a) the rotation of earth about its axis.
(b) motion of an oscillating mercury column in a U-tube.
(c) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lowermost point.
(d) general vibrations of a polyatomic molecule about its equilibrium position.

4. Figure depicts four x-t plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic motion) ?