Simple Harmonic Motion Concept Page - 1

Definition

Oscillatory Motion

A phenomenon, process or motion, which repeats itself after equal intervals of time, is called periodic. If a body moves to and fro repeatedly about a mean position it is called oscillatory motion.

Definition

Periodic Motion

Examples of Periodic Motion those are not oscillatory:
by a rocking chair, a bouncing ball, a vibrating tuning fork, a swing in motion, the Earth in its orbit around the Sun, and a water wave.

Definition

Angular Simple Harmonic Motion

If we rotate the disk from its rest position (where the reference line is at 0) and release it, it will oscillate about that position in angular simple harmonic motion and q$q$ varies  in time according to the relationship:
q(t)=qmcos(wt+f)$q(t)=q_{m}cos(wt+f)$
in which qm,w,f$q_m,w,f$ are constants of the motion. By differentiating q(t) with respect to time twice, we obtain an expression for the angular acceleration of the  disk.
a(t)=−w2q(t)$a(t)=-w^{2}q(t)$

Definition

Periodic Motion

Periodic motion, motion repeated in equal intervals of time. Periodic motion is performed, for example, by a rocking chair, a bouncing ball, a vibrating tuning fork, a swing in motion, the Earth in its orbit around the Sun, and a water wave.

Definition

Periodic Motion Graph

Periodic Motion Graph is as shown:

Definition

Mathematical Condition of Simple Harmonic Motion

When the restoring force of any motion has equation of form:
F=−kx$F=-kx$
Then fundamentally this equation represents simple harmonic motion.

Example

how to identify periodic and non periodic motion

A motion that repeats itself at regular intervals of time is called periodic motion.
Example :  Explain whether Functions 1. sinωt+cosωt$sin\omega t+cos\omega t$ and 2. e−ωt$e^{-\omega t}$ are periodic or not?
1. we can write sinωt+cosωt$sin\omega t+cos\omega t$ as 2√sin(ωt+π4)$\sqrt{2}sin(\omega t+{\displaystyle \frac{\pi }{4}})$
 = 2√sin(ωt+π4+2π)$\sqrt{2}sin(\omega t+{\displaystyle \frac{\pi }{4}}+2\pi )$
 = 2√sin[ω(t+2πω)+π4]$\sqrt{2}sin[\omega (t+{\displaystyle \frac{2\pi }{\omega }})+{\displaystyle \frac{\pi }{4}}]$
The periodic time of function is 2πω${\displaystyle \frac{2\pi }{\omega }}$
So it is periodic motion.
2. The function e−ωt$e^{-\omega t}$ decreases monotonically with increasing time and tends to zero as t approaches infinity and thus never repeats its value.
Hence it is non periodic motion.

Definition

Define frequency of Periodic Motion

Frequency is the number of cycles completed per second. It is the reciprocal of period.
Note: The unit representing cycles per second is hertz.

Definition

Define Periodic Motion

Periodic motion, motion repeated in equal intervals of time. Periodic motion is performed, for example, by a rocking chair, a bouncing ball, a vibrating tuning fork, a swing in motion, the Earth in its orbit around the Sun, and a water wave.
In each case the interval of time for a repetition, or cycle, of the motion is called a period, while the number of periods per unit time is called the frequency.

Definition

Relation between frequency and time period

Relation between frequency and time period:
f=1T$f={\displaystyle \frac{1}{T}}$

Definition

Difference between Oscillatory and Simple Harmonic Motion

Simple Harmonic Motion: Any motion that repeats itself after equal interval of time is called Simple Harmonic Motion.
The relation F=−kx$F=-kx$ is always followed in this type of motion.

Periodic Motion: If an object in periodic motion moves back and forth over same path, the motion is called Oscillatory motion.

Definition

Non-Harmonic Oscillator

When introducing oscillations at the beginning typically only discuss the harmonic oscillator. This is natural, but this focus may disguise its special features (e.g., frequency amplitude independence). Using the air track, our students investigate oscillations experimentally and match experimental data with theoretical. An additional theoretical choice, along with the damped harmonic oscillator, is the solution for the non-harmonic force law: F=F0sin(x)v$F=F_{0}sin(x)v$

When there is no damping (=0) there is a difference of less than 0.06 between the non-harmonic and the harmonic oscillator curves if both have unit amplitude and the same frequency and phase. Still, the accuracy of the data allows discrimination between these models. However, the difference shows up most clearly when damping is present since then the period of the non-harmonic oscillator changes with the amplitude and curves that start in phase become out of phase. A simple approximate solution for this damped non harmonic oscillator has been obtained.

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