simple harmonic motions complete notes class 12
Simple harmonic motion (SHM) is a type of oscillatory motion that occurs when the restoring force acting on an object is directly proportional to the object's displacement from its equilibrium position and is directed towards the equilibrium position. In other words, the object oscillates back and forth around its equilibrium position.
Some key concepts and formulas related to simple harmonic motion are:
Displacement (x): the distance from the equilibrium position.
Amplitude (A): the maximum displacement of the object from its equilibrium position.
Frequency (f): the number of oscillations per unit time (usually measured in Hertz, Hz).
Period (T): the time taken for one complete oscillation (measured in seconds, s).
Angular frequency (ω): the rate of change of the phase angle (measured in radians per second, rad/s).
Phase angle (φ): the angle between the object's position and a fixed reference point.
Acceleration (a): the rate of change of velocity (measured in meters per second squared, m/s^2).
Velocity (v): the rate of change of displacement (measured in meters per second, m/s).
Some important formulas related to SHM are:
x = A cos(ωt + φ), where x is the displacement at time t.
ω = 2πf = 2π/T, where ω is the angular frequency, f is the frequency, and T is the period.
a = -ω^2x, where a is the acceleration and x is the displacement.
v = -ωA sin(ωt + φ), where v is the velocity.
Some other key points to keep in mind about SHM include:
The restoring force is always directed towards the equilibrium position.
The object's motion is periodic, meaning it repeats itself after a certain time interval.
The motion is sinusoidal, meaning it can be described by a sine or cosine function.
The period and frequency of the motion are independent of the amplitude.
The velocity of the object is zero at the equilibrium position and maximum at the amplitude.
The acceleration is maximum at the equilibrium position and zero at the amplitude.
Applications of SHM include pendulums, springs, and many other types of oscillators. Understanding simple harmonic motion is important for many fields, including physics, engineering, and mathematics.
MOREVER
(SHM) is a key idea in material science that depicts the movement of items that are dependent upon a reestablishing force that is relative to their removal from a harmony position. In this outline, we will examine the standards, attributes, and numerical formulae of SHM.
Principles of Simple Harmonic Motion:
The movement of an item dependent upon a restoring force that is relative to its relocation from a balance position is called Simple Harmonic Motion:
F = - kx
Where F is the power, x is the dislodging from the harmony position, and k is the spring consistent, which decides the strength of the reestablishing force.
The movement of an article going through SHM is described by its adequacy, recurrence, period, and stage. The abundancy is the greatest removal of the article from its balance position, while the recurrence is the quantity of motions per unit time. The period is the time taken for one complete pattern of swaying, and the stage is the place of the article as for its cycle.
Characteristics of Simple Harmonic Motion:
The movement of an article going through SHM has a few trademark properties that recognize it from different sorts of movement:
Restoring Force: The movement is intermittent, implying that it rehashes the same thing after a specific timeframe.
velocity: The movement is driven by a reestablishing force that is relative to the relocation from the balance position.
Speed: The speed of the item is greatest at the balance position and least at the limits of its movement.
Speed increase: The speed increase of the article is greatest at the limits of its movement and zero at the harmony position.
Energy: The all out energy of the framework is steady, and it is divided among dynamic energy and likely energy.
Numerical Formulae:
The movement of an article going through SHM can be portrayed utilizing numerical formulae:
Displacement: : x = A cos(ωt + φ)
Where x is the removal of the item, An is the sufficiency, ω is the rakish recurrence, t is the time, and φ is the stage point.
Speed: v = - Aω sin(ωt + φ)
Where v is the speed of the article.
a = - Aω^2 cos(ωt + φ)
Where an is the speed increase of the item.
period: T = 2π/ω
Where T is the time of the movement.
Recurrence or frequency: f = 1/T = ω/2π
Where f is the recurrence of the movement.
Utilizations of SHM
SHM has various applications in science and designing, including:
Pendulum clocks and springs in mechanical frameworks.
Waves, for example, sound waves and light waves, which can be displayed as SHM.
Motions in electrical circuits, for example, LC circuits.
Vibrations in instruments and speakers.
Sub-atomic vibrations in compound responses.
at last,
Simple harmonic motion
is a major idea in physical science that depicts the movement of items that are dependent upon a reestablishing force that is corresponding to their removal from a balance position. It is described by its amplitude, wavelength, period, and stage, and it has various applications in science and designing. Understanding the standards and numerical formulae of SHM is fundamental for a large number of logical and innovative applications.
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