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SIMPLE HARMONIC MOTION

Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring.

Additionally, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum and

molecular vibration.

Simple harmonic motion provides the basis of the characterization of more complicated motions through the techniques of

Fourier analysis.

Simple harmonic motion had shown both in real space and phase space. The orbit is periodic. (Here the velocity and

position axes have been reversed from the standard convention in order to align the two diagrams)

A simple harmonic oscillator is attached to the spring, and the other end of the spring is connected to a rigid support such

as a wall. If the system is left at rest at the equilibrium position then there is no net force acting on the mass. However, if the mass isdisplaced from the equilibrium position, a restoring elastic force which obeys Hooke's law is exerted by the spring.

Mathematically, the restoring force F is given by:

where F is the restoring elastic force exerted by the spring (in SI units: N), k

is the spring constant (Nm1

), and x is the displacement from the

equilibrium position (in m).

For any simple harmonic oscillator:

When the system is displaced from its equilibrium position, arestoring force which obeys Hooke's law tends to restore the system to

equilibrium.

Once the mass is displaced from its equilibrium position, itexperiences a net restoring force. As a result, it accelerates and starts

going back to the equilibrium position. When the mass moves closer to the

equilibrium position, the restoring force decreases. At the equilibrium

position, the net restoring force vanishes. However, at x= 0, the

momentum of the mass does not vanish due to the impulse of the restoring force that has acted on it. Therefore, the mass

continues past the equilibrium position, compressing the spring. A net restoring force then tends to slow it down, until its velocity

vanishes, whereby it will attempt to reach equilibrium position again. As long as the system has no energy loss, the mass will

continue to oscillate. Thus, simple harmonic motion is a type ofperiodic motion.

Dynamics of simple harmonic motion

For one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential

equation with constant coefficients, could be obtained by means ofNewton's second law and Hooke's law.

where m is the inertial mass of the oscillating body, xis its displacement from the equilibrium (or mean) position, and kis the spring

constant.

Therefore,

Solving the differential equation above, a solution which is a sinusoidal function is obtained.

Where

In the solution, c1 and c2 are two constants determined by the initial conditions, and the origin is set to be the equilibrium

position.[A]

Each of these constants carries a physical meaning of the motion: A is the amplitude (maximum displacement from the

equilibrium position), = 2fis the angular frequency, and is the phase.[B]

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Using the techniques ofdifferential calculus, the velocity and acceleration as a function of time can be found:

Acceleration can also be expressed as a function of displacement:

Then since = 2f,

And since T= 1/fwhere T is the time period,

These equations demonstrate that the simple harmonic motion is isochronous (the period and frequency are independent of the

amplitude and the initial phase of the motion).

Energy of simple harmonic motion

The kinetic energy Kof the system at time tis

And the potential energy is

The total mechanical energy of the system therefore has the constant value

The following physical systems are some examples ofsimple harmonic oscillator.

Mass on a spring

(DRAWING?)

A mass m attached to a spring of spring constant kexhibits simple harmonic motion in space. The equation

Shows that the period of oscillation is independent of both the amplitude and gravitational acceleration.

Uniform circular motion

Simple harmonic motion can in some cases be considered to be the one-dimensional projection ofuniform circular motion.

If an object moves with angular velocity around a circle of radius rcentered at the origin of the x-yplane, then its motion along

each coordinate is simple harmonic motion with amplitude rand angular frequency .

Mass on a simple pendulum

In the small-angle approximation, the motion of a simple pendulum is approximated by simple harmonic motion. The

period of a mass attached to a string of length with gravitational acceleration g is given by

This shows that the period of oscillation is independent of the amplitude and mass of the pendulum but not the acceleration due to

gravity (g), therefore a pendulum of the same length on the Moon would swing more slowly due to the Moon's lower gravitational

acceleration.

This approximation is accurate only in small angles because of the expression for angular acceleration being proportional to

the sine of position:

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Where I is the moment of inertia. When is small, sin and therefore the expression becomes

This makes angular acceleration directly proportional to , satisfying the definition of simple harmonic motion.

Pendulum

A pendulum is a weight suspended from a pivot so that it can swing freely.[1]

When a pendulum is displaced from its resting

equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position.

When released, the restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium position,

swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period. A pendulum swings

with a specific period which depends (mainly) on its length.

From its discovery around 1602 by Galileo Galilei the regular motion of pendulums was used for timekeeping, and was the

world's most accurate timekeeping technology until the 1930s.[2]

Pendulums are used to regulate pendulum clocks, and are used in

scientific instruments such as accelerometers and seismometers. Historically they were used as gravimeters to measure the

acceleration of gravity in geophysical surveys, and even as a standard of length. The word 'pendulum' is new Latin, from the Latin

pendulus, meaning 'hanging'.[3]

The simple gravity pendulum is an idealized

mathematical model of a pendulum. This is a weight (or

bob) on the end of a massless cord suspended from a pivot,

without friction. When given an initial push, it will swing

back and forth at constant amplitude. Real pendulums are

subject to friction and air drag, so the amplitude of their

swings declines.

A simple pendulum is one which can be considered to be apoint mass suspended from a string or rod of negligible

mass. It is a resonant system with a single resonant

frequency. For small amplitudes, the period of such a

pendulum can be approximated by:

If the rod is not of negligible mass, then it must be treated as a physical pendulum

Pendulum Motion

The motion of a simple pendulum is like simple harmonic motion in that the equation for the angular displacement is

which is the same form as the motion of a mass on a spring:

The anglular frequency of the motion is then given by

compared to for a mass on a spring.

"Simple gravity

pendulum" model

assumes no friction or

air resistance.

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The frequency of the pendulum in Hz is given by

and the period of motion is then

Period of Simple Pendulum

A point mass hanging on a massless string is an idealized example of a simple pendulum. When displaced from its equilibrium point,

the restoring force which brings it back to the center is given by:

For small angles , we can use the approximation

in which case Newton's 2nd law takes the form

Even in this approximate case, the solution of the equation uses calculus and differential equations. The

differential equation is

and for small angles the solution is:

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