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SIMPLE HARMONIC MOTION

Oscillation

.Oscillation is a periodic, to and fro, bounded motion about a reference, usually the position of equilibrium.

Simple harmonic motion

.A simple harmonic motion can be conceived as a "to and fro" motion along an axis (say x-axis).

.A body will undergo SIMPLE HARMONIC MOTION when the force that tries to restore the object to its rest position is proportional to the displacement of the object.

.In order to simplify the matter, we choose origin as the point of reference about which particle oscillates.

.If we start our observation from positive extreme of the motion, then displacement of the particle "x" at a time "t" is given by :

x=Acos w t

where " w" is angular frequency and "t" is the time.

http://ecommons.uwinnipeg.ca/archive/00000030/02/shmani2.gif

.The figure shows the positions of the particle executing SHM

.The important thing to note here is that displacements in different intervals are not equal, suggesting that velocity of the particle is not uniform.

. An object is said to be in simple harmonic motion if the following occurs:
It moves in a straight line.

A variable force acts on it.

The magnitude of force is proportional to the displacement of the mass.

The force is always opposite in direction to the displacement direction.

The motion is repetitive and a round trip, back and forth, is always made in equal time periods.

.A pendulum and a mass on a spring both undergo this type of motion which can be described by a SINE WAVE .

.Thus ,there are two basic examples of simple harmonic motion:

springs and pendulums.

.Springs

.A spring that is oscillates an attached mass back and forth on a frictionless surface is an example of simple harmonic motion.

An undamped spring-mass system is a simple harmonic oscillator.

An undamped spring-mass system is a simple harmonic oscillator.

.The motion of the spring wrt time can be expressed as a function of Sine wave.

The image

.The magnitude and direction of the force in terms of its displacement is:

F= -kx

.k is a constant for the particular spring called the spring constant and x is the displacement.

.The right side of the equation is negative because the force always acts opposite the displacement

. When certain force is applied on the particle, the energy is stored in the form of potential energy which accelerates the particle from position 2.

.No force is acting when the mass returns to its original position (positions 1 and 3), but the mass keeps moving because it was moving before.

.In addition, at the original position, the mass is moving at its highest speed.

.Once it passes the original position, the force acts to slow it down and move it in the opposite direction. The entire cycle repeats again and again in a certain period of time.

.The period (T) of one full cycle can be calculated with the following equation:

T=2(m/k)

. The equation for the period, is independent of the maximum displacement.

.No matter how far you stretch or compress the spring, the period will remain the same.

.Pendulums

.The motion of a pendulum can be considered simple harmonic motion even though the bob hanging at the end of the string moves in a curve.

.This is because if the string is relatively long compared to the initial displacement, the curve made by the bob is close enough to a straight line.

. The pendulum works almost like the spring.

.The force is always pointing opposite to the displacement.

Pendulum Illustration

.The bob is moving the fastest when it passes its lowest point . The pendulum continuously moves back and forth.

.Also, the equation for the period of a pendulum is similar to that of the spring:

T=2(l/g)

L is the length of the string and g is the gravitation acceleration .

.The period does not depend at all on the mass or the initial displacement.

. A simple pendulum exhibits simple harmonic motion under the conditions of no damping and small amplitude.

. SHM and uniform circular motion

. One of the simplest of periodic motions is uniform circular motion.

.By shadow projecting both uniform circular motion and oscillatory simple harmonic motion onto a screen, one can show that these two seemingly different kinds of motion are actually identical.

.Simple harmonic motion can be visualized as the projection of uniform circular motion onto one axis.

Uniform Circular Motion
(radius A, angular velocity w)

Simple Harmonic Motion
(amplitude A, angular frequency w)

.Thus when a particle moves with uniform circular motion, its projection on a diameter moves with SHM.

Source:

http://library.thinkquest.org/16600/intermediate/simpleharmonicmotion.shtml
http://physics.mtsu.edu/~wmr/fourier_2.htm
http://www.physics.uoguelph.ca/tutorials/shm/phase0.html

Community Contributions - Articles by goIITians

.Springs

.Pendulums