The Science of Sound and Music
PHYS-013, Fall 2002
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DEFINITION : Periodic
Motion consists in the indefinite repetition of a certain kinematic
pattern so that the body has exactly the same position and velocity at equal
time intervals.
DEFINITION : Simple
Harmonic Motion is the simplest type of periodic motion, in which the
driving force is always proportional and opposite to the displacement.
In other words, SHM is produced when an elastic or a quasi-elastic force acts
upon a body.
Note:
Mathematically, SHM is a motion in which the displacement can be expressed as a
trigonometric function of time of the type 'sine' or 'cosine': x = A sin (2pi´t)
EXAMPLES: Motion of an elastic
pendulum (an object hanging from a spring), a simple gravitational pendulum (a
body suspended from a string), a child’s swing, a torsional pendulum, vibration
of a blade or a stick with a fixed end, vibration of a guitar or piano string,
the bouncing of an elastic floor or a bridge caused by walking, the bouncing of
a diving board of a pool, etc.
DEFINITION Amplitude,
A , is the maximum displacement from the equilibrium position. 'SI'
Unit = meter (m)
DEFINITION : Period
,P, is the shortest time
interval between any pair of identical repetitions of the motion . 'SI',
or the duration of a single cycle of the periodic motion. Unit = second
(s)
DEFINITION : Frequency
, f , is the number of repetitions of the motion each second. Unit
f
= 1/P or P = 1/f , or Pf = 1
'SI'
Unit for frequency = Hz (hertz) or
1/s or s-1 – is the frequency of a periodic motion that makes a
complete cycle in one second
NOTE
: any object submitted to an elastic force, when perturbed from the
equilibrium position and then released, begins to oscillate due to a
combination of the elastic restoring force trying to bring the body to its
equilibrium position on one hand, and, on the other hand, inertia making the
body pass the equilibrium position without stopping, until it reaches the
maximum position on the other side of equilibrium, then starts to muve in the
opposite direction.
For a single
oscillator, the oscillation is a simple harmonic motion (SHM). However, if more oscillators
connected together are present, the motion can be very complex, while still
being periodic.
Fourier’s Theorem: Any periodic motion, however
complex, is the result of the superposition of harmonic oscillations, whose
frequencies are integer multiples of a
given ‘fundamental frequency’.
DEMO
1.
Oscillations of a cart between two springs
2.
Oscillations of the piston of a syringe
the period of oscillation, P, is given by
the equation: P = (2 pi)´Sqr(k/m)
where ‘Sqr’ means ‘square
root of’, k is the stiffness of the spring, m is the mass of the object, and
[pi]= 3.14.
DEMO
1.
Effect of stiffness on the period of an elastic pendulum
2.
Effect of mass on the period of an elastic pendulum
DEFINITIION : The
simple harmonic motion (SHM) is the motion produced by an elastic
force acting on o single object, or the expression for the total energy of a
harmonic oscillator:
Eosc
= ½kA²
Since , k = 4[pi]²mf²
it follows that the energy is:
Eosc = 2[pi]²f ²mA²
Coupled Oscillators
DEMO
1. Individual modes of two elastically coupled carts
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