simple harmonic motion (SHM)
A form of wave motion in which a moving particle
traces a path symmetric about a midpoint or equilibrium position, through
which it passes at regular intervals of time. A particle is said to move
in simple harmonic motion when it oscillates along a line about a central
or equilibrium point, O, so that its acceleration
towards O is always proportional to its distance from O.
Thus, if a point P moves in a circle,
center O and radius r, with a constant angular
velocity ω, the projection of P on any diameter
will move in simple harmonic motion. If the distance from O of
the projection of P on a vertical diameter is y, at time
t, then a graph of y against t will give a sine
wave of amplitude r and equation
y = r sinωt. This equation may be rewritten
in the more general form:
|Hypothetical example of simple harmonic motion.
A body is suspended on springs midway between two hooks. The body
is displaced from its equilibrium position, taking care that it remains
equidistant from both hooks. If gravitational effects are ignored,
the body will described an SHM about the midline when it is released.
The force (F1) toward the center of motion is the resultant
of the components toward the midline of the tensions (T1,
T2) in the springs. Undamped, the motion will continue
indefintely, potential energy in the springs being converted to kinetic
energy in the motion of the body as the body approaches the midline,
and the reverse occurring as the body approaches its maximum amplitude
(A) again. The graphical expression of the motion, displacement
(y) against time (t), is shown in the diagram above.
It is the graph of y = A sin2πft, where
f is the frequency of the motion.
y = r sin2π(t /T - x/λ)
where T is the period of the wave,
λ its wavelength and
x the distance it has traveled from O in time t.
Simple harmonic motion derives its name from the fact that the vibrations
produced by musical instruments (e.g., a string of a violin, the legs of
a tuning fork), and hence the sound waves
they propagate, approximate to it. In fact these, as all other vibrations
and wave motions, may be treated as compounded of a number of SHMs.
AND WAVE PHENOMENA