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Now that we have established the theory and equations behind harmonic motion, we
will examine various physical situations in which objects move in simple
harmonic motion. Previously, we worked with a mass-spring
system,
and will
examine other harmonic oscillators in a similar manner. Finally, after
establishing these applications, we can examine the similarity between simple
harmonic motion and uniform circular motion.

The Torsional Oscillator

Consider a circular disk suspended from a wire fixed to a ceiling. If the disk
is rotated, the wire will twist. When the disk is released, the twisted wire
exerts a restoring
force
on
the disk, causing it to rotate past its equilibrium point, twisting the wire the
other direction, as shown below. This system is called a torsional
oscillator.

It has been found experimentally that the torque exerted on the disk is
proportional to the angular displacement of the disk, or:

τ = - κθ

where κ is a proportionality constant, a property of the wire. Note the
similarity to our spring equation F = - kx. Since τ = Iα for any
rotational motion we can state that

- κθ = Iα = I

If we substitute m for I, k for κ, and x for θ we can see that
this is the exact same differential equation we had for our spring system. Thus
we may skip to the final solution, describing the angular displacement of the
disk as a function of time:

θ = θ_{m}cos(σt)

where θ_{m} is defined as the maximum angular displacement and σ is
the angular
frequency
given by
σ = . Note: It is important not to confuse
angular frequency and angular velocity. σ in this case refers to the
angular frequency of the oscillation, and cannot be used for angular velocity.

From our expression for angular frequency we can derive that

T = 2Π

This equation for the period of a torsional oscillator has a significant
experimental use. Suppose a body of unknown moment of inertia is placed on a
wire of known constant κ. The period of oscillation can be measured, and
the moment of inertia of the body can be determined experimentally. This is
quite useful, as the rotational inertia of most bodies cannot be easily
determined using the traditional calculus-based method.

From our examination of the torsional oscillator we have derived that its motion
is simple harmonic. This oscillator can almost be seen as the rotational
analogue of the mass-spring system: just as with the mass-spring we substituted
θ for x, I for m and κ for k. Not all simple harmonic oscillators
have such close correlation.

The Pendulum

Another common oscillation is that of the simple pendulum. The classic pendulum
consists of a particle suspended from a light cord. When the particle is pulled
to one side and released, it swings back past the equilibrium point and
oscillates between two maximum angular displacements. It is clear that the
motion is periodic--we want to see if it is simple harmonic.

We do so by drawing a free body diagram and examining the forces on the pendulum
at any given time.

The two forces acting on the pendulum at any given time are tension from the
rope and gravity. At the equilibrium point the two are antiparallel and cancel
exactly, satisfying our condition that there must be no net force at the
equilibrium point. When the pendulum is displaced by an angle θ, the
gravitational force must be resolved into radial and tangential components. The
radial component, mg cosθ, cancels with the tension, leaving net
tangential force;

F = - mg sinθ

In this case the restoring force is not proportional to the angular
displacement θ, but is rather proportional to the sine of the angular
displacement, sinθ. Strictly speaking, then, the pendulum does not
engage in simple harmonic motion. However, most pendulums function at very
small angles. If the angle is small we may make the approximation sinθθ. With this approximation we can rewrite our force
expression:

F = - mgθ

This equation does predict simple harmonic motion, as force is proportional to
angular displacement. We can simplify by noticing that the linear displacement
of the particle corresponding to an angle of θ is given by x = Lθ.
Substituting this in, we see that:

F = - mg = - x

Thus we have an equation in the same form as our mass-spring equation; in this
case k = . We can skip the calculus and simply state the period
of the pendulum:

pendulum

T = 2Π = 2Π

Note that the period, and thus the frequency, of the pendulum is independent of
the mass of the particle on the cord. It only depends on the length of the
pendulum and the gravitational constant. Keep in mind, also, that this is only
an approximation. If the angle exceeds more than fifteen degrees or so, the
approximation breaks down.