Waves
Vibrations
Things wiggle. They do the
back and forth. They vibrate; they shake; they oscillate. These phrases
describe the motion of a variety of objects. They even describe the motion of
matter at the atomic level. Even atoms wiggle - they do the back and forth.
Wiggles, vibrations, and oscillations are an inseparable part of nature. In
this chapter of The Physics Classroom Tutorial, we will make an effort to
understand vibrational motion and its relationship to waves. An understanding
of vibrations and waves is essential to understanding our physical world. Much
of what we see and hear is only possible because of vibrations and waves. We
see the world around us because of light waves. And we hear the world around us
because of sound waves. If we can understand waves, then we will be able to
understand the world of sight and sound.
To begin our ponderings of vibrations and waves, consider one
of those crazy bobblehead dolls that you've likely seen at baseball stadiums or novelty
shops. A bobblehead doll consists of an oversized replica of a person's head attached
by a spring to a body and a stand. A light tap to the oversized head causes it
to bobble. The head wiggles; it vibrates; it oscillates. When pushed or somehow
disturbed, the head does the
back and forth. The back and forth doesn't happen forever. Over time, the
vibrations tend to die off and the bobblehead stops
bobbing and finally assumes its usual resting position.
The bobblehead doll is a good illustration of many of
the principles of vibrational motion. Think about how you would describe the
back and forth motion of the oversized head of a bobblehead doll.
What words would you use to describe such a motion? How does the motion of
the bobblehead change over time? How does the motion of one bobblehead differ
from the motion of another bobblehead? What quantities could you measure to describe
the motion and so distinguish one motion from another motion? How would you
explain the cause of such a motion? Why does the back and forth motion of the bobblehead finally
stop? These are all questions worth pondering and answering if we are to
understand vibrational motion. These are the questions we will attempt to
answer in Section 1 of this chapter.
Like any object that undergoes vibrational motion, the bobblehead has
a resting position. The resting
position is the position assumed by the bobblehead when it is not vibrating. The resting
position is sometimes referred to as the equilibrium position. When an object is positioned at its equilibrium position, it is in a state of equilibrium. As discussed in
the Newton's Law Chapter of the Tutorial, an object which
is in a state of equilibrium is experiencing a balance of forces. All the
individual forces - gravity, spring, etc. - are balanced or add up to an
overall net force of 0 Newtons. When a bobblehead is at the equilibrium position, the forces on the bobblehead are
balanced. The bobblehead will remain in this position until somehow disturbed from its
equilibrium.
If a force is applied to the bobblehead, the
equilibrium will be disturbed and the bobblehead will begin vibrating. We could use the
phrase forced vibration to
describe the force which sets the otherwise resting bobblehead into
motion. In this case, the force is a short-lived, momentary force that begins
the motion. The bobblehead does its back and forth, repeating the motion over and over. Each
repetition of its back and forth motion is a little less vigorous than its
previous repetition. If the head sways 3 cm to the right of its equilibrium
position during the first repetition, it may only sway 2.5 cm to the right of
its equilibrium position during the second repetition. And it may only sway 2.0
cm to the right of its equilibrium position during the third repetition. And so
on. The extent of its displacement from the equilibrium position becomes less
and less over time. Because the forced vibration that initiated the motion is a
single instance of a short-lived, momentary force, the vibrations ultimately
cease. The bobblehead is said to experience damping. Damping
is the tendency of a vibrating object to lose or to dissipate its energy over
time. The mechanical energy of the bobbing head is lost to other objects.
Without a sustained forced vibration, the back and forth
motion of the bobblehead eventually ceases as energy is dissipated to other objects. A
sustained input of energy would be required to keep the back and forth motion
going. After all, if the vibrating object naturally loses energy, then it must
continuously be put back into the system through a forced vibration in order to
sustain the vibration.
A vibrating bobblehead often does the back and forth a number of
times. The vibrations repeat themselves over and over. As such, the bobblehead will
move back to (and past) the equilibrium position every time it returns from its
maximum displacement to the right or the left (or above or below). This begs a
question - and perhaps one that you have been thinking of yourself as you've
pondered the topic of vibration. If the forces acting upon the bobblehead are
balanced when at the equilibrium position, then why does the bobblehead sway
past this position? Why doesn't the bobblehead stop the first time it returns to the
equilibrium position? The answer to this question can be found in Newton's first law
of motion. Like any moving object, the motion of a
vibrating object can be understood in light of Newton's laws. According to Newton's
law of inertia, an object which is moving will continue its motion if the
forces are balanced. Put another way, forces, when balanced, do not stop moving
objects. So every instant in time that the bobblehead is at the equilibrium position, the
momentary balance of forces will not stop the motion. The bobblehead keeps
moving. It moves past the equilibrium position towards the opposite side of its swing. As the bobblehead is
displaced past its equilibrium position, then a force capable of slowing it down
and stopping it exists. This force that slows the bobblehead down
as it moves away from its equilibrium position is known as a restoring force. The restoring
force acts upon the vibrating object to move it back to its original
equilibrium position.
Vibrational motion is often contrasted with
translational motion. In translational motion, an object is permanently
displaced. The initial force that is imparted to the object displaces it from
its resting position and sets it into motion. Yet because there is no restoring
force, the object continues the motion in its original direction. When an
object vibrates, it doesn't move permanently out of position. The restoring
force acts to slow it down, change its direction and force it back to its
original equilibrium position. An object in translational motion is permanently
displaced from its original position. But an object in vibrational motion
wiggles about a fixed position - its original equilibrium position. Because of
the restoring force, vibrating objects do the
back and forth. We will explore the restoring force in more detail later in this
lesson.
As you know, bobblehead dolls are not the only objects that
vibrate. It might be safe to say that all objects in one way or another can be
forced to vibrate to some extent. The vibrations might not be large enough to be visible. Or the amount of
damping might be so strong that the object scarcely completes a full cycle of
vibration. But as long as a force persists to restore the object to its
original position, a displacement from its resting position will result in a vibration. Even a large massive
skyscraper is known to vibrate as winds push upon its structure. While held
fixed in place at its foundation (we hope), the winds force the length of the
structure out of position and the skyscraper is forced into vibration.
A pendulum is a classic example of an object
that is considered to vibrate. A simple pendulum consists of a relatively
massive object hung by a string from a fixed support. It typically hangs
vertically in its equilibrium position. When the mass is displaced from
equilibrium, it begins its back and forth
vibration about its fixed equilibrium position. The motion is regular and
repeating. In the next part of this lesson, we will
describe such a regular and repeating motion as a periodic motion. Because of
the regular nature of a pendulum's motion, many clocks, such as grandfather
clocks, use a pendulum as part of its timing mechanism.
An inverted pendulum is another classic example
of an object that undergoes vibrational motion. An inverted pendulum is simply
a pendulum which has its fixed end located below the vibrating mass. An
inverted pendulum can be made by attaching a mass (such as a tennis ball) to the
top end of a dowel rod and then securing the bottom end of the dowel rod to a
horizontal support. This is shown in the diagram below. A gentle force exerted
upon the tennis ball will cause it to vibrate about a fixed, equilibrium
position. The vibrating skyscraper can be thought of as a type of inverted
pendulum. Tall trees are often displaced from their usual vertical orientation
by strong winds. As the winds cease, the trees will vibrate back and forth
about their fixed positions. Such trees can be thought of as acting as inverted
pendula. Even the tines of a tuning fork can be considered a type of inverted
pendulum
Another classic example of an object that
undergoes vibrational motion is a mass on a spring. The animation at the right
depicts a mass suspended from a spring. The mass hangs at a resting position. If the mass is
pulled down, the spring is stretched. Once the mass is released, it begins to
vibrate. It does the back and
forth, vibrating about a fixed position. If the spring is rotated
horizontally and the mass is placed upon a supporting surface, the same back
and forth motion can be observed. Pulling the mass to the right of its resting
position stretches the spring. When released, the mass is pulled back to the
left, heading towards its resting position. After passing by its resting
position, the spring begins to compress. The compressions of the coiled spring
result in a restoring force that again pushes rightward on the leftward moving
mass. The cycle continues as the mass vibrates back and forth about a fixed
position. The springs inside of a bed mattress, the suspension systems of some
cars, and bathroom scales all operated as a mass on a spring system.
In all the vibrating systems just mentioned,
damping is clearly evident. The simple pendulum doesn't vibrate forever; its
energy is gradually dissipated through air resistance and loss of energy to the
support. The inverted pendulum consisting of a tennis ball mounted to the top
of a dowel rod does not vibrate forever. Like the simple pendulum, the energy
of the tennis ball is dissipated through air resistance and vibrations of the
support. Frictional forces also cause the mass on a spring to lose its energy
to the surroundings. In some instances, damping is a favored feature.
Car suspension systems are intended to dissipate vibrational energy, preventing
drivers and passengers from having to do the back and forth as they also do the
down the road.
Hopefully a lot of our original questions have been answered.
But one question that has not yet been answered is the question pertaining to
quantities that can be measured. How can we quantitatively describe a vibrating
object? What measurements can be made of vibrating objects that would
distinguish one vibrating object from another? We will ponder this question on vibrational motion.