Part C : Vibration & Oscillation

Part C: Vibration & Oscillation

Chapter 10. Repeating Motion
10a. Period & Frequency
Although vibrations are not themselves sound (in the sense of the objective entity that travels through air),
the two are intimately connected. Most sounds can be traced back to some object vibrating. In many
situations, sounds cause an object to vibrate. Indeed, when we discuss perception of sound, we really mean
perception of a vibration in the ear. And later chapters will describe how the air itself is vibrating as sound
passes through it.
Sound travels across distances, and also involves changes with time. That makes the vibration of air in
sound more complicated than the vibration of objects, which do not travel. So, we will first discuss objects.
The words vibration and oscillation both refer to back-and-forth motion. Usually, “vibration” is used for
motion that is quite fast, while “oscillation” is used for slower motion. But their characteristics are the
same, and the words will be used interchangeably in this book.
To have a vibration, there must be an object that vibrates, and there must be something causing it to move.
If the cause is complicated, or even sentient (like a person shaking a tambourine), then the possible motions
could be almost anything. In the physics spirit, in this Part we’ll focus on relatively simple causes for the
motion. The cause of motion and the object are considered together as a system, which can have properties
that neither of the components have on their own. The way the system moves (without any outside
influence) is called its natural motion.
In general, in order for a system to oscillate on its own, you need two conditions. First, there must be a
special position for the object (or possibly a range of positions) where the cause of motion has no effect, so
that the object could remain stationary there indefinitely. This is the equilibrium position for the system.
Second, if the object ever moves from the equilibrium position, there must be something that pushes the
object back towards equilibrium, called the restoring force. The concept of force is detailed starting in
Chapter 15.
A key feature of simple vibrations is that their motion repeats. Each repeating unit is called a cycle. That
is, a cycle is the shortest set of motions which, if replayed over and over, would describe the vibration for
as long as it lasts. Notice that a cycle is not something that can be measured with a single number. Also
note that for a given vibration, you can choose to start a cycle at any point. Suppose a vibration is
represented by the repeating pattern …ABCDEABCDEABCDE…, where each letter might represent a
small motion that is part of the vibration. Then ABCDE would be a cycle, but so would CDEAB and
EABCD.
The time required for one cycle is called the period of the oscillation, and it is given the particular symbol
𝑇. In conversational English, “period” can mean any time interval of interest, but in physics “period” has
a much more specific meaning. Anything that is repeating can also be called periodic, meaning that is has
a period.
You might notice that we have already used 𝑇 for temperature in Eq. 7.1. We try to avoid having the same
default variable for different concepts, but in the end there are just too many concepts and too few letters.
While working a physics question, you certainly must have a different variable for every individual
measurement. If you were working a question involving both temperature and period, that would be a good
time to distinguish them by subscripts.
Since it is a particular interval of time, the root unit for period is the second. For audible sounds,
milliseconds are typically handy. However, it is often helpful to consider period to be measured in the unit
seconds/cycle. A cycle is what physicists call a dimensionless unit. This book will not address the details,

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but the result is that “cycle” can be inserted or removed from a derived unit whenever it makes sense. It
functions outside the rules about units combining algebraically. As a result, both s and s⁄cycle are valid
units for period.
If you want to observe some number of cycles 𝑁, the elapsed time 𝑡 required is proportional to 𝑁. Therfore,
for any particular vibrating motion, the ratio of time interval to number of cycles is always the same. A
particularly useful corresponding pair is that 1 cycle lasts for one period, giving
𝑡 𝑇
= =𝑇 . (10.1)
𝑁 1
There is another often-used parameter that is sort of the alter ego of the period, in that it conveys the same
information about the motion. Especially when dealing with very rapid motion, rather than how much time
each cycle takes, it can be easier to consider how many cycles are completed in each second. This is called
the frequency of the oscillation, denoted by f. The root unit of frequency is cycles/second, which hints at
the mathematical relationship between frequency and period,
1
𝑓= . (10.2)
𝑇
Because it is used so much, cycles/second is given the name hertz, abbreviated Hz, named after a student
of the sound scientist Helmholtz. More about Helmholtz is in Chapter 28.
Often when considering a vibration, we imagine that it goes on forever, with no beginning or end. But any
real vibration does have a beginning and end, and the time interval during which the motion occurs is called
the duration. Duration doesn’t have any special algebraic symbol. Take particular notice of the difference
between the duration and the period of an oscillation.
10b. Displacement
Before getting into how systems oscillate, we need more tools for
describing motion. Our definition of speed in Eq. 4.21 relied on the fact
that sound always travels at a steady speed, but oscillating objects keep
changing speed. We need to distinguish between a few kinds of speed,
which first requires distinguishing between a few kinds of distance.
Consider the motion of the tip of the tine of a tuning fork (see Figure
10.1). Suppose that the extremes of motion are 2 mm apart and that the
motion repeats itself every 𝑇 = 3.8 ms (which makes this a tuning fork
for the pitch “middle C”). In particular, consider the cycle during which
the tine moves from the far left to the far right, and then back to the far
left.
Distance
There is a sense in which the tine has moved 4 mm during that time.
This is what physicists call distance traveled, or just distance. Distance
is always a positive number. Often d is chosen as the algebraic symbol.
Displacement
If you compare the beginning and ending positions, without regard for
where the tine was in between, there is another sense in which the tine Figure 10.1
has not moved at all. Physicists call this idea displacement, and in this Tuning fork vibrating. Arrows
case the displacement is zero. When an object does not end in the same relate to the motion of the dot on the
place as it started, completely describing the result (but not the full upper right corner.
motion) requires giving both the straight-line distance between the

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beginning and ending positions, and the direction of that line. It turns out to be handy to lump both distance
and direction together, and to call the combination the displacement. The fact that direction is included
makes displacement a vector.
In general, to describe a direction with a number, you need to first specify a special reference direction, for
instance “towards North” or “to the right on the page.” Then any other direction can be specified by angles
away from the reference direction.
But luckily, for oscillations there are only two directions involved, because the motion is along a single
axis. That is, the motion is one-dimensional. If we choose to call one direction positive and the other
negative, then the vector idea reduces to simply a signed number. In principle, either direction can be
chosen as positive. But nearly always, positive is chosen to be rightward or upward.
The magnitude of a vector is its size. That is, it is all the information in the vector except its direction. So,
we could correctly say, “For any motion, the magnitude of the displacement due to that motion is the
straight-line distance from the beginning point to the ending point.” For our one-dimensional case,
magnitude is obtained just by taking the absolute value.
It is not necessary to have an actual object moving in order to refer to a displacement. The displacement
from point A to point B is still just a distance and direction, imagining the points to be the start and end
points of a hypothetical motion.
Position
If we want to quantify where an object is, we can do so by first choosing a special point of reference, called
an origin, and choosing a reference direction. We can then specify a position as a displacement from that
origin. Since a position is a special kind of displacement, it is a vector. A common choice for the algebraic
symbol of position is 𝑟. But again, for oscillations, choosing a reference direction reduces to choosing
which direction along the axis of motion to call positive. Positions are then just signed numbers. In that
situation, the usual variable name choice is 𝑥, or sometimes 𝑦 or 𝑧.
Once an origin has been chosen, the displacement between two other places is given by the difference of
their positions. For this reason, a common algebraic symbol for displacement is 𝛥𝑥. The relation to position
is then given by
Δ𝑥 = 𝑥end − 𝑥begin . (10.3)
Notice that this has the pattern “later value minus earlier value,” or “final minus initial” or “ending minus
beginning.” This pattern usually requires a little attention because the parts are reversed from the order in
which they occurred.
Very likely, you once learned in a math class to use the variable 𝑥 to represent “the unknown” in a question.
In physics, that is never done. Algebra is already abstract enough; it’s a bad plan to make the equations
even more obscure by using a generic 𝑥. In principle, algebra allows you to use any symbol that you desire
to represent any quantity. But in practice, it is strongly recommended to use variables that remind you, as
well as anyone reviewing your work, of what type of physical quantity it represents. Reserve 𝑥 for position.

Chapter 11. Position Graphs

In order to describe motions like oscillations, in which positions and speeds are continually changing, we
need more than a single number. One way to describe the continual changes is with graphs, pictorial
representations of the relationship between two variables.

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To show how things change, we need to use one axis to

represent time, as in Figure 11.1. Nearly always, the horizontal
axis is chosen for time. A time axis is more abstract than a
position axis, as we are representing intervals of time with
horizontal displacements on paper. However, the same basic
principles apply as for any graph, including that there is a
horizontal scale factor, which now might have units like
seconds per centimeter (s/cm). That scale might be indicated
by tick marks along the axis, but Figure 11.1 has been left
without them because it will be used to illustrate ideas that hold Figure 11.1
for any scale. Graph of position versus time.
Only the vertical axis remains to represent position
information. But since our goal is to describe one-dimensional oscillation, position along one line will be
sufficient for our purposes. The result is a position versus time graph (often shortened to position-time
graph). Notice that the labels for the axes are not x and 𝑦, as they would be for position axes; the labels
change to reflect the quantity they represent. Since the two axis scales are different (indeed, they have
different units), the length of a diagonal line on the graph has no useful meaning.

Chapter 12. Velocity

If we take the idea of speed, and replace the distance in Eq. 5.2 with displacement, we get a quantity that
physicists call velocity,
Δ𝑥
𝑣= . (12.1)
Δ𝑡
Only lowercase 𝑣 is ever used to represent velocity. Its units are, not surprisingly, the same as for speed.
Even though the denominator represents the same sort of elapsed time as in Eq. 5.2, it is represented with
Δ to emphasize the parallel with the displacement.
Velocity includes the direction that it inherits from the displacement Δ𝑥. For this book, that just means that
velocity can be positive or negative, indicating whether the object’s motion is towards or against the chosen
positive direction. If something moves with constant speed and always in the same direction, as sound
often does, that is called uniform motion. In that case, the magnitude of the velocity equals the speed.
Average Velocity
But for oscillations, the speed, and even the direction, of motion are not constant. Similar to how average
speed was defined, Eq. 12.1 can still be applied to such inconstant motion, with a result called average
velocity. As with average speed, the average velocity depends not only on the motion itself, but also on
how the initial and final times are chosen. And again, the calculation will result in a velocity that is roughly
in the middle of the velocities that occur between the chosen moments, but this sort of average is not simply
related to an arithmetic mean.
For instance, consider the average velocity of the tuning fork tine in Figure 10.1 as it moves from extreme
right to extreme left. That’s one half of a cycle, taking a time Δ𝑡 = 𝑇⁄2 = 1.9 ms. Making the usual
choice that rightward is positive, the average velocity is
−2 mm m
𝑣1 = = −1.053 . (12.2)
1.9 ms s
For some time intervals, the magnitude of the average velocity will be different from the average speed.
Consider the tine’s average velocity calculated over a single cycle. No matter how we choose the start of
the cycle, the tine’s position at the end of the cycle must be the same as at the beginning, so that the motion

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is ready to start into the next cycle. Thus 𝛥𝑥 = 0 m and 𝑣2 = 0 m⁄s exactly. Compare that to the tine’s
average speed over one cycle
4 mm m
𝑠2 = = 1.053 . (12.3)
3.8 ms s
This occurs as a result of the change in direction, which results in a distance traveled that is different from
the magnitude of the displacement. So in general, the average speed will equal the magnitude of the average
velocity only when there is no change in direction between the start and finish.
We might calculate the average velocity over a very large number of cycles. Although the tine travels over
a greater and greater distance as time goes on, the displacement can never be larger than 2 mm. So as the
𝛥𝑡 in the denominator gets larger, the average velocity must get very small. The longer the time interval,
the closer the average velocity is to zero. This result is not unreasonable. Certainly, there is a sense in
which the vibrating tuning fork never actually travels anywhere. And depending on the choice of initial
and final positions, the velocity can be positive or negative, so that an average of zero is not a surprise
mathematically. But these observations don’t help us understand the details of that motion. For that, we
need the next few chapters.

Chapter 13. Graphs and Velocity

Position versus time graphs are useful for more than simply illustrating how an object moves in time. For
instance, consider the motion in Figure 11.1 from point A to point B. Because the graph is a straight line
between them, for any times between A and B the distance traveled is proportional to the time of travel.
That is, during that time the motion has a constant, uniform velocity (and speed). To calculate the velocity,
we have
𝛥𝑥𝐴𝐵 𝑥𝐵 − 𝑥𝐴 change on vertical axis
𝑣= = = , (13.1)
𝛥𝑡𝐴𝐵 𝑡𝐵 − 𝑡𝐴 change on horizontal axis
which is the definition of the slope of a graph.
Slope on a position versus time graph gives the velocity of the motion.
This idea is useful even when velocity is not constant. For the motion from point A to point C, with two
parts having different velocities, the same considerations show us that the average velocity between A and
C is given by the slope of a straight line between those points, even though that straight line does not
describe the details of the motion.
Instantaneous Velocity
In fact, this slope-velocity relationship provides us with a new way to handle non-constant velocity, one
that would have been quite difficult to use without the graph. Consider the motion near point D, where the
curving line tells us that the velocity is not constant for any time interval. We could calculate a variety of
average velocities by choosing different beginning and ending points near D. However, there is only one
tangent line that matches the curve’s slope exactly at point D. It is the dashed line in Figure 11.1. The
slope of that line is the instantaneous velocity at point D. The instantaneous velocity is a bit harder to
calculate from a graph than an average velocity, since it requires determining the tangent line. But
conceptually it has a big advantage over average velocity in that it is specific to a single moment in time,
instead of depending on both a beginning and ending time.
Notice that when the velocity is constant, the tangent line at a point (for instance, at point B in Figure 11.1)
passes exactly along the actual graph line. Thus, the instantaneous velocity and the constant velocity match
perfectly.

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The magnitude of the instantaneous velocity is the instantaneous speed. This brings us back to the
situation where speed equals the magnitude of the velocity. For ease of use, the word “instantaneous” is
often dropped from these names. However, the word “average” is never dropped from the terms average
velocity or average speed.

Chapter 14. Acceleration

Position, displacement, and velocity are all elements of the subject of kinematics, the description of motion.
In all sciences, accurately describing the subjects of study is an important first step. However, sciences
then make progress by finding organizing principles and, if possible, instances of cause and effect.
Therefore, the question arises, “What causes motions, in a general sense?” Answering this question is the
subject of dynamics. Everyday experience provides a ready answer: motions are caused by pushing,
pulling, or more generally applying a force. A later chapter will consider quantifying force; for now, the
everyday concept will suffice.
Do forces directly cause objects to have velocity? For a very long era in Western thought, that was the
prevalent model, attributed to Aristotle. It was thought that as long as an object is moving, it must be
experiencing a force. However, this model has difficulty describing many situations. For instance, if you
throw a ball, it continues to move long after your hand has stopped pushing it. And if this book starts to
slide off your lap, you could stop it by applying a force, which would be an example of a force removing
velocity. Aristotle’s model needed extra rules to account for these things.
A new model was eventually found, of which Galileo was the most famous proponent. This model said
that forces cause changes in velocity, and ultimately it explains a much wider range of situations with many
fewer assumptions. In fact, this model has been extraordinarily successful, accurately predicting results in
all physical situations except on the very smallest scales, where quantum mechanics is required.
Being successful, or at least useful, is necessary for a model to gain credibility in physics. But the fact that
it has fewer assumptions is also important—physicists would call the model “elegant.” In fact, as physics
has advanced, it has generally turned out that the more elegant a model, the more likely it is to be successful.
In this respect, it seems that nature tends to be simple. This doesn’t necessarily mean easier to understand.
Changes in velocity are more abstract than velocity itself, and therefore harder to think about. But it does
mean that there are fewer underlying rules.
In order to use this model for dynamics, we must first expand our knowledge of kinematics to include a
way to describe changes in velocity. We have already seen how velocity is a useful description of the rate
at which position changes. So, it is a natural extension to use the same mathematical structure to describe
the rate at which velocity changes, which physicists name acceleration. For instance, if the velocity of a
car changes during an interval of time, we can define its average acceleration as
𝛥𝑣 𝑣end − 𝑣start
𝑎= = , (14.1)
𝛥𝑡 𝑡end − 𝑡start
Since velocity is a vector idea, this equation says that acceleration is also. But again, for our purposes of
describing motion along a single line, this simply means that acceleration is a signed quantity. Only
lowercase 𝑎 is ever used for the algebraic variable. As always, the defining equation tells us what units can
be used for acceleration; the example with SI root units is (m⁄s)⁄s = m⁄s 2 .
Acceleration is an idea that we all have direct experience with, for instance while being in a vehicle that
changes its velocity over a relatively short interval of time. And we all know that the experience of
acceleration is very different from the experience of traveling down a highway at constant speed (high
velocity, but zero acceleration). Nevertheless, there are many situations where it is easy to confuse these
two aspects of motion.

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One such situation is in determining the direction of acceleration. One approach is to carefully use Eq. 14.1
to determine the sign of 𝑎. This can be done as long as you know the signs and relative sizes of the
velocities; actual numbers are not necessary. For example, suppose that you throw a ball up into the air and
consider a time interval from just after the ball leaves your hand to just before the ball reaches its highest
point. Taking upwards to be positive, 𝑣begin > 𝑣end > 0 (ball is moving in positive direction but slowing
down). This implies that 𝑣end − 𝑣begin < 0, so the acceleration is in the negative direction.
Another approach which may be useful is to anthropomorphize the moving object, and ask, “In which
direction is it trying to move?” For instance, even though the thrown ball is moving upwards, we can
imagine that it is “trying” to go down, again giving a negative acceleration.
One of the original motivations for defining acceleration was the following very useful model, first studied
thoroughly by Galileo Galilei.5
Any object that has a reasonably large weight, and is reasonably compact
in shape, when near the surface of the earth and not in contact with
anything else, moves with an acceleration of 9.8 m⁄s 2 in the downwards
direction.
Such an object is said to be in free-fall. This special value is named the acceleration due to gravity,
denoted by 𝑔. The next digit, the 0.01 m⁄s 2 place, depends on where you are on the earth, but it is quite
close to 0, so that using just two significant figures gives you better than 1% accuracy. The usual custom
is to not treat 𝑔 as a vector quantity, but as a positive numerical quantity. Thus, if we choose the usual axis
for vertical motion, with the upward direction being positive, the above principle is expressed by the
equation
𝑎free-fall = −𝑔 . (14.2)
Be careful to use this value only in appropriate cases. If there is anything other than gravity influencing an
object’s motion, then its acceleration is highly unlikely to be 𝑔 . Inappropriately assuming that an
acceleration is 9.8 m⁄s 2 is a common error in introductory physics courses.
In a typical introductory physics course, because of the constant acceleration due to gravity, the kinematics
of constantly accelerated motion (also called projectile motion) is studied extensively. However, our goal
is to describe oscillations, for which even the acceleration is not constant. Gravity will be useful in a few
parts of this book. But to make progress, we need to move directly to the cause of acceleration: force and
the subject of dynamics.

Chapter 15. Force and Acceleration

Force is a pretty familiar, everyday idea. But we need to be able to quantify forces, so that we can write
equations about them. One element we can notice right away is that forces push or pull in particular
directions: they are vector quantities. For the purposes of describing motion along a line this means, once
again, that we can expect forces to be signed quantities.
In the early 1600s, two important English scientists invented two different ways to quantify forces.
Isaac Newton chose the relationship between force and acceleration as his basis. Since what we would
intuitively describe as larger forces result in larger accelerations, Newton proposed that force be defined as
being proportional to the resulting acceleration, all other things being equal. This is expressed by
𝐹on ∝ 𝑎 , (15.1)

5
Galileo Galilei, Dialogs Concerning Two New Sciences, trans. Henry Crew and Alfonso de Salvio (New York:
Dover Publications Inc., (1638) 1954).

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where 𝐹 is the standard symbol for force, and the subscript “on” reminds us that the force is applied on the
object.
While one can always choose to make this definition for accelerating a single object, it is conceivable that
it might fail to be consistent if you apply the same forces to different objects. If force A accelerates a car
twice as much as force B, does that guarantee that force A also accelerates a ball twice as much as force B?
Is force A twice B for any object at all? Centuries of experience give us the answer, “Yes.” This model
has been spectacularly successful.
Of course, any given force will give the ball a much larger acceleration that it will give the car; cars are
much harder to move. As usual with proportional quantities, it is useful to take their ratio. In this case, the
ratio of force applied on an object to the resulting acceleration describes how hard it is to accelerate that
object, and is defined as the object’s inertial mass, usually shortened to just mass and denoted by 𝑚.
Rewriting that ratio gives one of the most famous equations of physics, Newton’s Second Law
𝐹
= 𝑚 ⇒ 𝐹 = 𝑚𝑎 . (15.2)
𝑎
Although it arises as a sort of side-effect in Eq. 15.2, mass is a fundamental type of quantity in the SI
system, like distance and time. More intuitively, it describes how much material, or how much “stuff,” an
object has. Linking directly to Eq. 15.2, mass describes how hard it is to shake the object back and forth.
There is also an extremely close connection to how hard it is to lift the object, which is explored in
Section 18b.
These equations require some new units to be defined. The SI root unit of mass is the gram, with the symbol
g. Other units of mass can be obtained by adding prefixes to gram in the usual way. However, the SI base
unit of mass is the kilogram (kg), which means that the kilogram is used to build named derived units. This
is the only place in SI where the base and root units are different, which can be a source of difficulty.
Usually, the safest method is to always use kilograms in calculations. Something with a mass of 1 kg
weighs a little more than two pounds.
Equation 15.2 tells us that the SI basic unit for force must be (kg ⋅ m⁄s 2 ). Because this is such a mouthful,
this derived unit is given a name of its own, the root unit newton (N). A newton of force is roughly equal
to a quarter pound. Keep in mind that the newton has the kilogram (not gram) inside. When working with
small objects, with masses given in grams, you must remember to convert to kilograms before combining
with newtons.
Notice that when a force pushes in one direction, the pushed object always accelerates in that direction
(even when its velocity is in the opposite direction, such as when the object is being slowed to a stop).
Therefore, 𝐹 and 𝑎 will either both be positive or both be negative, making 𝑚 always positive in Eq. 15.2.

Chapter 16. Force and Springs

If a force is applied to a spring (or anything springy), then it deforms, either stretching or compressing or
deflecting. Larger forces result in larger deformations. In the early 1600s, Robert Hooke chose this
relationship as his basis for quantifying force. Hooke proposed that force be defined as being proportional
to the resulting deformation, all other things being equal. This is expressed by
𝐹on ∝ 𝛥𝐿 , (16.1)
where 𝐹on is the force on the spring and 𝛥𝐿 is, as a measure of its deformation, the resulting change in the
length of the spring. See Figure 16.1.
While one can always choose to make this definition for a single spring, it is conceivable that it might fail
to be consistent if you apply the same forces to different springs. If force A compresses a bed spring twice

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as much as force B, does that guarantee that force A also stretches a rubber band twice as much as force B?
Is force A twice B for anything springy? Centuries of experience give us the answer, “Yes, within limits.”
Hooke’s model has been very successful, but two drawbacks prevent it from being considered universal.
The first is that if real springy things are stretched or compressed too far, the consistency of Hooke’s model
between different springs starts to fail. Indeed, real springs will eventually break. The second drawback is
that there exist a few oddball springy things that never fit the model. But avoiding those oddball “springs”
and keeping 𝛥𝐿 small-to-moderate is not a severe restriction.
The same force will deform different things by different amounts. For a given spring, the ratio of force to
resulting displacement describes how difficult it is to deform the spring. Rewriting the ratio gives Hooke’s
Law, traditionally written as
𝐹by
= −𝑘 ⇒ 𝐹by = −𝑘 𝛥𝑥 . (16.2)
𝛥𝑥
Here 𝑘 is called the spring stiffness constant, often shortened
to spring constant. The new quantities need new units to be
defined. The SI unit for force is the newton (N), which is roughly
equal to a quarter pound. SI defines the newton in terms of more
fundamental quantities, but that isn’t needed here. (For checking
calcualtions, N = kg ⋅ m/s 2 . For the reasons behind this, see
Chapter 15.) Spring stiffness is not a root quantity in SI either.
From the equation we can see that the derived basic unit for the
spring constant is N⁄m.
In Eq. 16.2 the deformation 𝛥𝐿 has been replaced by the
displacement 𝛥𝑥 of one end of the spring. With the assumption
that the other end is held unmoving, the two are exactly equal,
so this is a change in perspective, not in the math. It is handy, Figure 16.1
because most often we are interested in the displacement of some Spring before and while it is stretched.
object that is attached to the spring. But keep in mind that at its
foundation, Hooke’s Law is about deformation, not displacement. Notice that 𝛥𝑥 refers to displacement of
the spring’s end from a very specific place, namely from the position where no external force is acting to
deform the spring. This is called the equilibrium position for the end of the spring.
For a stretched spring, there are two forces available to focus on. One is the force applied to the spring, for
instance by the hand in Figure 16.1, which was the 𝐹on in Eq. 16.1. The other is the force with which the
spring is pulling back, for instance on the hand, which is the 𝐹by in Eq. 16.2. Hooke’s Law is usually written
with the second option in mind, in order to keep the definition all about the spring, regardless of what is
stretching it. It turns out that 𝐹on and 𝐹by are always equal in magnitude and exactly opposite in direction
(that is Newton’s Third Law, which won’t get much attention in this book), so that the proportion in Eq. 16.1
holds for either force.
Because a spring always wants to return to its undeformed state, the position-dependent force that it
provides is called a restoring force. 𝐹by and 𝛥𝑥 will always be in opposite directions. In our treatment of
physics along one axis this means they have opposite signs, making the ratio in Eq. 16.2 negative. The
minus sign on the right side makes that negative explicit, so that the spring constant 𝑘 will always be
positive.

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Figure 16.2 shows how this restoring characteristic means that

a force-displacement graph only passes through the top-left
and bottom-right quadrants. For springs that obey Hooke’s
Law, this graph forms a straight line, so that the force is called
a linear restoring force.
Unstretched coiled springs often have their coils touching
each other, so that the spring can be stretched but not
compressed. But some coiled springs, along with many other
springy things, can be deformed in either of two directions
from equilibrium, such as either stretched or compressed. In
those situations, it’s not obvious that the difficulty of
deformation (that is, the spring stiffness 𝑘) will be the same in Figure 16.2
both directions. Nevertheless, that is the case for the large Object (gray rectangle) attached to spring, and
majority of springs. Graphically, this means that a graph like graph of the linear restoring force that the spring
Figure 16.2 has the same slope on both sides of the origin. applies to the object.

Chapter 17. Force Laws

Which model is better, Newton’s or Hooke’s? It turns out that we don’t have to choose, because Newton’s
and Hooke’s definitions of force are usually consistent with each other. The only situations in which they
are not consistent are cases in which Hooke’s Law fails to be internally consistent anyhow, and therefore
Newton’s Law is considered more fundamental. But they are both extremely important to physics, in a
wide variety of situations.
It is Hooke’s model that provides the most practical way to measure forces. This is because it is much
easier and more convenient to measure the displacement of the end of a spring than to measure the
acceleration of a massive object. The majority of devices for measuring forces (including most common
weight scales, which essentially measure a force due to gravity) operate by deforming a springy object.
You will notice that both models have been elevated to the status of “law.” There is no hard and fast rule
about what it takes for a model to be considered a scientific law. It does not mean that the model is always
true — Hooke’s Law has its limitations. It does mean that the model has proven to be very, very useful,
which would only happen if the model were often very accurate. A law is also usually pretty fundamental,
that is, simple and based more on observations than on logical reasoning.

Chapter 18. Practical Forces

18a. Multiple Forces
Sometimes an object is pushed or pulled by several forces. Observations have shown that the resulting
motion of the object, in particular its acceleration, is the same as if it were experiencing a single force that
equals the sum of the individual forces. That imagined total force is called the net force,
𝐹net = 𝐹1 + 𝐹2 + 𝐹3 + ⋯ . (18.1)
This model is considered a part of Newton’s Second Law, so that in Eq. 15.2 the 𝐹 should really be 𝐹net .
The sum here must include the directionality of the vector forces. When several children pull in different
directions on a toy, it doesn’t accelerate very much in any direction (unless it breaks!). For our description
of one-dimensional motion, this means that some of the individual forces may be negative. The best way
to handle this is not to change some of the additions in Eq. 18.1 into subtractions, but rather to include the
negative within the variables. The distinction is subtle, so here is an example. Suppose that a rock is
hanging motionless on a string, and we focus on the forces applied to the rock. The upward force of the

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string is balanced by the downward pull of gravity. With the usual choice of upwards as positive, the math
might be
𝐹net = 𝐹string + 𝐹grav = (3 N) + (−3 N) = 0 . (18.2)
Notice that the negative is inside the parentheses.
Whenever it works to add things together to get the total effect, physicists call it superposition. Perhaps
this gets such a grandiose name because the addition is often of things fancier than simple numbers. For
instance, here we are dealing with superposition of vectors. In any case, superposition is a useful model
that will show up multiple times, for instance in Chapter 37.
18b. Gravitational Forces
A hanging rock is actually very useful.
In Chapter 14, it is noted that near the earth’s surface all large, dense objects in free-fall are observed to
accelerate downwards at a certain rate 𝑔. Newton’s Second Law then implies that any such object is feeling
a force which causes that acceleration, a force due to the phenomenon of gravity. Since the acceleration is
always the same, Newton’s Second Law becomes a relationship between mass and this force,
𝐹grav = −𝑚𝑔 . (18.3)
The magnitude of this force of gravity |𝐹grav | is called the object’s weight (although there are a few minor
disagreements between physicists over precisely what the word “weight” means). Weight and mass are so
closely related by Eq. 18.3 that in everyday life it is rarely necessary to distinguish between them. However,
they are not exactly the same concept. If an object were to move far from the earth’s surface, it would still
have an unchanged mass. But if it were dropped back towards earth, it would no longer accelerate at the
rate 𝑔, and Eq. 18.3 would not be correct.
The last paragraph only really considered objects in free-fall. However, there has never been any indication
that the force on an object due to gravity depends on how the object is moving. Even stationary objects
experience the same gravitational force given by Eq. 18.3, and this provides an extremely practical way to
produce well-known forces. Just assemble some rocks of well-known mass and hang one from some
support, like a string.
1. If the rock is stationary, then it is not accelerating: 𝑎 = 0 .
2. Eq. 15.2 therefore tells us that the net force on the rock is zero: 𝐹net = 𝑚𝑎 = 0 .
3. Eq. 18.1 therefore tells us that the forces from the string and from gravity have equal magnitude:
0 = 𝐹net = 𝐹string + 𝐹grav .
4. Eq. 18.3 tells us the magnitude of the force from the string: 𝐹string = −𝐹grav = 𝑚𝑔 .
5. Finally, Newton’s Third Law tells us that if the string is applying 𝐹 = 𝑚𝑔 upwards on the rock, then
the rock is pulling downward on the string (or other support) with a force of the same magnitude.
Thus, you can create well-known forces by hanging well-known masses. Of course, this force is always
downwards, but by running the string over a pulley, we can create a string-pulling force in any direction
we like.

Chapter 19. Hanging Spring

There are many kinds of springs. But the type that most people think of first is a coiled spring, and most
coiled springs are tightly coiled, so that they can be stretched but not compressed. Nevertheless, they can
allow for motion to either side of an equilibrium point if an object is hung from the spring, as illustrated on

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the right side in Figure 19.1. With nothing on the spring, its lower end is at position 𝑦0 . When stretched,
the displacement of the end is
𝛥𝑦 = 𝑦 − 𝑦0 , (19.1)
which is negative.
When an object is hung from the spring, gravity pulls down on the
object with a constant force, while the spring pulls up on the object
with a force that depends on the object’s position. So, if the object
is lowered gently, it will hang motionless with the spring stretched
just enough to balance the gravitational force. The new equilibrium
position 𝑦1 is determined by the fact that the forces on the object
cancel each other
−𝑚𝑔 = 𝐹grav = −𝐹spring = −(−𝑘 𝛥𝑦)
(19.2)
= 𝑘(𝑦1 − 𝑦0 ) .
If the object is then pulled below this new equilibrium position, then
the spring force increases to be larger than the gravitational force,
pulling the object up. If the object is moved above its equilibrium
position then the spring force decreases, and the larger gravitational
force pulls the object down. In fact, the total force on the object is
now Figure 19.1
𝐹grav + 𝐹spring = −𝑚𝑔 − 𝑘(𝑦 − 𝑦0 ) Spring being stretched down from
(19.3) equilibrium position 𝑦0 .
𝐹net = 𝑘(𝑦1 − 𝑦0 ) − 𝑘(𝑦 − 𝑦0 )
𝐹net = 𝑘(𝑦 − 𝑦1 ) .
where a substitution has been made in the second line using Eq. 19.2. Thus, we have a new result: this
mass-and-spring system behaves as if gravity is not involved at all! Now that 𝑦1 is the new equilibrium
position for this spring, the spring can be either compressed or stretched away from that equilibrium.
While being able to solve questions with numerical answers (like the one in Figure 9.1) is nice, this
conclusion about hanging springs is the kind of thing that a physicist really likes. By carefully combining
a few equations, we find a result that applies in more than just one specific situation, and the result simplifies
how we can think about those situations. Best of all, it is a little surprising, in that the spring stiffness
constant is completely unaffected by the change in equilibrium.

Chapter 20. The “Circle of Physics”

20a. Velocity Graphs
Of course, anything that changes with time can be described
with a time graph. In particular, we could graph
(instantaneous) velocity versus time. This is more abstract than
a position-time graph. Given a velocity-time graph, it is quite
a bit more difficult to understand what motion it represents.
For instance, it is important to recognize that the motion
described by Figure 20.1 does not reverse direction at any
point. For the early and late times the velocity is 𝑣 = 0 m⁄s,
which means that the object is stationary. In between, the Figure 20.1
object moves in a positive direction at constant speed. Graph of velocity versus time, with velocity
Nowhere is the velocity negative. always positive.

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Part C : Vibration & Oscillation

Notice that a velocity graph is missing one piece of information that is not missing for position graphs: it
does not tell anything about the starting position. However, given a velocity graph and the physical position
at the time origin (or, really, any particular time), it is possible to reconstruct the complete position versus
time, at least in principle. On the other hand, while the origin of a position axis needs to be associated with
a physical location, the origin of a velocity axis needs no clarification; it just means “not moving.”
Going the other way is a bit easier. Given a position-time
graph, we can obtain the velocity at any point by finding the
slope. Thus, it is possible to find the entire velocity-time graph
for a motion from its position-time graph.
For instance, Figure 20.2 shows two graphs describing the
motion of a ball bouncing on the ground. First, consider the
position-time graph. Notice that although the graph looks
similar to what you would see if a ball bounced across a room,
that is not necessarily the motion it describes. The graph only
tells us the vertical position of the ball (described by the
variable y), with the floor chosen as the y origin. There is no
horizontal position information. The ball might be bouncing
straight up and down. In fact, that would be nice, since it’s
Figure 20.2
most similar to oscillating motions that we are interested in.
Position (height) and velocity graphs of ball
The velocity-time graph has been positioned so that its time bouncing on the ground.
axis lines up with the time axis from the position-time graph.
The velocity is positive while the ball is moving up, and negative while the ball is moving down. At the
bounces, a corner in the position-time graph indicates that the velocity changes suddenly. As a result, the
velocity-time graph has an almost-vertical line at that point.
The general rule for creating a velocity-time graph from a position-time
graph is slope to value.
The following are some useful steps to follow when implementing that general rule.
1. Mark times where the position-time graph has zero slope (that is, it is horizontal).
2. Between the zero marks, label the time intervals according to whether the position-time slope is positive
or negative. This tells you where the velocity-time graph should be above or below the horizontal axis.
3. Look for where the position-time graph is steepest. At those points in time, the velocity-time graph
should be farthest from the horizontal axis.
4. Look for places where the position-time graph is a straight line. Over those time intervals, the velocity
is constant, so the velocity-time graph should be horizontal.
5. If the overall motion keeps returning to the same position,
then the object must experience an equal “amount” of
positive and negative velocity. Although the details are
beyond the scope of this book, the consequence is that if
the space between the velocity curve and the horizontal
axis were colored in, then the colored areas above and
below the axis must be equal.
Often the places of zero velocity found in step 1 are points in
time. Sometimes, however, there are whole time intervals with
Figure 20.3
no motion. Figure 20.3 shows an example, which might
Graphs of frog jumping, with pauses between
represent the height of a frog as it jumps. Notice that in this jumps.
graph, the 𝑦 position (or height) origin was not placed at the

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ground. The ground is a natural choice, but it is not required. Also notice that during the times with no
motion, the rule is still “horizontal position graph gives zero on velocity graph.”
20b. Acceleration Graphs
As with position and velocity, you can graph acceleration of an
object versus time as well. And since the defining relationship
between acceleration and velocity (Eq. 14.1) has exactly the
same pattern as the defining relationship between velocity and
position (Eq. 12.1), everything in Chapter 13 and Section 20a
has an analogous rule for acceleration. Once again, slope of a
line on a velocity-time graph gives acceleration. If the line
connects two points on the velocity-time graph, then its slope
is the average acceleration between those points. But it is more
useful to define instantaneous acceleration as the slope of a
tangent line at a single point on a velocity-time graph. Once
again, an acceleration-time graph can be obtained by applying
the slope-to-value rule to a velocity-time graph. Figure 20.4
extends the example of a bouncing ball to an acceleration-time
graph.
In the abstract, given a position-time graph, you could now find
the corresponding acceleration-time graph. First make the
velocity graph, and then all the steps from Section 20a apply
for getting the acceleration graph from the velocity graph. But
in practice, trying to do slope-to-value twice in a row can get Figure 20.4
sloppy. To get cleaner and more accurate graphs, it helps to Position, velocity, and acceleration graphs of a
consider the following additional clues. bouncing ball.
1. If the moving object is ever in free-fall, then the
acceleration is known to be downwards and constant, in particular 𝑎 = −𝑔 = −9.8 m⁄s 2. Figure 20.4
has examples of this. Working backwards, this means that the velocity-time graph is a straight line
with negative slope.
If the moving object is not in free-fall, then cases of constant acceleration are rather rare.
2. Sharp corners in the position-time graph indicate brief intervals of time with large “pulses” of
acceleration.
3. Considering the forces that the object feels may be helpful.
Because of Newton’s Second Law, the direction and relative
magnitudes of forces will be the same as those of
accelerations. That is, a graph of the net force on the object
versus time would look exactly the same as the acceleration
graph, although the scale on the vertical axis would be
different.
20c. The “Circle of Physics”
Attaching an object to the end of a spring that can be both
compressed and stretched, as in Figure 16.2, applies a restoring
force on the object, with the result that the object can oscillate.
Figure 20.5
This is why Hooke’s Law is particularly important to the subject
of sound. It is true that a restoring force does not have to be Circle of physics equations.

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linear in order to cause oscillation. But it is a very common situation. In this situation, exactly what sort
of motion will occur?
Combining our equations from kinematics and dynamics, we
find that the four main variables are linked by four equations,
as illustrated in Figure 20.5. The figure uses the assumption
that the displacement of the object and the stretch or
compression of the spring are exactly the same, Δ𝐿 = 𝛥𝑥. The
variables 𝛥𝑥, 𝑣, and 𝑎 all refer to the object, while 𝐹 refers to
the force applied on the object by the spring.
When equations link variables in a “circle of physics” like this,
physicists get very excited. It means that only certain types of
motion can satisfy the equations. When solving a numerical
question, as in Chapter 9, finding four equations relating four
unknowns would mean that you have found enough information
to answer the question. The current situation is similar, except
that some of these equations involve changes over time. As a
result, the “solution” is not just a number; the solution is an
entire type of motion, that is, a particular way that position
varies in time.
Calculus is necessary to derive exactly the kind of motion that
works, but graphing techniques can well illustrate how the
equations constrain the possible motions. Our goal is to find a
Figure 20.6
displacement-time graph such that, after obtaining velocity-,
An oscillation that fails to satisfy circle of
acceleration-, and force-time graphs, we will come back to the
physics, because bottom 𝛥𝑥 graph doesn’t
same position-time graph after working all the way around the match top graph.
circle of physics.
For instance, the top of Figure 20.6 shows a plausible
motion for this oscillation. It is constant velocity motion
(the dotted lines) alternating with reversals of direction (the
solid curves). In the next two graphs, slope-to-value is used
to obtain the corresponding velocity and acceleration
graphs. A graph of the force F would look the same as the
graph of a, because Newton’s Second Law tells us that they
are proportional with the same sign. Finally, Hooke’s Law
tells us that displacement should be proportional to the
force, but with the opposite sign.
Since the bottom graph of Figure 20.6 does not match the
top graph, this motion does not satisfy the circle of physics.
The most significant failing is that the straight segments in
the top graph result in intervals of zero in the bottom graph.
The displacement-time graph that does successfully satisfy
the circle of physics is illustrated in Figure 20.7. Graphs of
this shape are called sinusoidal graphs because the
mathematical functions that generate them are the sine or
cosine functions.
Figure 20.7
This particular, extremely common natural motion, arising
Oscillation that does satisfy the circle of physics.
from a linear restoring force, is named Simple Harmonic

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Motion, or SHM for short. SHM and sinusoidal motion more generally are extremely important to the
study of sound.

Chapter 21. Sinusoidal Motion Parameters

Sinusoidal motion, also called simple harmonic motion or SHM, is of fundamental importance to the study
of sound. The shape of this motion is described by either the sine or cosine function; in this book, cosine
will be used. The algebraic equation relating displacement to time is
𝛥𝑥 = ∎ sin(∎𝑡 + ∎) or 𝛥𝑥 = ∎ cos(∎𝑡 + ∎) . (21.1)
In these equations, black squares indicate where
additional numerical quantities must be added to
fill in details and to make the units work out. A
particular sinusoidal motion is described by just
a few of these parameters, as illustrated in the
position-time graph of Figure 21.1.
Time Parameters
SHM is a repeating motion, and the parameters Figure 21.1
from Chapter 10 apply to describe the “size of Graph illustrating the parameters of SHM.
the motion” along the time axis. The sine and
cosine mathematical functions describe curves that have no beginning and no end, so their duration is
infinite. Figure 21.1, like real examples of SHM, has a finite duration indicated by 𝛥𝑡, although precisely
when the repeating motion begins and ends is not perfectly well defined.
The most obvious way to measure the period is from one maximum (or minimum) to the next. But on a
drawn graph, intersections can be more precisely identified. Since a cycle can start anywhere (an example
is highlighted as a heavy curve in the figure), any horizontal line can be used to measure the period between
intersections. Just remember to skip one intersection, where the motion is in the opposite direction (upward
or downward). The most convenient horizontal line is likely to be the time axis.
Amplitude
Simple harmonic motion oscillates to either side of the equilibrium position Δ𝑥 = 0, a place where the
moving object could indefinitely rest stationary without any external influence. The maximum degree to
which the moving object is displaced from equilibrium is the amplitude. In the field or lab, it’s often the
most practical to measure the distance from one extreme of the motion to the other. This is called the peak-
to-peak amplitude, labeled in the figure as 𝑥𝑝𝑝 . When working mathematically, however, it is usually
more convenient to refer to the distance from the equilibrium to either of the extremes, which is simply
called the amplitude (no preceding adjective) and is labeled in the figure as 𝑥𝑚 . The subscript m is short
for “maximum.” The variable 𝐴 is also commonly used for amplitude, but it will not be used in this book
due to conflicts with other concepts.
Notice that amplitude and displacement are not the same thing. This pair of concepts are so closely related
that many students find them difficult to keep straight. Since amplitude equals the maximum displacement,
they both have the same root unit of meters.
Phase
Finally, there is a parameter of sinusoidal motion that is in some sense less fundamental. In Figure 21.2,
graphs a) and b) show motions that are identical (assuming that the two time axes are synchronized). But
the graphs differ in one respect: they have different choices for the moment in time designated as 𝑡 = 0.
On the other hand, graphs b) and c) show motions that really are different, in that one is shifted along the
time axis.

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To describe these things, we start with the idea of

phase. Phase is a way to describe how far through a
cycle some point is. Since a cycle can start
anywhere, we first need to choose a convention for
where a cycle starts. Throughout this book, we will
choose the convention of the cosine function, which
is that a cycle starts at a maximum, which therefore
has a phase of zero. Each cycle is then divided into
360 degrees (symbol °) that are equally spaced along
the time axis, with numbers increasing in the same
direction as increasing time. The resulting numbers
measure phase along the sinusoidal graph, usually
represented algebraically by the symbol 𝜙 (the
Greek letter phi).
Several phases are indicated in Figure 21.2. Notice
that, although the degrees indicate equal spacing
along the horizontal axis, the phase is associated
with the curve, not the axis. Also notice that there is
no rule about which maximum should be 0°, so that Figure 21.2
there is some built-in ambiguity as to how to label Graphs illustrating phase in SHM.
the graph. Phases 𝜙 = 0° , 𝜙 = 360° , and 𝜙 =
720° all mean the same thing, and 𝜙 = −90° is the same as 𝜙 = 270°. You can always add or subtract
360° without changing the meaning. It is therefore sometimes convenient to convert all phases into the
range 0° to 360°; in this book, that will be called the reduced phase when we need to be explicit.
Having the degrees of phase equally spaced along the time axis is another way to say that changes of phase
are proportional to changes of time, 𝛥𝜙 ∝ 𝛥𝑡. On a given sinusoidal motion, a pair of points defines a time
difference and a phase difference; the ratio of those differences will be the same, regardless of which pair
of points you choose. One particular choice to consider is points separated by one cycle, making the
proportion equation
𝛥𝜙 360°
= . (21.2)
𝛥𝑡 𝑇
The proportion means that the left side of that equation can refer to any pair of points; only the right side is
specifically for one cycle.
Using the phase idea, we can specify how a sinusoidal graph relates to the time origin by specifying the
phase which occurs at 𝑡 = 0. This is called the initial phase or the phase constant and is usually denoted
by 𝜙0 . The initial phase of sinusoidal motion is less fundamental than the other parameters because it
depends on when we choose the time origin, rather than on the physical reality. In Figure 21.2, the top two
graphs show the same physical reality but have different initial phases.
However, the phase itself does have physical meaning. The physically real shift between graphs (b) and
(c) of Figure 21.2 can be described by their phase difference: at any moment in time, subtract the phases
to get 𝛥𝜙 = 287°. This works no matter which moment in time you choose. Because the time origins of
the two graphs match, the easiest choice is to use the initial phases, so that the phase difference can be
calculated as the difference of the initial phases, 𝛥𝜙 = 𝜙0𝑏 − 𝜙0𝑐 = 107° − (−180°) = 287°.
As a final note, it is no accident that phase and angle are both measured in degrees. Sinusoidal motion
repeats every cycle just as the angle between a clock’s minute hand and vertical repeats ever hour. In fact,
a graph of the vertical position of the minute hand’s point would be precisely sinusoidal motion. On the
other hand, the fact that temperature units are also called degrees is an accident of etymology.

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Chapter 22. Sinusoidal Motion Function

The algebraic function that describes the shape of SHM is cosine (or sine). That is, this function allows the
calculation of vertical information (e.g., displacement) from horizontal information (time). The correct
formula for a specific simple harmonic motion contains all the information about it, so in theory the formula
can be used to answer any question about SHM. In practice, however, this often is not the easiest way to
answer a question, so do not consider the following to be a panacea.
Given the phase 𝜙 at some point in the oscillation, one can calculate the displacement using the formula
𝛥𝑥 = 𝑥𝑚 cos 𝜙 . (22.1)
The phase, in turn, can be related to time through Eq. 21.2. By choosing the time origin as the reference
point for the differences in the left side of that equation, it becomes
𝜙 − 𝜙0 360°
= , (22.2)
𝑡−0 𝑇
which together with Eq. 22.1 yields the basic equation for sinusoidal motion that fills in the gaps in Eq. 21.1:
360°
𝛥𝑥 = 𝑥𝑚 cos ( 𝑡 + 𝜙0 ) . (22.3)
𝑇

Chapter 23. Importance of SHM

There are a number of reasons why SHM is so important to the study of sound. Here are a few.
• Sinusoidal motion causes sounds that pleasant to hear, or at least not harsh. In fact, a sound produced
by sinusoidal motion is called a pure tone.
• Sinusoidal functions are relatively easy to work with mathematically.
• An object attached to a spring creates a system that moves in a way that is well modeled by SHM.
If you have read Section 20c, you have seen how SHM satisfies the circle of physics. In brief, the
spring’s resting shape defines an equilibrium, and deforming the spring provides a linear restoring
force. Springy things are very common in the real world, and SHM is their natural motion.
• Even in cases where springy things provide a restoring force that is not linear, SHM is often a very
good approximation for the resulting motion. This is especially true if the amplitude is small. (As
detailed in Chapter 16, this is precisely because a straight line is a good approximation for a very
short piece of a curving force-versus-displacement graph.) SHM can also be a good starting place
from which to describe the actual motion with a non-linear restoring force, although that is beyond
the scope of this book.
• But possibly the biggest reason that SHM is important can’t be described until Chapter 42.
A sinusoidal function goes on forever, with the amplitude always remaining the same. This is certainly not
true of real oscillations, which all decay in size if observed for long enough. Given that, why should simple
harmonic motion remain so important? One reason is that, for oscillations whose amplitude decays slowly
enough, SHM is a good model of the motion for comparatively short intervals of time. Another is that, as
covered in Chapter 26, several properties of oscillation are unaffected by changes in amplitude.

Chapter 24. SHM Speed

As an object executes SHM, how fast does it go? As the object moves from a minimum (that is, most
negative) to a maximum of displacement, since there is no change in direction of travel, the average speed
is the magnitude of the average velocity. The average velocity is equal to the slope of a line connecting
those points on a displacement-time graph,

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𝛥𝑥 2𝑥𝑚 𝑥𝑚
𝑠𝑎𝑣𝑔,𝑝𝑝 = |𝑣𝑎𝑣𝑔,𝑝𝑝 | = = 1 =4 . (24.1)
𝛥𝑡 2
𝑇 𝑇
However, that is not the maximum speed achieved during the motion. The steepest point on the sinusoidal
graph is where the displacement is zero. It turns out that a tangent line at that point has the slope
𝜋 𝑥𝑚
𝑠𝑚 = 𝑠𝑎𝑣𝑔,𝑝𝑝 = 2𝜋 . (24.2)
2 𝑇
Deriving that result is most easily done with calculus, and thus is beyond the scope of this book. But it
should be apparent on a graph that the maximum slope is roughly 50% higher than the average found in
Eq. 24.1.

Chapter 25. Sinusoids Beget Sinusoids

When the displacement of an object varies sinusoidally with time, it turns out that many other characteristics
of the motion also vary sinusoidally with time as well. For instance, the velocity of the object, the
acceleration of the object, and the force applied on the object are all sinusoids, as is apparent in Figure 20.7.
We will see other sinusoids cropping up.
When the graph of any quantity versus time has a sinusoidal shape, it can be described by essentially the
same parameters as are used in Chapter 21. The concept of period is completely unchanged. The amplitude
idea is adjusted to whatever is on the vertical axis. For instance, for a sinusoidal velocity-time graph, one
might refer to the maximum velocity achieved as the velocity amplitude 𝑣𝑚 . The phase idea is mostly
unchanged; the only possible modification is that it might be desirable to choose a different point on the
sinusoidal shape as 𝜙 = 0°.
The equations in Chapter 24 are really just relationships about slopes of the sinusoid shape. They could
equally well be applied to get the rate of change for any vertical axis quantity. For instance, applied to a
velocity-time graph, they can be used to find the accelerations.

Chapter 26. SHM Frequency and Amplitude

When an object has SHM as its natural motion, it oscillates at a very specific period and frequency. In
particular, the frequency of the oscillation is the same regardless of the motion’s amplitude. This may seem
very counterintuitive. A larger amplitude motion requires traveling larger distances with each cycle, and
one might well expect those to take a longer time, resulting in a longer period for the motion.
However, larger displacements also mean that the spring is more deformed, so that the forces involved are
larger. These cause higher accelerations at the extremes of motion, so that the maximum speed is also
larger. In the end, it turns out that these two effects (larger distances and larger speeds) exactly cancel each
other out, so that the period and frequency are the same for all amplitudes.
This is quite a special property. Non-sinusoidal oscillations, called anharmonic, will vary in frequency if
the amplitude changes.

Chapter 27. SHM Frequency Formula

When an object moves with natural SHM because it is attached to a spring, the strength of the spring and
the mass of the object control the period and frequency. A stiffer spring (higher value of k) will result in a
larger force (all other things being equal), which moves the object faster, resulting in a shorter period and
a higher frequency. Conversely, a more massive object is more difficult to accelerate, so that (all other

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things being equal) the object moves more slowly, resulting in a longer period and a lower frequency. The
exact relationship is
𝑚
𝑇0 = 2𝜋√ , (27.1)
𝑘

1 𝑘
𝑓0 = √ . (27.2)
2𝜋 𝑚
Although these equations are offered without derivation, you should still be able to verify that they do,
indeed, exhibit the relationships described above. The results are called the natural period and natural
frequency of that particular system. The subscript zero is commonly used to indicate this natural status,
i.e., that the variable represented arises is a property of the vibrating system with no external influence.

Chapter 28. Helmholtz Resonators

One unusual form of a mass on a spring is composed almost
entirely of air. It’s called a Helmholtz resonator, named after
the German physicist who pioneered the use of such devices in
sound analysis.6 All that is needed is an enclosed volume that
is connected to the outside by a hole or short tube. Whenever
you make a hooting sound by blowing across the mouth of a
bottle, or whenever you whistle, you are using a Helmholtz
resonator, where the bottle or your oral cavity, respectively,
make the enclosed volume. The surprising part is that these
can be understood as a mass on a spring.
This book assumes that you have a good familiarity with the
idea of volume. Its traditional variable is capital V. The SI unit
for volume is meters cubed, also called cubic meters,
symbolized by m3.
Figure 28.1
The details of how a Helmholtz resonator works will be Dimensions of a bottle, considered as a
covered in Chapter 135. But it is easiest to conceptualize for a Helmholtz resonator.
bottle with a single, straight-sided neck, as shown in Figure
28.1. The air in the neck of the bottle (the gray chunk) has a mass, albeit very small. It can oscillate up
and down, which causes the air in the body of the bottle to expand or compress. The air in the body acts
like a spring, as described in Chapter 134, pushing the neck air back towards an equilibrium position.
The end result is a system that vibrates with a natural frequency given by the equation

𝑠 𝐴
𝑓0 = √ , (28.1)
2𝜋 𝑉ℎ

where 𝑠 is the speed of sound for the air in the bottle, and ℎ, 𝐴, and 𝑉 are dimensions of the bottle,
respectively the length of the bottle neck (along the axis of vibration), the cross-sectional area of the bottle
neck (perpendicular to the axis of vibration), and the volume of the bottle body. You can see in the form
of this equation hints that it came from Eq. 27.2, although exactly how the two relate has become obscured.

6
Hermann von Helmholtz, On the sensations of tone as a physiological basis for the theory of music, trans.
Alexander J. Ellis (London: Longmans, Green, and Co., 1885), 43.

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Beyond where Eq. 28.1 comes from, there are other complications that can arise. When whistling, your
oral cavity is open at both your lips and the back of your throat—how does that change things? And why
should a steady blowing over the mouth of a bottle cause it to vibrate? (See Chapter 196 for a partial
answer.) In any case, the fact that it does vibrate in response to an external cause is the reason that the setup
is called a “resonator” instead of a “vibrator.”
You have probably experienced the Helmholtz resonator in another
way, when someone opens a single window in a travelling car and a
throbbing sound results. Here the cabin of the car forms the volume,
and the open window forms the neck. However, Eq. 28.1 can’t be
applied properly because there is no apparent length to the neck. The
opening in the window provides a clear area A, but what should you
use for the variable h?
When faced with such a situation in a physics question, one usually
needs to look back at where the equation came from. Certainly, it is
not a recipe for success to cast about for another value to use. To Figure 28.2
think, “I’m missing a length… maybe it will work if I just use the Dimensions of Helmholtz resonator with
neck diameter,” is guessing, not science. no neck.
The length of the neck was used in Figure 28.1 because that is also
the length of the oscillating chunk of air. With an opening instead of a neck, there can still be an oscillating
chunk of air, as in Figure 28.2, but how thick is it? Answering that question requires physics well beyond
the scope of this book. So instead, use this rule of thumb: the effective thickness / neck length is 85% of
the smallest width of the opening,
ℎeff = 0.85 ⋅ 𝑤min . (28.2)
For a circular opening, 𝑤min is the diameter. For an oblong opening, 𝑤min is the width, not the length.
Many similar shapes act as Helmholtz resonators, even though the shape makes it difficult to say precisely
how large the “neck,” or even the “body,” is. In fact, a detailed investigation of even a perfectly cylindrical
neck will reveal that it has an effective length that is slightly longer than its physical length. Nevertheless,
judicious use of the two models above can give quite acceptable results.

Chapter 29. Energy

In preceding chapters, the motion of objects has been considered in light of the forces they feel. But there
is an alternative viewpoint which is sometimes easier to use, and which makes some relationships easier to
understand: the viewpoint of energy. As a starting point, here is a qualitative definition of energy, as used
in physics:
Energy is a property possessed by a system, quantifying the system’s
ability to affect some other physical system.
Energy can come in different forms. The most visible form of energy is energy of motion, called kinetic
energy (from the Greek word for motion). For example, a moving golf club can hit a ball, causing the ball
to fly off. A moving hammer can shove a nail further into wood. In both cases, the kinetic energy of the
moving thing allows it to affect something else. A less obvious form of energy is energy due to shape,
called potential energy. For instance, a stretched slingshot can be released, flinging (affecting) the stone
it is holding. Potential energy is often thought of as stored energy. Energy associated with visibly large
objects moving or being deformed is called mechanical energy. Thus, some mechanical energy is kinetic,
and some mechanical energy is potential.

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 Part C : Vibration & Oscillation

This qualitative idea is fine as far as it goes, but it took many talented scientists several centuries during the
Renaissance to figure out and implement a very special property of energy. If energy is quantified in the
right way, then the total amount of energy around us is constant. Energy is not “stuff” in the same sense as
objects in the material world. But it nevertheless shares some of the properties of “stuff”: it can be moved
around, but it cannot be created or destroyed, although it can be converted from one form to another. In
technical language, energy is conserved.
In order to make this energy conservation idea work, from time to time physicists have had to invent (or
discover, depending on your perspective) new forms of energy. For instance, a book held high in the air
can be dropped, with the result that its kinetic energy increases. If energy is to be conserved, then the book
must have had that energy all along, simply by virtue of being high in the air; this is another form of
potential energy. (An even more enlightened understanding of this energy is that is belongs to the
combination of the book and the earth, which are pulling each other together with gravity.) A hot pan from
an oven can burn you, which is certainly affecting you, so it must have energy by virtue of being hot; this
is named thermal energy. (It was not until the 1840s that James Joule convincingly showed that thermal
energy was the same sort of thing as mechanical energy.) Several other forms of energy are known, but
they are beyond the scope of this book.
This model of conserved energy has been so successful over history, that it is now considered one of the
most fundamental tenants of physics, even more basic than Newton’s Second Law (see Chapter 15). Every
time a new form of energy has been found, the guiding principle of energy conservation has helped us to
understand that new form. Sometimes it is useful to approach a physics question in a way that doesn’t make
energy conservation evident. For example, when a soccer ball rolls to a stop on level grass, its mechanical
energy clearly is not conserved, but useful information can be obtained without identifying where that
energy went. Nevertheless, you can also be assured that the energy did go somewhere, either to a different
object, or into a non-mechanical form, or both.
When energy is transferred from one object or system to another in a macroscopic way, the amount of
energy transferred is called the work done in the process. Be warned that this terminology only sometimes
aligns with the English use of the word. Simply holding a heavy object stationary in the air could well be
called work in the everyday sense, but in the physics sense no work is being done, because the object is not
receiving any additional energy.
The SI root unit for energy is the joule, abbreviated J. A joule is not a huge amount of energy. If you drop
a 5 pound bag of sugar or flour a distance of 5 cm, it gains roughly 1 J of kinetic energy. In a thermal form,
it would take about 130 J to warm a thimbleful of water from room temperature to hot tap water temperature.
How the joule derives from other units will be seen in Chapter 30.

Chapter 30. Mechanical Energy

How should the various types of energy be quantified, so that conservation of energy works? Some insight
into the equations can be obtained by considering work in the colloquial sense of “effort.” Consider, for
instance, how much work must be done to lift a book into the air, thereby giving it potential energy. More
massive books require more effort to lift. If gravity were weaker (for instance, by being on the moon), it
would require less effort to lift anything. And the farther the book is lifted, the more effort is required. The
formula for gravitational potential energy (near the earth’s surface) indeed includes all these factors:
𝑃𝐸grav = 𝑚𝑔ℎ , (30.1)
where 𝑚 is the mass of the object, ℎ is the height that the object has been lifted, and 𝑔 = 9.8 m⁄s 2 is the
acceleration due to gravity from Chapter 14.
How much work must be done to stretch a spring, thereby giving it potential energy? Again, the more the
spring is to be stretched, the more effort is required. But here, there are two separate reasons for that added

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Part C : Vibration & Oscillation

effort. Stretching farther requires more effort, just as with gravity. But also, the force against which you
are pulling gets larger, due to Hooke’s law from Chapter 16. As a result, the spring potential energy is
proportional to the square of the distance stretched 𝛥𝑥,
𝑃𝐸spring ∝ 𝛥𝑥 ⋅ 𝛥𝑥 = 𝛥𝑥 2 . (30.2)
Stiffer springs are also harder to stretch, so the spring stiffness constant 𝑘 should show up. The final
formula turns out to be
1
𝑃𝐸spring = 2𝑘 𝛥𝑥 2 . (30.3)
Notice that the square means that spring potential energy is always positive, whether the spring is stretched
or compressed. This correctly corresponds to the observation that one must do work on a spring to deform
it either way. Notice also the 12 , which tends to show up in equations that have quantities squared.
The equation for kinetic energy has a similar form to that of Eq. 30.3, but it is harder to rationalize because
it involves velocity 𝑣 instead of position. The equation for an object with mass 𝑚 is
1 1
𝐾𝐸 = 2𝑚𝑣 2 = 2𝑚𝑠 2 . (30.4)
Because the velocity is squared, the kinetic energy is again positive regardless of the sign (that is, the
direction) of the velocity. This allows for the second form, using speed 𝑠 instead of velocity. Even when
it is necessary to specify the direction of velocity with a fully-fledged vector, the square of a vector is
defined so that it is a positive number.
As usual, these equations tell us not only how to quantify energy, but also how the SI units relate. Working
from Eq. 30.4, we have
m 2 m2
1 J = 1 kg ( ) = 1 kg 2 . (30.5)
s s
Notice that a 1 kg object moving at 1 m⁄s does not have 1 J of kinetic energy, but rather 0.5 J. When doing
the calculation, the numbers combine separately from the units.
This book always uses a variable for energy that contains a capital 𝐸. Other references, in an effort to avoid
two-letter variables, use 𝑈 for potential energy and 𝑇 for kinetic energy.

Chapter 31. Energy in SHM

Ideal SHM never dies away. This together with the conservation of energy imply that the total mechanical
energy in the system is always the same. But energy is continually traded back and forth between object
and spring. It’s also continually changing type, between KE (of the object) and PE (in the spring),
𝐸tot = 𝑃𝐸spring + 𝐾𝐸object = a constant . (31.1)
For instance, at the moment in time represented by the open circle in Figure 31.1(a), the displacement-time
graph is horizontal, so that the velocity and kinetic energy of the object are both zero. But the spring is
stretched, so that it is storing potential energy. In fact, both the stretching and the PE are at a maximum.

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 Part C : Vibration & Oscillation

As time progresses, the spring uses that potential energy

to do work on the object, pulling it toward the
equilibrium position. By the time ¼ period has passed
(filled circle in Figure 31.1(a) ) the displacement is 𝛥𝑥 =
0, so that the spring has used up all of its energy. The
object, however, has reached its maximum speed, and
therefore its maximum kinetic energy.
As the mass continues to move, it starts to do work on
the spring by compressing it. Eventually, all the energy
is transferred into potential energy of spring
compression, and the object once again has zero velocity.
At this point, half a cycle past where this description
started, the energy situation is almost the same as when
we started. The only difference is that the spring
potential energy is due to compression instead of
stretching.
These observations qualitatively explain the energy
graphs in Figure 31.1. Notice that the energy graphs
repeat their shape twice as frequently as the displacement
graph. A surprising fact is that, for sinusoidal motion, Figure 31.1
the shapes of the energy graphs are also sinusoidal Displacement and energy in SHM. In bottom graph,
(although they are shifted up so that they are always constant total energy is broken into potential energy
positive). The bottom graph in Figure 31.1 shows how (white) and kinetic energy (gray).
these two sinusoids fit together to make the total energy
constant; think of flipping the 𝐾𝐸 graph upside-down to make the gray part of the 𝐸 graph.
Consider the energy balance at two specific times. At the open circle in Figure 31.1(a), since 𝐾𝐸 = 0, we
can write
𝐸tot = 𝑃𝐸max + 0 . (31.2)
On the other hand, at the filled circle in Figure 31.1(a), the spring has no potential energy, so that all of the
energy is in kinetic form, leading to
𝐸tot = 0 + 𝐾𝐸max . (31.3)
Equation 31.1 tells us that the total energy 𝐸tot is the same at both times, so that we may set Eq. 31.2 and
Eq. 31.3 equal to obtain
𝐸tot = 𝑃𝐸max = 𝐾𝐸max , (31.4)
which might look like it contradicts Eq. 31.1. But it doesn’t because of the subscripts.
We can use Eq. 30.3 and Eq. 30.4 to express this energy equality in terms of displacement and velocity.
But we must be careful, because we want to refer to the displacement and the velocity at different times.
Happily, we have already defined variables for these specific quantities. The maximum PE results from
the maximum stretch of the spring, which is the amplitude of the motion, and the maximum KE results
from the maximum speed. Substituting those into Eqs. 30.3 and 30.4 gives the relations
1 2
𝑃𝐸max = 2 𝑘 𝑥𝑚 , (31.5)
1 2
𝐾𝐸max = 2 𝑚 𝑣𝑚 . (31.6)

All of this allows us to discover the following

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Part C : Vibration & Oscillation

1 2 1 2
𝐸tot = 2 𝑘 𝑥𝑚 = 2 𝑚 𝑣𝑚
𝑥𝑚 2 2
𝑥𝑚 (31.7)
2
𝑘 𝑥𝑚 = 𝑚 (2𝜋 ) = (2𝜋)2 𝑚 2
𝑇 𝑇
𝑚 (31.8)
𝑇 = 2𝜋√ ,
𝑘
where in the second line the maximum velocity from Eq. 24.2 was substituted.
So, by combining basic energy ideas, we have derived the special property of SHM that is given with no
justification in Chapter 27. This is an example of the power of the energy viewpoint. To obtain the same
result considering only forces and accelerations would be far more difficult.
This derivation is also an example of the power of physics in general. When reading about this formula in
Chapter 27, it probably seems to come out of nowhere. At that point, there is no reason to expect squares,
let alone square roots, in the equations relating to SHM. But now we have derived that formula using
relatively simple ideas about energy. In physics we often find that relatively straightforward relationships
can lead to surprisingly complex behavior or can imply surprisingly complex relationships.

Chapter 32. Energy and Amplitude

32a. Energy in Non-Natural Sinusoidal Motions
An object can move sinusoidally in time for reasons other than being attached to a spring. For instance, a
person might be holding it, and shaking it back and forth. Or, the object might be attached to something
more mechanical, such as a motor designed to push and pull the object. This is not natural motion, because
an external agent is causing it. Call it driven motion.
If the driven motion is sinusoidal, then the object’s kinetic energy behaves the same as if it were on a spring.
Figure 31.1(a) and (c) would still apply. However, with no spring to hold potential energy, conservation of
energy is no longer helpful in describing the motion. Instead of energy being stored in a spring and then
returned to the object, energy is continually transfering to and from the driver in a way that is independent
of the motion itself.
Even if an object is subject to a spring-like restoring force, an external agent can drive the system in
sinusoidal motion. An object could be attached to both a spring and a motor. But the driving force might
oscillate at a frequency that’s different from the natural frequency of the spring and mass alone. Again, the
kinetic energy would behave as in Figure 31.1, varying at the actual frequency rather than the natural
frequency. The potential energy in the spring would also vary sinusoidally. This motion differs from
natural motion in that the kinetic and potential energies in Figure 31.1(b) and (c) would not have the same
maximum values. As a result, the total energy would vary in each cycle.
The energy of the vibrating system is not constant, but in the bigger picture energy is still always conserved.
As the object moves, the external agent is doing work on it, transferring energy to and from it. (Removing
energy from the object is considered negative work.) But since the system is not self-contained, many of
the conclusions in Chapter 31 do not apply.
32b. Energy and Amplitude
Equations 31.5–31.7 are examples of a very general principle. Whenever the topic is natural motion of
oscillations or waves, all quantities associated with energy are proportional to the square of any quantity
measuring the amplitude of the motion:
energy-related quantity ∝ (amplitude of any sort)2 . (32.1)

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 Part C : Vibration & Oscillation

The amplitude need not be the usual “maximum displacement” amplitude. In Chapter 24 and in Eq. 31.6,
𝑠𝑚 and 𝑣𝑚 are velocity amplitudes. Even if you do not know the equation relatting your amplitude to an
energy-related quantity, you can still use this proportionality to compare two natural oscillations or waves.
What about oscillating motion that is not natural motion? You can often identify this situation if the
frequency of the motion is determined by something outside the system. Then this proportion can still be
useful, but it’s necessary to be more careful that the energy and amplitude are directly related. For example,
the maximum kinetic energy would always be proportional to the square of the velocity amplitude, because
of Eq. 31.6. However, the kinetic energy might not be proportional to the maximum displacement.

Chapter 33. Damped Harmonic Motion

Oscillations in the real world decay in amplitude, sometimes very slowly and sometimes quickly. A slow
decay of an oscillating system is called damping. The reason for damping is that forces, other than the
restoring force required for vibrations, act on the moving object. The detailed nature of those forces can be
complicated, but often they continuously impede the motion; friction is the most well-known example.
These additional forces were not included in the model used for Section 20c, which is why the “Circle of
Physics” predicted everlasting SHM.
Because the damping forces tend to impede the oscillatory motion, you might guess that damping would
slow the rate of oscillation. This is indeed true. If damping forces are added to an oscillator, they will
increase its period and reduce its frequency. However, this effect is generally extremely small. For
example, in the specific case of DHM (described below), damping forces that are sufficient to change the
frequency by only 0.1% will also cause the amplitude to be cut in half after less than 2.5 cycles! So, in
most cases, it’s an excellent approximation that the damped frequency is equal to the natural frequency.
One common example is a particular damped motion called damped harmonic motion, or DHM. DHM
is illustrated by the solid curve in Figure 33.1. This requires a particular type of damping force. For
example, rubbing friction does not result in DHM, while air resistance does. Many damped motions that
are technically not DHM are nevertheless well modeled by DHM.
DHM preserves some of the properties from SHM. Although it is not truly a repeating motion, it clearly
has a cyclical aspect, and in fact the mathematical function describing the motion contains a sinusoid. A
period can be measured for those cycles. If it is measured between points at the equilibrium position, where
𝛥𝑥 = 0, then the period and frequency stay precisely the same throughout the motion. Thus, one can say
that the frequency of oscillation is still independent of amplitude. That is, although the period in Figure
33.1 is approximately the same as it would be without damping, the period is exactly the same on the left
and right sides of the figure. The time interval between pairs of maxima or pairs of minima is not exactly
equal to the period, although it is very close.

Figure 33.1
Damped harmonic motion.

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Figure 33.2
Shrinking amplitude of damped harmonic motion. DHM itself is not shown.

The independence of frequency and amplitude is extraordinarily important for the production of music.
Consider what would happen if, as the vibration of a guitar string fades away, the frequency (and hence
pitch) from the string changed!
The amplitude of DHM also decays in a very specific way. Although the amplitude is only evident where
the DHM reaches an extreme displacement, we can imagine the amplitude as a continuous function that
forms an envelope around the oscillation, as shown by the dotted lines in Figure 33.1.
For DHM, over equal intervals of time (or an equal number of cycles) the
oscillation amplitude is reduced by a specific constant fraction.
For example, suppose that for some system the amplitude is reduced by 10% for every five cycles. If it
were to start with an amplitude of 5 cm, as time goes on the amplitude would change as illustrated in Figure
33.2. For equal-sized steps along the time axis, the amplitude is reduced to a specific fraction of its previous
value. This would work for any time interval that you might choose; for a different one (say, three cycles),
the reducing fraction would simply be different.
Notice that this means that the oscillation will never die away completely, because the amplitude will
always be larger than zero. Of course, it will eventually become imperceptibly small.

Chapter 34. Energy in DHM

As the energy in a damped harmonic motion decreases, the energy must be going elsewhere in order for
energy to be conserved. The small additional forces that cause the damping also cause the spring-object
system to do work on its surroundings, transferring the energy out. Where does this energy go? There are
many possibilities for different situations. Some important examples are heat energy (in the surroundings
or in the spring), air turbulence, and, most important for this book, sound! If sound is to be “mechanical
radiant energy,” then its energy must come from somewhere. In order for a vibrating object to create sound,
it must be at least a little damped as the sound energy leaves it.
The energy-amplitude proportion can be used to determine how the energy of DHM decreases. Pick two
times, 𝑡1 earlier than 𝑡2 , and use the variable 𝑅 for the ratio of the amplitudes at those two times
𝛥𝑥max,2 = 𝑅 𝛥𝑥max,1 . (34.1)
For instance, if the two times are two neighboring labeled points in Figure 33.2, then 𝑡2 − 𝑡1 = 5𝑇 and 𝑅 =
0.9. Using 𝐸 to represent energy, proportion 32.1 gives
2
𝐸2 𝛥𝑥max,2
= 2 = 𝑅2 . (34.2)
𝐸1 𝛥𝑥max,1
From this we have two conclusions. First, energy decreases in the same sort of way that amplitude does.

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For DHM, over equal intervals of time (or an equal number of cycles) the
total energy is reduced by a specific constant fraction.
Second, for a specific time interval, the reducing fraction for energy is the square of the reducing fraction
for amplitude.

Chapter 35. Math for DHM

This chapter lists a few more mathematical details for DHM, which are not required in the rest of the book
but which may be of interest.
In order to get damped harmonic motion, as opposed to any other type of damped motion, the damping
force must be proportional to the velocity of the moving object,
𝐹damp ∝ −𝑣 . (35.1)
That proportion would have been equally correct without the minus sign, but the minus sign is included as
a reminder that the force must be in the opposite direction from the velocity. Otherwise, the force would
speed up the motion instead of slowing it down.
As mentioned, this is a good model for many situations, including air resistance. In other common
situations, is it not. A block sliding back and forth while supported on a table feels a frictional force from
the table which does not obey proportion 35.1, and the resulting motion is not DHM.
We can think of the dotted-line envelope as giving the amplitude as a function of time. The equation for
this envelope is an exponential,
𝑥𝑚 = 𝑥𝑚0 10(−𝑡⁄𝑡0 ) , (35.2)
where 𝑥𝑚0 is the amplitude at the beginning (that is, when 𝑡 = 0) and 𝑡0 is the time interval over which the
amplitude decreases to one tenth of its starting value. The 10 is called the base of the exponent, and it is
an arbitrary choice. If another base is chosen, then 𝑡0 just needs to be adjusted to the time corresponding
to the amplitude shrinking by a factor of 1⁄base. But in this book, only the base 10 will be used.

59