Physics For Scientists And Engineers 6E - part 114

eriodic motion is motion of an object that regularly repeats—the object returns to a

given position after a fixed time interval. With a little thought, we can identify several
types of periodic motion in everyday life. Your car returns to the driveway each after-
noon. You return to the dinner table each night to eat. A bumped chandelier swings
back and forth, returning to the same position at a regular rate. The Earth returns to
the same position in its orbit around the Sun each year, resulting in the variation
among the four seasons. The Moon returns to the same relationship with the Earth
and the Sun, resulting in a full Moon approximately once a month.

In addition to these everyday examples, numerous other systems exhibit periodic

motion. For example, the molecules in a solid oscillate about their equilibrium posi-
tions; electromagnetic waves, such as light waves, radar, and radio waves, are character-
ized by oscillating electric and magnetic field vectors; and in alternating-current elec-
trical circuits, voltage, current, and electric charge vary periodically with time.

A special kind of periodic motion occurs in mechanical systems when the force act-

ing on an object is proportional to the position of the object relative to some equilib-
rium position. If this force is always directed toward the equilibrium position, the mo-
tion is called simple harmonic motion, which is the primary focus of this chapter.

15.1 Motion of an Object Attached to a Spring

As a model for simple harmonic motion, consider a block of mass m attached to the
end of a spring, with the block free to move on a horizontal, frictionless surface
(Fig. 15.1). When the spring is neither stretched nor compressed, the block is at the
position called the

equilibrium position of the system, which we identify as x ! 0. We

know from experience that such a system oscillates back and forth if disturbed from its
equilibrium position.

We can understand the motion in Figure 15.1 qualitatively by first recalling that

when the block is displaced to a position x, the spring exerts on the block a force that
is proportional to the position and given by

Hooke’s law (see Section 7.4):

! "

(15.1)

We call this a

restoring force because it is always directed toward the equilibrium posi-

tion and therefore opposite the displacement from equilibrium. That is, when the block
is displaced to the right of x ! 0 in Figure 15.1, then the position is positive and the
restoring force is directed to the left. When the block is displaced to the left of x ! 0,
then the position is negative and the restoring force is directed to the right.

Applying Newton’s second law F

to the motion of the block, with Equation

15.1 providing the net force in the x direction, we obtain

kx ! ma

(15.2)

! "

453

Hooke’s law

(a)

x = 0

(b)

x = 0

= 0

(c)

x = 0

Active Figure 15.1 A block

attached to a spring moving on a

frictionless surface. (a) When the

block is displaced to the right of

equilibrium (x # 0), the force

exerted by the spring acts to the

left. (b) When the block is at its

equilibrium position (x ! 0), the

force exerted by the spring is zero.

the left of equilibrium (x $ 0), the

force exerted by the spring acts to

the right.

At the Active Figures link,

at http://www.pse6.com, you

can choose the spring constant

and the initial position and

velocities of the block to see

the resulting simple harmonic

motion.

That is, the acceleration is proportional to the position of the block, and its direction is
opposite the direction of the displacement from equilibrium. Systems that behave in
this way are said to exhibit

simple harmonic motion. An object moves with simple

harmonic motion whenever its acceleration is proportional to its position and is
oppositely directed to the displacement from equilibrium.

If the block in Figure 15.1 is displaced to a position x ! A and released from rest,

its initial acceleration is " kA/m. When the block passes through the equilibrium posi-
tion x ! 0, its acceleration is zero. At this instant, its speed is a maximum because the
acceleration changes sign. The block then continues to travel to the left of equilibrium
with a positive acceleration and finally reaches x ! " A, at which time its acceleration
is % kA/m and its speed is again zero, as discussed in Sections 7.4 and 8.6. The block
completes a full cycle of its motion by returning to the original position, again passing
through x ! 0 with maximum speed. Thus, we see that the block oscillates between the
turning points x ! &A. In the absence of friction, because the force exerted by the
spring is conservative, this idealized motion will continue forever. Real systems are gen-
erally subject to friction, so they do not oscillate forever. We explore the details of the
situation with friction in Section 15.6.

As Pitfall Prevention 15.1 points out, the principles that we develop in this chapter

are also valid for an object hanging from a vertical spring, as long as we recognize that
the weight of the object will stretch the spring to a new equilibrium position x ! 0. To
prove this statement, let x

represent the total extension of the spring from its equilib-

rium position without the hanging object. Then, x

! "

(mg/k) % x, where " (mg/k) is

the extension of the spring due to the weight of the hanging object and x is the instan-
taneous extension of the spring due to the simple harmonic motion. The magnitude
of the net force on the object is then F

! "

k(" (mg/k) % x) " mg ! " kx. The

net force on the object is the same as that on a block connected to a horizontal spring
as in Equation 15.1, so the same simple harmonic motion results.

454

C H A P T E R 15 • Oscillatory Motion

Quick Quiz 15.1

A block on the end of a spring is pulled to position x ! A

and released. In one full cycle of its motion, through what total distance does it travel?
(a) A/2 (b) A (c) 2A (d) 4A

15.2 Mathematical Representation

of Simple Harmonic Motion

Let us now develop a mathematical representation of the motion we described in the
preceding section. We model the block as a particle subject to the force in Equation
15.1. We will generally choose x as the axis along which the oscillation occurs; hence,
we will drop the subscript-x notation in this discussion. Recall that, by definition,
a ! dv/dt ! d

x/dt

, and so we can express Equation 15.2 as

(15.3)

If we denote the ratio k/m with the symbol '

(we choose '

rather than ' in order to

make the solution that we develop below simpler in form), then

(15.4)

and Equation 15.3 can be written in the form

(15.5)

! "

▲

PITFALL PREVENTION

15.1 The Orientation of

the Spring

Figure  15.1  shows  a  horizontal
spring,  with  an  attached  block
sliding  on  a  frictionless  surface.
Another  possibility  is  a  block
hanging from a vertical spring. All
of  the  results  that  we  discuss  for
the  horizontal  spring  will  be  the
same  for  the  vertical  spring,  ex-
cept  that  when  the  block  is
placed  on  the  vertical  spring,  its
weight will cause the spring to ex-
tend.  If  the  resting  position  of
the block is defined as x ! 0, the
results  of  this  chapter  will  apply
to this vertical system also.

▲

PITFALL PREVENTION

15.2 A Nonconstant

Acceleration

Notice that the acceleration of
the particle in simple harmonic
motion is not constant. Equation
15.3 shows that it varies with posi-
tion x. Thus, we cannot apply the
kinematic equations of Chapter 2
in this situation.

What we now require is a mathematical solution to Equation 15.5—that is, a func-

tion x(t) that satisfies this second-order differential equation. This is a mathematical
representation of the position of the particle as a function of time. We seek a function
x(t) whose second derivative is the same as the original function with a negative sign
and multiplied by '

. The trigonometric functions sine and cosine exhibit this behav-

ior, so we can build a solution around one or both of these. The following cosine func-
tion is a solution to the differential equation:

(15.6)

where A, ', and ( are constants. To see explicitly that this equation satisfies Equation
15.5, note that

(15.7)

(15.8)

Comparing Equations 15.6 and 15.8, we see that d

x/dt

! "

x and Equation 15.5 is

satisfied.

The parameters A, ', and ( are constants of the motion. In order to give physical

significance to these constants, it is convenient to form a graphical representation of
the motion by plotting x as a function of t, as in Figure 15.2a. First, note that A, called
the

amplitude of the motion, is simply the maximum value of the position of the

particle in either the positive or negative x direction. The constant ' is called the
angular frequency, and has units of rad/s.

It is a measure of how rapidly the oscilla-

tions are occurring—the more oscillations per unit time, the higher is the value of '.
From Equation 15.4, the angular frequency is

(15.9)

The constant angle ( is called the

phase constant (or initial phase angle) and,

along with the amplitude A, is determined uniquely by the position and velocity of
the particle at t ! 0. If the particle is at its maximum position x ! A at t ! 0, the
phase constant is ( ! 0 and the graphical representation of the motion is shown in
Figure 15.2b. The quantity ('t % () is called the

phase of the motion. Note that

the function x(t) is periodic and its value is the same each time 't increases by 2)
radians.

Equations 15.1, 15.5, and 15.6 form the basis of the mathematical representation

of simple harmonic motion. If we are analyzing a situation and find that the force
on a particle is of the mathematical form of Equation 15.1, we know that the motion
will be that of a simple harmonic oscillator and that the position of the particle is
described by Equation 15.6. If we analyze a system and find that it is described by a
differential equation of the form of Equation 15.5, the motion will be that of a sim-
ple harmonic oscillator. If we analyze a situation and find that the position of a
particle is described by Equation 15.6, we know the particle is undergoing simple
harmonic motion.

' !

√

! "

sin('t % () ! "

A cos('t % ()

cos('t % () ! "'A sin('t % ()

x(t) ! A cos('t % ()

S E C T I O N 15 . 2 • Mathematical Representation of Simple Harmonic Motion

455

We have seen many examples in earlier chapters in which we evaluate a trigonometric function of

an angle. The argument of a trigonometric function, such as sine or cosine, must be a pure number.
The radian is a pure number because it is a ratio of lengths. Angles in degrees are pure numbers simply
because the degree is a completely artificial “unit”—it is not related to measurements of lengths. The
notion of requiring a pure number for a trigonometric function is important in Equation 15.6, where
the angle is expressed in terms of other measurements. Thus, ' must be expressed in rad/s (and not,
for example, in revolutions per second) if t is expressed in seconds. Furthermore, other types of func-
tions such as logarithms and exponential functions require arguments that are pure numbers.

▲

PITFALL PREVENTION

15.3 Where’s the Triangle?

Equation 15.6 includes a trigono-
metric  function,  a  mathematical
function that can be used whether
it  refers  to  a  triangle  or  not.  In
this  case,  the  cosine  function
happens  to  have  the  correct  be-
havior  for  representing  the  posi-
tion  of  a  particle  in  simple  har-
monic motion.

Position versus time for an

object in simple harmonic

motion

–A

(b)

–A

(a)

Active Figure 15.2 (a) An x -vs.-t

graph for an object undergoing

simple harmonic motion. The

amplitude of the motion is A, the

period (page 456) is T, and the

phase constant is

(

. (b) The x -vs.-t

graph in the special case in which

x ! A at t ! 0 and hence

(

At the Active Figures link

at http://www.pse6.com, you

can adjust the graphical repre-

sentation and see the resulting

simple harmonic motion of the

block in Figure 15.1.

An experimental arrangement that exhibits simple harmonic motion is illustrated

in Figure 15.3. An object oscillating vertically on a spring has a pen attached to it.
While the object is oscillating, a sheet of paper is moved perpendicular to the direction
of motion of the spring, and the pen traces out the cosine curve in Equation 15.6.

Let us investigate further the mathematical description of simple harmonic motion.

The period T of the motion is the time interval required for the particle to go through
one full cycle of its motion (Fig. 15.2a). That is, the values of x and v for the particle at
time t equal the values of x and v at time t % T. We can relate the period to the angular
frequency by using the fact that the phase increases by 2) radians in a time interval of

Simplifying this expression, we see that 'T ! 2), or

(15.10)

T !

['(t % T

) % (] " ('t % () ! 2)

456

C H A P T E R 15 • Oscillatory Motion

Quick Quiz 15.2

Consider a graphical representation (Fig. 15.4) of simple

harmonic motion, as described mathematically in Equation 15.6. When the object is at
point ! on the graph, its (a) position and velocity are both positive (b) position and ve-
locity are both negative (c) position is positive and its velocity is zero (d) position is neg-
ative and its velocity is zero (e) position is positive and its velocity is negative (f ) position
is negative and its velocity is positive.

Quick Quiz 15.3

Figure 15.5 shows two curves representing objects undergo-

ing simple harmonic motion. The correct description of these two motions is that the
simple harmonic motion of object B is (a) of larger angular frequency and larger ampli-
tude than that of object A (b) of larger angular frequency and smaller amplitude than
that of object A (c) of smaller angular frequency and larger amplitude than that of
object A (d) of smaller angular frequency and smaller amplitude than that of object A.

Motion
of paper

Figure 15.3 An experimental

apparatus for demonstrating

simple harmonic motion. A pen

attached to the oscillating object

traces out a sinusoidal pattern on

the moving chart paper.

Figure 15.4 (Quick Quiz 15.2) An x-t graph for an object undergoing simple harmonic

motion. At a particular time, the object’s position is indicated by ! in the graph.

Object A

Object B

Figure 15.5 (Quick Quiz 15.3) Two x-t graphs for objects undergoing simple harmonic

motion. The amplitudes and frequencies are different for the two objects.

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