Physics Including Human Applications 309
Chapter 15
SIMPLE HARMONIC MOTION
GOALS
When you have mastered the contents of this chapter, you will be able to achieve the
following goals:
Definitions
Define each of the following terms, and use it in an operational definition:
period frequency
simple harmonic motion restoring force
amplitude damping
phase angle
UCM and SHM
Correlate uniform circular motion and simple harmonic motion.
SHM Problems
Solve problems involving simple harmonic motion.
Energy Transformations
Analyze the transfer of energy in simple harmonic motion.
Superposition
Explain the application of the principle of superposition to simple harmonic motion.
Natural Frequencies
Calculate the natural frequencies of solids from their elastic moduli and density values.
PREREQUISITES
Before beginning this chapter you should have achieved the goals of Chapter 5, Energy,
Chapter 7, Rotational Motion, and Chapter 13, Elastic Properties of Materials.
, Physics Including Human Applications 310
Chapter 15
SIMPLE HARMONIC MOTION
15.1 Introduction
You are familiar with many examples of repeated motion in your daily life. If an object
returns to its original position a number of times, we call its motion repetitive. Typical
examples of repetitive motion of the human body are heartbeat and breathing. Many
objects move in a repetitive way, a swing, a rocking chair, and a clock pendulum, for
example. Probably the first understanding the ancients had of repetitive motion grew
out of their observations of the motion of the sun and the phases of the moon.
Strings undergoing repetitive motion are the physical basis of all stringed musical
instruments. What are the common properties of these diverse examples of repetitive
motion?
In this chapter we will discuss the physical characteristics of repetitive motion, and
we will develop techniques that can be used to analyze this motion quantitatively.
15.2 Kinematics of Simple Harmonic Motion
One common characteristic of the motions of the heartbeat, clock pendulum, violin
string, and the rotating phonograph turntable is that each motion has a well-defined
time interval for each complete cycle of its motion. Any motion that repeats itself with
equal time intervals is called periodic motion. Its period is the time required for one cycle
of the motion.
Let us analyze the periodic motion of the
turntable of a phonograph. Suppose that we place
a marker on a turntable that is rotating about a
vertical axis at a uniform rate in a counterclockwise
direction when viewed from above. If you observe
the motion of the marker in a horizontal plane-that
is, viewing the turntable edge-on-the marker will
seem to be moving back and forth along a line. The
motion you see is the projection of uniform circular
motion onto a diameter and is called simple
harmonic motion (Figure 15.1). To derive the
equation for simple harmonic motion, project the
motion of the marker upon the diameter AB . The
displacement is given relative to the center of the
path O and is represented by x = OC. From Figure
15.1 we see that x = R cosθ, where R is the distance
of the marker from the axis of rotation. The
maximum displacement of the motion is called the
amplitude of the motion and is represented by the
symbol A.
, Physics Including Human Applications 311
The displacement of the marker in a direction parallel to the diameter AOB is then given
by the following equation,
x = A cosθ (15.1)
where θ is the angle through which the marker on the turntable has turned. Since we
know that the turntable is rotating with a constant angular velocity ω , we recall from
Chapter 7 that we can write an expression for the angular displacement θ as the angular
speed times the time plus the starting angle,
θ=ωt+φ
where t is the time of rotation and φ, the phase angle, is the angular displacement at the
beginning, t = 0. If we choose the starting position along the line DOB, then φ = 0 at t =
0. In general, the equation for the x-displacement is given by
x = A cos (ωt +φ) (15.2)
The velocity of the marker for the position
shown in Figure 15.1 is tangential to the circle
of motion of the marker and in the direction
shown in Figure 15.2 . You may recall from
Chapter 7 that the magnitude of the velocity is
given by
v = 2π R n =ω R = ω A (7.12)
where n is the number of revolutions of the
turntable in one second, ω is the angular
speed in radians per second, and R is the
radius of the circle of motion and is equal to
the amplitude of displacement A .
EXAMPLE
What is the velocity of a point on the rim of the standard 12-inch long-playing
phonograph record?
R = 15.2 cm
ω = 33 (1/3) rpm = 5.56 x 10-1 rev/sec = 3.49 rad/sec.
v = ωR = 53.1 cm/sec
The velocity of the marker in a direction parallel to the line DOB is shown by the
component υx in Figure 15.2 , where
vx = -v sinθ = -ωA sinθ (15.3)
The negative sign indicates that the direction of motion is in the negative x-direction.
When sinθ is positive, the velocity vx is in the negative x-direction. Notice sinθ is always
Chapter 15
SIMPLE HARMONIC MOTION
GOALS
When you have mastered the contents of this chapter, you will be able to achieve the
following goals:
Definitions
Define each of the following terms, and use it in an operational definition:
period frequency
simple harmonic motion restoring force
amplitude damping
phase angle
UCM and SHM
Correlate uniform circular motion and simple harmonic motion.
SHM Problems
Solve problems involving simple harmonic motion.
Energy Transformations
Analyze the transfer of energy in simple harmonic motion.
Superposition
Explain the application of the principle of superposition to simple harmonic motion.
Natural Frequencies
Calculate the natural frequencies of solids from their elastic moduli and density values.
PREREQUISITES
Before beginning this chapter you should have achieved the goals of Chapter 5, Energy,
Chapter 7, Rotational Motion, and Chapter 13, Elastic Properties of Materials.
, Physics Including Human Applications 310
Chapter 15
SIMPLE HARMONIC MOTION
15.1 Introduction
You are familiar with many examples of repeated motion in your daily life. If an object
returns to its original position a number of times, we call its motion repetitive. Typical
examples of repetitive motion of the human body are heartbeat and breathing. Many
objects move in a repetitive way, a swing, a rocking chair, and a clock pendulum, for
example. Probably the first understanding the ancients had of repetitive motion grew
out of their observations of the motion of the sun and the phases of the moon.
Strings undergoing repetitive motion are the physical basis of all stringed musical
instruments. What are the common properties of these diverse examples of repetitive
motion?
In this chapter we will discuss the physical characteristics of repetitive motion, and
we will develop techniques that can be used to analyze this motion quantitatively.
15.2 Kinematics of Simple Harmonic Motion
One common characteristic of the motions of the heartbeat, clock pendulum, violin
string, and the rotating phonograph turntable is that each motion has a well-defined
time interval for each complete cycle of its motion. Any motion that repeats itself with
equal time intervals is called periodic motion. Its period is the time required for one cycle
of the motion.
Let us analyze the periodic motion of the
turntable of a phonograph. Suppose that we place
a marker on a turntable that is rotating about a
vertical axis at a uniform rate in a counterclockwise
direction when viewed from above. If you observe
the motion of the marker in a horizontal plane-that
is, viewing the turntable edge-on-the marker will
seem to be moving back and forth along a line. The
motion you see is the projection of uniform circular
motion onto a diameter and is called simple
harmonic motion (Figure 15.1). To derive the
equation for simple harmonic motion, project the
motion of the marker upon the diameter AB . The
displacement is given relative to the center of the
path O and is represented by x = OC. From Figure
15.1 we see that x = R cosθ, where R is the distance
of the marker from the axis of rotation. The
maximum displacement of the motion is called the
amplitude of the motion and is represented by the
symbol A.
, Physics Including Human Applications 311
The displacement of the marker in a direction parallel to the diameter AOB is then given
by the following equation,
x = A cosθ (15.1)
where θ is the angle through which the marker on the turntable has turned. Since we
know that the turntable is rotating with a constant angular velocity ω , we recall from
Chapter 7 that we can write an expression for the angular displacement θ as the angular
speed times the time plus the starting angle,
θ=ωt+φ
where t is the time of rotation and φ, the phase angle, is the angular displacement at the
beginning, t = 0. If we choose the starting position along the line DOB, then φ = 0 at t =
0. In general, the equation for the x-displacement is given by
x = A cos (ωt +φ) (15.2)
The velocity of the marker for the position
shown in Figure 15.1 is tangential to the circle
of motion of the marker and in the direction
shown in Figure 15.2 . You may recall from
Chapter 7 that the magnitude of the velocity is
given by
v = 2π R n =ω R = ω A (7.12)
where n is the number of revolutions of the
turntable in one second, ω is the angular
speed in radians per second, and R is the
radius of the circle of motion and is equal to
the amplitude of displacement A .
EXAMPLE
What is the velocity of a point on the rim of the standard 12-inch long-playing
phonograph record?
R = 15.2 cm
ω = 33 (1/3) rpm = 5.56 x 10-1 rev/sec = 3.49 rad/sec.
v = ωR = 53.1 cm/sec
The velocity of the marker in a direction parallel to the line DOB is shown by the
component υx in Figure 15.2 , where
vx = -v sinθ = -ωA sinθ (15.3)
The negative sign indicates that the direction of motion is in the negative x-direction.
When sinθ is positive, the velocity vx is in the negative x-direction. Notice sinθ is always
