Periodic motion, in physics, motion repeated in equal intervals of time. Periodic motion is
performed, for example, by a rocking chair, a bouncing ball, a vibrating tuning fork, a
swing in motion, the Earth in its orbit around the Sun, and a water wave. In each case the
interval of time for a repetition, or cycle, of the motion is called a period, while the number
of periods per unit time is called the frequency. Thus, the period of the Earth’s orbit is one
year, and its frequency is one orbit per year. A tuning fork might have a frequency of 1,000
cycles per second and a period of 1 millisecond (1 thousandth of a second).
Simple harmonic motion is a special case of periodic motion. In the examples given
above, the rocking chair, the tuning fork, the swing, and the water wave execute simple
harmonic motion, but the bouncing ball and the Earth in its orbit do not.
Waves that can be represented by sine curves are periodic. If the wave is propagated with
a velocity v and has a wavelength λ, then the period T is equal to wavelength divided by
velocity, or T= λ/v. The frequency f is the reciprocal of the period; thus, f = 1/T = v/λ.
Key Points

Motion that repeats itself regularly is called periodic motion. One complete repetition of the
motion is called a cycle. The duration of each cycle is the period.

The frequency refers to the number of cycles completed in an interval of time. It is the reciprocal
of the period and can be calculated with the equation f=1/T.

Some motion is best characterized by the angular frequency (ω). The angular frequency refers
to the angular displacement per unit time and is calculated from the frequency with the equation
ω=2πf.
Key Terms



period: The duration of one cycle in a repeating event.
angular frequency: The angular displacement per unit time.
frequency: The quotient of the number of times n a periodic phenomenon occurs over the time
t in which it occurs: f = n / t.
Period and Frequency
The usual physics terminology for motion that repeats itself over and over is periodic motion, and the
time required for one repetition is called the period, often expressed as the letter T. (The symbol P is
not used because of the possible confusion with momentum. ) One complete repetition of the motion
is called a cycle. The frequency is defined as the number of cycles per unit time. Frequency is usually
denoted by a Latin letter f or by a Greek letter ν (nu). Note that period and frequency are reciprocals
of each other.
Sinusoidal Waves of Varying Frequencies: Sinusoidal waves of various frequencies; the bottom waves have higher
frequencies than those above. The horizontal axis represents time.
f=1/Tf=1/T
For example, if a newborn baby’s heart beats at a frequency of 120 times a minute, its period (the
interval between beats) is half a second. If you calibrate your intuition so that you expect large
frequencies to be paired with short periods, and vice versa, you may avoid some embarrassing
mistakes on physics exams.
Units
Locomotive Wheels: The locomotive’s wheels spin at a frequency of f cycles per second, which can also be described as ω
radians per second. The mechanical linkages allow the linear vibration of the steam engine’s pistons, at frequency f, to drive the
wheels.
In SI units, the unit of frequency is the hertz (Hz), named after the German physicist Heinrich Hertz: 1
Hz indicates that an event repeats once per second. A traditional unit of measure used with rotating
mechanical devices is revolutions per minute, abbreviated RPM. 60 RPM equals one hertz (i.e., one
revolution per second, or a period of one second). The SI unit for period is the second.
Angular Frequency
Often periodic motion is best expressed in terms of angular frequency, represented by the Greek
letter ω (omega). Angular frequency refers to the angular displacement per unit time (e.g., in rotation)
or the rate of change of the phase of a sinusoidal waveform (e.g., in oscillations and waves), or as the
rate of change of the argument of the sine function.
y(t)=sin(θ(t))=sin(ωt)=sin(2πft)y(t)=sin(θ(t))=sin(ωt)=sin(2πft)
ω=2πfω=2πf
Angular frequency is often represented in units of radians per second (recall there are 2π radians in a
circle).
Period of a Mass on a Spring
The period of a mass m on a spring of spring constant k can be calculated as T=2π√mkT=2πmk.

If an object is vibrating to the right and left, then it must have a leftward force on it when it is on
the right side, and a rightward force when it is on the left side.

The restoring force causes an oscillating object to move back toward its stable equilibrium
position, where the net force on it is zero.

The simplest oscillations occur when the restoring force is directly proportional to displacement.
In this case the force can be calculated as F=-kx, where F is the restoring force, k is the force
constant, and x is the displacement.

The motion of a mass on a spring can be described as Simple Harmonic Motion (SHM):
oscillatory motion that follows Hooke’s Law.
The period of a mass on a spring is given by the equation T=2π√mkT=2πmk

Hooke’s Law
The simplest oscillations occur when the restoring force is directly proportional to displacement. The
name that was given to this relationship between force and displacement is Hooke’s law:
F=kxF=kx
Here, F is the restoring force, x is the displacement from equilibrium or deformation, and k is a
constant related to the difficulty in deforming the system (often called the spring constant or force
constant). Remember that the minus sign indicates the restoring force is in the direction opposite to
the displacement. The force constant k is related to the rigidity (or stiffness) of a system—the larger
the force constant, the greater the restoring force, and the stiffer the system. The units of k are
newtons per meter (N/m). For example, k is directly related to Young’s modulus when we stretch a
string. A typical physics laboratory exercise is to measure restoring forces created by springs,
determine if they follow Hooke’s law, and calculate their force constants if they do.
Mass on a Spring
A common example of an objecting oscillating back and forth according to a restoring force directly
proportional to the displacement from equilibrium (i.e., following Hooke’s Law) is the case of a mass
on the end of an ideal spring, where “ideal” means that no messy real-world variables interfere with
the imagined outcome.
The motion of a mass on a spring can be described as Simple Harmonic Motion (SHM), the name
given to oscillatory motion for a system where the net force can be described by Hooke’s law. We can
now determine how to calculate the period and frequency of an oscillating mass on the end of an
ideal spring. The period T can be calculated knowing only the mass, m, and the force constant, k:
T=2π√mkT=2πmk
When dealing with f=1/Tf=1/T, the frequency is given by:
f=12π√kmf=12πkm
We can understand the dependence of these equations on m and k intuitively. If one were to increase
the mass on an oscillating spring system with a given k, the increased mass will provide more inertia,
causing the acceleration due to the restoring force F to decrease (recall Newton’s Second
Law: F=maF=ma). This will lengthen the oscillation period and decrease the frequency. In contrast,
increasing the force constant k will increase the restoring force according to Hooke’s Law, in turn
causing the acceleration at each displacement point to also increase. This reduces the period and
increases the frequency. The maximum displacement from equilibrium is known as the amplitude X.
Motion of a mass on an ideal spring: An object attached to a spring sliding on a frictionless surface is an uncomplicated
simple harmonic oscillator. When displaced from equilibrium, the object performs simple harmonic motion that has an amplitude
X and a period T. The object’s maximum speed occurs as it passes through equilibrium. The stiffer the spring is, the smaller the
period T. The greater the mass of the object is, the greater the period T. (a) The mass has achieved its greatest displacement X
to the right and now the restoring force to the left is at its maximum magnitude. (b) The restoring force has moved the mass back
to its equilibrium point and is now equal to zero, but the leftward velocity is at its maximum. (c) The mass’s momentum has
carried it to its maximum displacement to the right. The restoring force is now to the right, equal in magnitude and opposite in
direction compared to (a). (d) The equilibrium point is reach again, this time with momentum to the right. (e) The cycle repeats.
Simple Harmonic Motion
Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional
to the displacement.

Simple harmonic motion is often modeled with the example of a mass on a spring, where the
restoring force obey’s Hooke’s Law and is directly proportional to the displacement of an object
from its equilibrium position.

Any system that obeys simple harmonic motion is known as a simple harmonic oscillator.

The equation of motion that describes simple harmonic motion can be obtained by combining
Newton’s Second Law and Hooke’s Law into a second-order linear ordinary differential
equation: Fnet=md2xdt2=−kxFnet=md2xdt2=−kx.
Key Terms


simple harmonic oscillator: A device that implements Hooke’s law, such as a mass that is
attached to a spring, with the other end of the spring being connected to a rigid support, such as
a wall.
oscillator: A pattern that returns to its original state, in the same orientation and position, after a
finite number of generations.
Simple Harmonic Motion
Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional
to the displacement (i.e., it follows Hooke’s Law). It can serve as a mathematical model of a variety of
motions, such as the oscillation of a spring. In addition, other phenomena can be approximated by
simple harmonic motion, such as the motion of a simple pendulum, or molecular vibration.
Simple Harmonic Motion: A brief introduction to simple harmonic motion for calculus-based physics
students.
Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear
elastic restoring force given by Hooke’s Law. A system that follows simple harmonic motion is known
as a simple harmonic oscillator.
Using Newton’s Second Law, Hooke’s Law, and some differential Calculus, we were able to derive
the period and frequency of the mass oscillating on a spring that we encountered in the last section!
Note that the period and frequency are completely independent of the amplitude.
The below figure shows the simple harmonic motion of an object on a spring and presents graphs of
x(t),v(t), and a(t) versus time. You should learn to create mental connections between the above
equations, the different positions of the object on a spring in the cartoon, and the associated positions
in the graphs of x(t), v(t), and a(t).
Visualizing Simple Harmonic Motion: Graphs of x(t),v(t), and a(t) versus t for the motion of an object on a spring. The net
force on the object can be described by Hooke’s law, and so the object undergoes simple harmonic motion. Note that the initial
position has the vertical displacement at its maximum value X; v is initially zero and then negative as the object moves down;
and the initial acceleration is negative, back toward the equilibrium position and becoming zero at that point.
Simple Harmonic Motion and Uniform Circular Motion
Simple harmonic motion is produced by the projection of uniform circular motion onto one of the axes
in the x-y plane.
The Simple Pendulum
A simple pendulum acts like a harmonic oscillator with a period dependent only on L and g for
sufficiently small amplitudes.
LEARNING OBJECTIVES
Identify parameters that affect the period of a simple pendulum
KEY TAKEAWAYS
Key Points

A simple pendulum is defined as an object that has a small mass, also known as the pendulum
bob, which is suspended from a wire or string of negligible mass.

When displaced, a pendulum will oscillate around its equilibrium point due to momentum in
balance with the restoring force of gravity.

When the swings ( amplitudes ) are small, less than about 15º, the pendulum acts as a simple
harmonic oscillator with period T=2π√LgT=2πLg, where L is the length of the string and g is the
acceleration due to gravity.
Key Terms

simple pendulum: A hypothetical pendulum consisting of a weight suspended by a weightless
string.
The Simple Pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is
displaced sideways from its resting equilibrium position, it is subject to a restoring force; after it
reaches its highest point in its swing, gravity will accelerate it back toward the equilibrium position.
When released, the restoring force combined with the pendulum’s mass causes it to oscillate about
the equilibrium position, swinging back and forth.
Simple Pendulum: A simple pendulum has a small-diameter bob and a string that has a very small mass but is strong enough
not to stretch appreciably. The linear displacement from equilibrium is s, the length of the arc. Also shown are the forces on the
bob, which result in a net force of −mgsinθ toward the equilibrium position—that is, a restoring force.
For small displacements, a pendulum is a simple harmonic oscillator. A simple pendulum is defined to
have an object that has a small mass, also known as the pendulum bob, which is suspended from a
wire or string of negligible mass, such as shown in the illustrating figure. Exploring the simple
pendulum a bit further, we can discover the conditions under which it performs simple harmonic
motion, and we can derive an interesting expression for its period.
Pendulums: A brief introduction to pendulums (both ideal and physical) for calculus-based physics
students from the standpoint of simple harmonic motion.
We begin by defining the displacement to be the arc length s. We see from the figure that the net
force on the bob is tangent to the arc and equals −mgsinθ. (The weight mg has
components mgcosθ along the string and mgsinθ tangent to the arc. ) Tension in the string exactly
cancels the component mgcosθ parallel to the string. This leaves a net restoring force drawing the
pendulum back toward the equilibrium position at θ = 0.
Now, if we can show that the restoring force is directly proportional to the displacement, then we have
a simple harmonic oscillator. In trying to determine if we have a simple harmonic oscillator, we should
note that for small angles (less than about 15º), sinθ≈θ (sinθ and θdiffer by about 1% or less at
smaller angles). Thus, for angles less than about 15º, the restoring force F is
F≈−mgθF≈−mgθ.
The displacement s is directly proportional to θ. When θ is expressed in radians, the arc length in a
circle is related to its radius (L in this instance) by:
s=Lθs=Lθ
so that
θ=sLθ=sL.
For small angles, then, the expression for the restoring force is:
F≈mgLsF≈mgLs.
This expression is of the form of Hooke’s Law:
F≈−kxF≈−kx
where the force constant is given by k=mg/L and the displacement is given by x=s. For angles less
than about 15º, the restoring force is directly proportional to the displacement, and the simple
pendulum is a simple harmonic oscillator.
Using this equation, we can find the period of a pendulum for amplitudes less than about 15º. For the
simple pendulum:
T=2π√mk=2π√mmgLT=2πmk=2πmmgL.
Thus,
T=2π√LgT=2πLg
or the period of a simple pendulum. This result is interesting because of its simplicity. The only things
that affect the period of a simple pendulum are its length and the acceleration due to gravity. The
period is completely independent of other factors, such as mass. Even simple pendulum clocks can
be finely adjusted and accurate. Note the dependence of T on g. If the length of a pendulum is
precisely known, it can actually be used to measure the acceleration due to gravity. If θ is less than
about 15º, the period T for a pendulum is nearly independent of amplitude, as with simple harmonic
oscillators. In this case, the motion of a pendulum as a function of time can be modeled as:
θ(t)=θocos(2πtT)θ(t)=θocos(2πtT)
For amplitudes larger than 15º, the period increases gradually with amplitude so it is longer than
given by the simple equation for Tabove. For example, at an amplitude of θ0 = 23° it is 1% larger. The
period increases asymptotically (to infinity) as θ0 approaches 180°, because the value θ0 = 180° is an
unstable equilibrium point for the pendulum.
https://courses.lumenlearning.com/boundless-physics/chapter/periodic-motion/
Periodic Motion
If an object repeats its motion along a certain path, about a certain point, in a fixed interval of time, the motion
of such an object is known as periodic motion. Examples of periodic motions are the motion of a pendulum,
the motion of a spring, the vibration of a guitar string, the rotation of the Earth over its axis, the revolving of the
Earth around the Sun, the revolving of the Sun around the center of the Galaxy, etc.
Pendulum
Any weight which is suspended from a pivot and can swing freely when displaced from resting position can be
described as a pendulum.You must have come across many instances of swinging bodies and they all can be
described as pendulums: for example, someone on a swing, swinging chandeliers, or a pendulum clock. In this
article, we will be discussing an idealized mathematical model of pendulum called simple pendulum.
A simple pendulum is an idealized mechanical model comprising of a point mass (also called as bob)
suspended by a massless and inextensible string. When a simple pendulum is pulled to one side and released, it
oscillates about its equilibrium position in simple harmonic motion.
Simple Harmonic Motion
Simple harmonic motion (SHM) is a special type of oscillatory motion performed by the object when it is
subjected to a force which is directly proportional to its displacement from mean position or equilibrium
position and tends to accelerate it towards the mean position. The simplest example will be a spring block
system.
Terms related to SHM

Amplitude (A)(A): It is the maximum distance of a body from mean position.

Angular frequency (\omega)(ω): I will define it after we study the phase diagram of SHM.

Frequency of motion: It is the linear frequency, i.e. number of oscillations per second.

Phase constant (\phi)(ϕ): It tells us about the initial position of the object.
Oscillation refers to the repeated back and forth movement of something between two
positions or states. An oscillation can be a periodic motion that repeats itself in a regular
cycle, such as a sine wave—a wave with perpetual motion as in the side-to-side swing of a
pendulum, or the up-and-down motion of a spring with a weight. An oscillating movement
occurs around an equilibrium point or mean value. It is also known as periodic motion.
A single oscillation is a complete movement, whether up and down or side to side, over a
period of time.
Oscillators
An oscillator is a device that exhibits motion around an equilibrium point. In a pendulum
clock, there is a change from potential energy to kinetic energy with each swing. At the top of
the swing, potential energy is at maximum, and that energy is converted to kinetic energy as it
falls and is driven back up the other side. Now again at the top, kinetic energy has dropped to
zero, and potential energy is high again, powering the return swing. The frequency of the
swing is translated via gears to mark time. A pendulum will lose energy over time to friction if
the clock isn't corrected by a spring. Modern timepieces use the vibrations of quartz and
electronic oscillators, rather than the movement of pendulums.
Oscillating Motion
An oscillating motion in a mechanical system is swinging side to side. It can be translated into
a rotary motion (turning around in a circle) by a peg-and-slot. Rotary motion can be changed
to oscillating motion by the same method.
Oscillating Systems
An oscillating system is an object that moves back and forth, repeatedly returning to its initial
state after a period of time. At the equilibrium point, no net forces are acting on the object.
This is the point in the pendulum swing when it's in a vertical position. A constant force or a
restoring force acts on the object to produce the oscillating motion.
Variables of Oscillation



Amplitude is the maximum displacement from the equilibrium point. If a pendulum
swings one centimeter from the equilibrium point before beginning its return journey,
the amplitude of oscillation is one centimeter.
Period is the time it takes for a complete round trip by the object, returning to its
initial position. If a pendulum starts on the right and takes one second to travel all the
way to the left and another second to return to the right, its period is two seconds.
Period is usually measured in seconds.
Frequency is the number of cycles per unit of time. Frequency equals one divided by
the period. Frequency is measured in Hertz, or cycles per second.
Simple Harmonic Motion
The motion of a simple harmonic oscillating system—when the restoring force is directly
proportional to that of the displacement and acts in the direction opposite to that of
displacement—can be described using sine and cosine functions. An example is a weight
attached to a spring. When the weight is at rest, it's in equilibrium. If the weight is drawn
down, there's a net restoring force on the mass (potential energy). When it's released, it gains
momentum (kinetic energy) and keeps moving beyond the equilibrium point, gaining
potential energy (restoring force) that will drive it in oscillating down again.
An object is undergoing simple harmonic motion (SHM) if;
1. the acceleration of the object is directly proportional to its displacement from its
equilibrium position.
2. the acceleration is always directed towards the equilibrium position.
The frequency (f) of an oscillation is measure in hertz (Hz) it is the number of oscillations per second.
The time for one oscillation is called the period (T) it is measured in seconds.
Acceleration – we can calculate the acceleration of the object at any point in it’s oscillation using the
equation below.
In this equation; a = acceleration in ms-2, f = frequency in Hz, x = displacement from the central
position in m.
Displacement – When using the equation below your calculator must be in radians not degrees ! we
can calculate the displacement of the object at any point in it’s oscillation using the equation below.
The terms in this equation are the same as the equations above. The extra terms in this equation are:
A = the amplitude (maximum displacement) in m, t = the time since the oscillation began in s.
Velocity– we can calculate the velocity of the object at any point in it’s oscillation using the
equation below.
The terms in this equation are the same as the equations above. The extra term in this equation is: v =
the velocity in ms-1.
SHM graphs
When we plot the displacement, velocity and acceleration during SHM against time we get the graphs
below.
The velocity equation simplifies to the equation below when we just want to know the maximum
speed.
The acceleration equation simplifies to the equation below when we just want to know the maximum
acceleration.
Time period of a mass-spring system
Time period of a Pendulum
INTRODUCTION
Have you ever wondered why a grandfather clock keeps accurate time? The motion of the pendulum is a particular kind of repetitive or
periodic motion called simple harmonic motion, or SHM. The position of the oscillating object varies sinusoidally with time. Many objects
oscillate back and forth. The motion of a child on a swing can be approximated to be sinusoidal and can therefore be considered as
simple harmonic motion. Some complicated motions like turbulent water waves are not considered simple harmonic motion.When an
object is in simple harmonic motion, the rate at which it oscillates back and forth as well as its position with respect to time can be easily
determined. In this lab, you will analyze a simple pendulum and a spring-mass system, both of which exhibit simple harmonic motion.
DISCUSSION OF PRINCIPLES
A particle that vibrates vertically in simple harmonic motion moves up and down between two extremes y = ±A. The maximum
displacement A is called the amplitude. This motion is shown graphically in the position-versus-time plot in Figure 1.
Figure 1: Position plot showing sinusoidal motion of an object in SHM
One complete oscillation or cycle or vibration is the motion from, for example,
y = −A
to
y = +A
and back again to
y = −A.
The time interval T required to complete one oscillation is called the period. A related quantity is the frequency f, which is the
number of vibrations the system makes per unit of time. The frequency is the reciprocal of the period and is measured in units of
Hertz, abbreviated Hz;
1 Hz = 1 s−1.
(1)
f = 1/T
If a particle is oscillating along the y-axis, its location on the y-axis at any given instant of time t, measured from the start of the
oscillation is given by the equation
(2)
y = A sin(2πft).
Recall that the velocity of the object is the first derivative and the acceleration the second derivative of the displacement function with
respect to time. The velocity v and the acceleration a of the particle at time t are given by the following.
(3)
v = 2πfA cos(2πft)
(4)
a = −(2πf)2[A sin(2πft)]
Notice that the velocity and acceleration are also sinusoidal. However, the velocity function has a 90° or π/2 phase difference while the
acceleration function has a 180° or π phase difference relative to the displacement function. For example, when the displacement is
positive maximum, the velocity is zero and the acceleration is negative maximum.Substituting from equation 2 into equation 4 yields
(5)
a = −4π2f2y.
From equation 5, we see that the acceleration of an object in SHM is proportional to the displacement and opposite in sign. This is a
basic property of any object undergoing simple harmonic motion.Consider several critical points in a cycle as in the case of a springmass system in oscillation. A spring-mass system consists of a mass attached to the end of a spring that is suspended from a stand.
The mass is pulled down by a small amount and released to make the spring and mass oscillate in the vertical plane. Figure 2 shows
five critical points as the mass on a spring goes through a complete cycle. The equilibrium position for a spring-mass system is the
position of the mass when the spring is neither stretched nor compressed.
Figure 2: Five key points of a mass oscillating on a spring
The mass completes an entire cycle as it goes from position A to position E. A description of each position is as follows.
Position A: The spring is compressed; the mass is above the equilibrium point at
y=A
and is about to be released.Position B: The mass is in downward motion as it passes through the equilibrium point.Position C: The
mass is momentarily at rest at the lowest point before starting on its upward motion.Position D: The mass is in upward motion as it
passes through the equilibrium point.Position E: The mass is momentarily at rest at the highest point before starting back down again.
By noting the time when the negative maximum, positive maximum, and zero values occur for the oscillating object's position, velocity
and acceleration, you can graph the sine (or cosine) function. This is done for the case of the oscillating spring-mass system in the
table below and the three functions are shown in Figure 3. Note that the positive direction is typically chosen to be the direction that the
spring is stretched. Therefore, the positive direction in this case is down and the initial position A in Figure 2 is actually a negative value.
The most difficult parameter to analyze is the acceleration. It helps to use Newton's second law, which tells us that a negative maximum
acceleration occurs when the net force is negative maximum, a positive maximum acceleration occurs when the net force is positive
maximum and the acceleration is zero when the net force is zero.
Figure 3: Position, velocity and acceleration vs. time
For this particular initial condition (starting position at A in Figure 2), the position curve is a cosine function (actually a negative cosine
function), the velocity curve is a sine function, and the acceleration curve is just the negative of the position curve.
Mass and Spring
A mass suspended at the end of a spring will stretch the spring by some distance y. The force with which the spring pulls upward on the
mass is given by Hooke's law
(6)
F = −ky
where k is the spring constant and y is the stretch in the spring when a force F is applied to the spring. The spring constant k is a
measure of the stiffness of the spring.The spring constant can be determined experimentally by allowing the mass to hang
motionless on the spring and then adding additional mass and recording the additional spring stretch as shown below. In Figure
4a, the weight hanger is suspended from the end of the spring. In Figure 4b, an additional mass has been added to the hanger and
the spring is now extended by an amount
Δy.
This experimental set-up is also shown in the photograph of the apparatus in Figure 5.
Figure 4: Set up for determining spring constant
Figure 5: Photo of experimental set-up
When the mass is motionless, its acceleration is zero. According to Newton's second law, the net force must therefore be zero. There
are two forces acting on the mass: the downward gravitational force and the upward spring force. See the free-body diagram in Figure 6
below.
Figure 6: Free-body diagram for the spring-mass system
So Newton's second law gives us
(7)
Δmg − kΔy = 0
where
Δm
is the change in mass and
Δy
is the change in the stretch of the spring caused by the change in mass, g is the gravitational acceleration, and k is the spring constant.
Equation 7 can also be expressed as
(8)
Δm =
k
g
Δy.
Newton's second law applied to this system is
ma = F = −ky.
Substitute from equation 5 for the acceleration to get
(9)
m(−4π2f2y) = −ky
from which we get an expression for the frequency f and the period T.
( 10 )
f=
1
2π
k
m
( 11 )
T = 2π
m
k
Using equation 11, we can predict the period if we know the mass on the spring and the spring constant. Alternately, knowing the mass
on the spring and experimentally measuring the period, we can determine the spring constant of the spring.Notice that in equation 11,
the relationship between T and m is not linear. A graph of the period versus the mass will not be a straight line. If we square both sides
of equation 11, we get
( 12 )
T2 = 4π2
m
k
.
Now, a graph of
T2
versus m will be a straight line, and the spring constant can be determined from the slope.
Simple Pendulum
The other example of simple harmonic motion that you will investigate is the simple pendulum. The simple pendulum consists of a
mass m, called the pendulum bob, attached to the end of a string. The length L of the simple pendulum is measured from the point of
suspension of the string to the center of the bob as shown in Figure 7 below.
Figure 7: Experimental set-up for a simple pendulum
If the bob is moved away from the rest position through some angle of displacement θ as in Figure 8, the restoring force will return the
bob back to the equilibrium position. The forces acting on the bob are the force of gravity and the tension force of the string. The
tension force of the string is balanced by the component of the gravitational force that is in line with the string (i.e. perpendicular to the
motion of the bob). The restoring force here is the tangential component of the gravitational force.
Figure 8: Simple pendulum
When we apply trigonometry to the smaller triangle in Figure 8, we get the magnitude of the restoring force |F| = mg sin θ. This force
depends on the mass of the bob, the acceleration due to gravity g, and the sine of the angle through which the string has been pulled.
Again Newton's second law must apply, so
( 13 )
ma = F = −mg sin θ
where the negative sign implies that the restoring force acts opposite to the direction of motion of the bob.Since the bob is moving
along the arc of a circle, the angular acceleration is given by
α = a/L.
From equation 13 we get
( 14 )
α=−
g
L
sin θ.
In Figure 9, the blue solid line is a plot of sin(θ) versus θ, and the straight line is a plot of θ in degrees versus θ in radians. For small
angles, these two curves are almost indistinguishable. Therefore, as long as the displacement θ is small, we can use the
approximation sin θ ≈ θ.
Figure 9: Graphs of sin θ versus θ
With this approximation, equation 14 becomes
( 15 )
α=−
g
L
θ.
Equation 15 shows the (angular) acceleration to be proportional to the negative of the (angular) displacement, and therefore the motion
of the bob is simple harmonic and we can apply equation 5 to get
( 16 )
α = −4π2f2 θ.
Combining equation 15 and equation 16 and simplifying, we get
( 17 )
f=
1
2π
g
L
and
( 18 )
T = 2π
L
g
.
Note that the frequency and period of the simple pendulum do not depend on the mass.
https://www.webassign.net/question_assets/ncsucalcphysmechl3/lab_7_1/manual.html
When you pluck a guitar string, the resulting sound has a steady tone and lasts a long time
((Figure)). The string vibrates around an equilibrium position, and one oscillation is completed when
the string starts from the initial position, travels to one of the extreme positions, then to the other
extreme position, and returns to its initial position. We define periodic motion to be any motion that
repeats itself at regular time intervals, such as exhibited by the guitar string or by a child swinging on
a swing. In this section, we study the basic characteristics of oscillations and their mathematical
description.
Figure 15.2 When a guitar string is plucked, the string oscillates up and down in periodic motion. The vibrating string causes the
surrounding air molecules to oscillate, producing sound waves. (credit: Yutaka Tsutano)
Period and Frequency in Oscillations
In the absence of friction, the time to complete one oscillation remains constant and is called
the period (T). Its units are usually seconds, but may be any convenient unit of time. The word
‘period’ refers to the time for some event whether repetitive or not, but in this chapter, we shall deal
primarily in periodic motion, which is by definition repetitive.
A concept closely related to period is the frequency of an event. Frequency (f) is defined to be the
number of events per unit time. For periodic motion, frequency is the number of oscillations per unit
time. The relationship between frequency and period is
f=1T.f=1T.
The SI unit for frequency is the hertz (Hz) and is defined as one cycle per second:
1Hz=1cyclesecor1Hz=1s=1s−1.1Hz=1cyclesecor1Hz=1s=1s−1.
A cycle is one complete oscillation.
EXAMPLE
Characteristics of Simple Harmonic Motion
A very common type of periodic motion is called simple harmonic motion (SHM). A system that
oscillates with SHM is called a simple harmonic oscillator.
Simple Harmonic Motion
In simple harmonic motion, the acceleration of the system, and therefore the net force, is proportional to the
displacement and acts in the opposite direction of the displacement.
A good example of SHM is an object with mass m attached to a spring on a frictionless surface, as
shown in (Figure). The object oscillates around the equilibrium position, and the net force on the
object is equal to the force provided by the spring. This force obeys Hooke’s law Fs=−kx,Fs=−kx, as
discussed in a previous chapter.
If the net force can be described by Hooke’s law and there is no damping (slowing down due to
friction or other nonconservative forces), then a simple harmonic oscillator oscillates with equal
displacement on either side of the equilibrium position, as shown for an object on a spring in (Figure).
The maximum displacement from equilibrium is called the amplitude (A). The units for amplitude and
displacement are the same but depend on the type of oscillation. For the object on the spring, the
units of amplitude and displacement are meters.
Figure 15.3 An object attached to a spring sliding on a frictionless surface is an uncomplicated simple harmonic oscillator. In the
above set of figures, a mass is attached to a spring and placed on a frictionless table. The other end of the spring is attached to
the wall. The position of the mass, when the spring is neither stretched nor compressed, is marked as x=0x=0 and is the
equilibrium position. (a) The mass is displaced to a position x=Ax=A and released from rest. (b) The mass accelerates as it
moves in the negative x-direction, reaching a maximum negative velocity at x=0x=0. (c) The mass continues to move in the
negative x-direction, slowing until it comes to a stop at x=−Ax=−A. (d) The mass now begins to accelerate in the positive xdirection, reaching a positive maximum velocity at x=0x=0. (e) The mass then continues to move in the positive direction until it
stops at x=Ax=A. The mass continues in SHM that has an amplitude A and a period T. The object’s maximum speed occurs as
it passes through equilibrium. The stiffer the spring is, the smaller the period T. The greater the mass of the object is, the greater
the period T.
What is so significant about SHM? For one thing, the period T and frequency f of a simple harmonic
oscillator are independent of amplitude. The string of a guitar, for example, oscillates with the same
frequency whether plucked gently or hard.
Two important factors do affect the period of a simple harmonic oscillator. The period is related to
how stiff the system is. A very stiff object has a large force constant (k), which causes the system to
have a smaller period. For example, you can adjust a diving board’s stiffness—the stiffer it is, the
faster it vibrates, and the shorter its period. Period also depends on the mass of the oscillating
system. The more massive the system is, the longer the period. For example, a heavy person on a
diving board bounces up and down more slowly than a light one. In fact, the mass m and the force
constant k are the only factors that affect the period and frequency of SHM. To derive an equation for
the period and the frequency, we must first define and analyze the equations of motion. Note that the
force constant is sometimes referred to as the spring constant.
Equations of SHM
Consider a block attached to a spring on a frictionless table ((Figure)). The equilibrium position (the
position where the spring is neither stretched nor compressed) is marked as x=0x=0. At the
equilibrium position, the net force is zero.
Figure 15.4 A block is attached to a spring and placed on a frictionless table. The equilibrium position, where the spring is
neither extended nor compressed, is marked as x=0.x=0.
Work is done on the block to pull it out to a position of x=+A,x=+A, and it is then released from rest.
The maximum x-position (A) is called the amplitude of the motion. The block begins to oscillate in
SHM between x=+Ax=+A and x=−A,x=−A, where A is the amplitude of the motion and T is the
period of the oscillation. The period is the time for one oscillation. (Figure) shows the motion of the
block as it completes one and a half oscillations after release. (Figure) shows a plot of the position of
the block versus time. When the position is plotted versus time, it is clear that the data can be
modeled by a cosine function with an amplitude A and a period T. The cosine
function cosθcosθ repeats every multiple of 2π,2π, whereas the motion of the block repeats every
period T. However, the function cos(2πTt)cos(2πTt)repeats every integer multiple of the period. The
maximum of the cosine function is one, so it is necessary to multiply the cosine function by the
amplitude A.
x(t)=Acos(2πTt)=Acos(ωt).x(t)=Acos(2πTt)=Acos(ωt).
Recall from the chapter on rotation that the angular frequency equals ω=dθdtω=dθdt. In this case, the
period is constant, so the angular frequency is defined as 2π2π divided by the period, ω=2πTω=2πT.
Figure 15.5 A block is attached to one end of a spring and placed on a frictionless table. The other end of the spring is anchored
to the wall. The equilibrium position, where the net force equals zero, is marked as x=0m.x=0m. Work is done on the block,
pulling it out to x=+Ax=+A, and the block is released from rest. The block oscillates between x=+Ax=+A and x=−Ax=−A. The
force is also shown as a vector.
Figure 15.6 A graph of the position of the block shown in (Figure) as a function of time. The position can be modeled as a
periodic function, such as a cosine or sine function.
The equation for the position as a function of time x(t)=Acos(ωt)x(t)=Acos(ωt) is good for modeling
data, where the position of the block at the initial time t=0.00st=0.00s is at the amplitude A and the
initial velocity is zero. Often when taking experimental data, the position of the mass at the initial
time t=0.00st=0.00s is not equal to the amplitude and the initial velocity is not zero. Consider 10
seconds of data collected by a student in lab, shown in (Figure).
Figure 15.7 Data collected by a student in lab indicate the position of a block attached to a spring, measured with a sonic range
finder.
The
data
are
collected
starting
at
time t=0.00s,t=0.00s, but
the
initial
position
is
near
position x≈−0.80cm≠3.00cmx≈−0.80cm≠3.00cm, so the initial position does not equal the amplitude x0=+Ax0=+A. The
velocity is the time derivative of the position, which is the slope at a point on the graph of position versus time. The velocity is
not v=0.00m/sv=0.00m/s at time t=0.00st=0.00s, as evident by the slope of the graph of position versus time, which is not zero
at the initial time.
The data in (Figure) can still be modeled with a periodic function, like a cosine function, but the
function is shifted to the right. This shift is known as a phase shift and is usually represented by the
Greek letter phi (φ)(φ). The equation of the position as a function of time for a block on a spring
becomes
x(t)=Acos(ωt+φ).x(t)=Acos(ωt+φ).
This is the generalized equation for SHM where t is the time measured in seconds, ωω is the angular
frequency with units of inverse seconds, A is the amplitude measured in meters or centimeters,
and φφ is the phase shift measured in radians ((Figure)). It should be noted that because sine and
cosine functions differ only by a phase shift, this motion could be modeled using either the cosine or
sine function.
Figure 15.9 (a) A cosine function. (b) A cosine function shifted to the right by an angle φφ. The angle φφ is known as the phase
shift of the function.
The velocity of the mass on a spring, oscillating in SHM, can be found by taking the derivative of the
position equation:
v(t)=dxdt=ddt(Acos(ωt+φ))=−Aωsin(ωt+ϕ)=−vmaxsin(ωt+φ).v(t)=dxdt=ddt(Acos(ωt+φ))=−Aωsin(ωt+
ϕ)=−vmaxsin(ωt+φ).
Because the sine function oscillates between –1 and +1, the maximum velocity is the amplitude times
the angular frequency, vmax=Aωvmax=Aω. The maximum velocity occurs at the equilibrium
position (x=0)(x=0) when the mass is moving toward x=+Ax=+A. The maximum velocity in the
negative direction is attained at the equilibrium position (x=0)(x=0) when the mass is moving
toward x=−Ax=−A and is equal to −vmax−vmax.
The acceleration of the mass on the spring can be found by taking the time derivative of the velocity:
a(t)=dvdt=ddt(−Aωsin(ωt+φ))=−Aω2cos(ωt+ϕ)=−amaxcos(ωt+φ).a(t)=dvdt=ddt(−Aωsin(ωt+φ))=−Aω2
cos(ωt+ϕ)=−amaxcos(ωt+φ).
The maximum acceleration is amax=Aω2amax=Aω2. The maximum acceleration occurs at the
position(x=−A)(x=−A), and the acceleration at the position (x=−A)(x=−A) and is equal
to −amax−amax.
Summary of Equations of Motion for SHM
In summary, the oscillatory motion of a block on a spring can be modeled with the following equations
of motion:
x(t)=Acos(ωt+φ)x(t)=Acos(ωt+φ)
v(t)=−vmaxsin(ωt+φ)v(t)=−vmaxsin(ωt+φ)
a(t)=−amaxcos(ωt+φ)a(t)=−amaxcos(ωt+φ)
xmax=Axmax=A
vmax=Aωvmax=Aω
amax=Aω2.amax=Aω2.
Here, A is the amplitude of the motion, T is the period, φφ is the phase shift,
and ω=2πT=2πfω=2πT=2πf is the angular frequency of the motion of the block.
Vertical Motion and a Horizontal Spring
When a spring is hung vertically and a block is attached and set in motion, the block oscillates in
SHM. In this case, there is no normal force, and the net effect of the force of gravity is to change the
equilibrium position. Consider (Figure). Two forces act on the block: the weight and the force of the
spring. The weight is constant and the force of the spring changes as the length of the spring
changes.
Figure 15.9 A spring is hung from the ceiling. When a block is attached, the block is at the equilibrium position where the weight
of the block is equal to the force of the spring. (a) The spring is hung from the ceiling and the equilibrium position is marked
as yoyo. (b) A mass is attached to the spring and a new equilibrium position is reached ( y1=yo−Δyy1=yo−Δy) when the force
provided by the spring equals the weight of the mass. (c) The free-body diagram of the mass shows the two forces acting on the
mass: the weight and the force of the spring.
When the block reaches the equilibrium position, as seen in (Figure), the force of the spring equals
the weight of the block, Fnet=Fs−mg=0Fnet=Fs−mg=0, where
−k(−Δy)=mg.−k(−Δy)=mg.
From the figure, the change in the position is Δy=y0−y1Δy=y0−y1 and
since −k(−Δy)=mg−k(−Δy)=mg, we have
k(y0−y1)−mg=0.k(y0−y1)−mg=0.
If the block is displaced and released, it will oscillate around the new equilibrium position. As shown
in (Figure), if the position of the block is recorded as a function of time, the recording is a periodic
function.
If the block is displaced to a position y, the net force
becomes Fnet=k(y−y0)−mg=0Fnet=k(y−y0)−mg=0. But we found that at the equilibrium
position, mg=kΔy=ky0−ky1mg=kΔy=ky0−ky1. Substituting for the weight in the equation yields
Fnet=ky−ky0−(ky0−ky1)=−k(y−y1).Fnet=ky−ky0−(ky0−ky1)=−k(y−y1).
Recall that y1y1 is just the equilibrium position and any position can be set to be the
point y=0.00m.y=0.00m. So let’s set y1y1 to y=0.00m.y=0.00m. The net force then becomes
Fnet=−ky;md2ydt2=−ky.Fnet=−ky;md2ydt2=−ky.
This is just what we found previously for a horizontally sliding mass on a spring. The constant force of
gravity only served to shift the equilibrium location of the mass. Therefore, the solution should be the
same form as for a block on a horizontal spring, y(t)=Acos(ωt+φ).y(t)=Acos(ωt+φ). The equations
for the velocity and the acceleration also have the same form as for the horizontal case. Note that the
inclusion of the phase shift means that the motion can actually be modeled using either a cosine or a
sine function, since these two functions only differ by a phase shift.
Figure 15.10 Graphs of y(t), v(t), and a(t) versus t for the motion of an object on a vertical spring. The net force on the object
can be described by Hooke’s law, so the object undergoes SHM. Note that the initial position has the vertical displacement at its
maximum value A; v is initially zero and then negative as the object moves down; the initial acceleration is negative, back toward
the equilibrium position and becomes zero at that point.
SUMMARY

Periodic motion is a repeating oscillation. The time for one oscillation is the period T and the
number of oscillations per unit time is the frequency f. These quantities are related by f=1Tf=1T.

Simple harmonic motion (SHM) is oscillatory motion for a system where the restoring force is
proportional to the displacement and acts in the direction opposite to the displacement.

Maximum displacement is the amplitude A. The angular frequency ωω, period T, and
frequency f of a simple harmonic oscillator are given
by ω=√kmω=km, T=2π√mk,andf=12π√kmT=2πmk,andf=12πkm, where m is the mass of the
system and k is the force constant.
Displacement as a function of time in SHM is given
byx(t)=Acos(2πTt+φ)=Acos(ωt+φ)x(t)=Acos(2πTt+φ)=Acos(ωt+φ).
The velocity is given
by v(t)=−Aωsin(ωt+φ)=−vmaxsin(ωt+φ),wherevmax=Aω=A√kmv(t)=−Aωsin(ωt+φ)=−vmaxsin(ω
t+φ),wherevmax=Aω=Akm.
The acceleration
is a(t)=−Aω2cos(ωt+φ)=−amaxcos(ωt+φ)a(t)=−Aω2cos(ωt+φ)=−amaxcos(ωt+φ),
where amax=Aω2=Akmamax=Aω2=Akm.



Conceptual Questions
What conditions must be met to produce SHM?
Show Solution
(a) If frequency is not constant for some oscillation, can the oscillation be SHM? (b) Can you think of any
examples of harmonic motion where the frequency may depend on the amplitude?
Give an example of a simple harmonic oscillator, specifically noting how its frequency is independent of
amplitude.
Show Solution
Explain why you expect an object made of a stiff material to vibrate at a higher frequency than a similar object
made of a more pliable material.
As you pass a freight truck with a trailer on a highway, you notice that its trailer is bouncing up and down
slowly. Is it more likely that the trailer is heavily loaded or nearly empty? Explain your answer.
Show Solution
Some people modify cars to be much closer to the ground than when manufactured. Should they install stiffer
springs? Explain your answer.
PROBLEMS
Prove that using x(t)=Asin(ωt+φ)x(t)=Asin(ωt+φ) will produce the same results for the period for the oscillations of a
mass and a spring. Why do you think the cosine function was chosen?
Show Solution
What is the period of 60.0 Hz of electrical power?
If your heart rate is 150 beats per minute during strenuous exercise, what is the time per beat in units of seconds?
Show Solution
Find the frequency of a tuning fork that takes
2.50×10−3s2.50×10−3s to complete one oscillation.
A stroboscope is set to flash every 8.00×10−5s8.00×10−5s. What is the frequency of the flashes?
Show Solution
A tire has a tread pattern with a crevice every 2.00 cm. Each crevice makes a single vibration as the tire moves. What is the
frequency of these vibrations if the car moves at 30.0 m/s?
Each piston of an engine makes a sharp sound every other revolution of the engine. (a) How fast is a race car going if its eightcylinder engine emits a sound of frequency 750 Hz, given that the engine makes 2000 revolutions per kilometer? (b) At how
many revolutions per minute is the engine rotating?
Show Solution
A type of cuckoo clock keeps time by having a mass bouncing on a spring, usually something cute like a cherub in a chair. What
force constant is needed to produce a period of 0.500 s for a 0.0150-kg mass?
A mass m0m0 is attached to a spring and hung vertically. The mass is raised a short distance in the vertical direction and
released. The mass oscillates with a frequency f0f0. If the mass is replaced with a mass nine times as large, and the
experiment was repeated, what would be the frequency of the oscillations in terms of f0f0?
Show Solution
A 0.500-kg mass suspended from a spring oscillates with a period of 1.50 s. How much mass must be added to the object to
change the period to 2.00 s?
By how much leeway (both percentage and mass) would you have in the selection of the mass of the object in the previous
problem if you did not wish the new period to be greater than 2.01 s or less than 1.99 s?
Show Solution
GLOSSARY
amplitude (A)
maximum displacement from the equilibrium position of an object oscillating around the equilibrium
position
equilibrium position
position where the spring is neither stretched nor compressed
force constant (k)
characteristic of a spring which is defined as the ratio of the force applied to the spring to the
displacement caused by the force
frequency (f)
number of events per unit of time
oscillation
single fluctuation of a quantity, or repeated and regular fluctuations of a quantity, between two extreme
values around an equilibrium or average value
periodic motion
motion that repeats itself at regular time intervals
period (T)
time taken to complete one oscillation
phase shift
angle, in radians, that is used in a cosine or sine function to shift the function left or right, used to match
up the function with the initial conditions of data
simple harmonic motion (SHM)
oscillatory motion in a system where the restoring force is proportional to the displacement, which acts
in the direction opposite to the displacement
simple harmonic oscillator
a device that oscillates in SHM where the restoring force is proportional to the displacement and acts in
the direction opposite to the displacement
When you pluck a guitar string, the resulting sound has a steady tone and lasts a long time
((Figure)). The string vibrates around an equilibrium position, and one oscillation is completed when
the string starts from the initial position, travels to one of the extreme positions, then to the other
extreme position, and returns to its initial position. We define periodic motion to be any motion that
repeats itself at regular time intervals, such as exhibited by the guitar string or by a child swinging on
a swing. In this section, we study the basic characteristics of oscillations and their mathematical
description.
Figure 15.2 When a guitar string is plucked, the string oscillates up and down in periodic motion. The vibrating string causes the
surrounding air molecules to oscillate, producing sound waves. (credit: Yutaka Tsutano)
Period and Frequency in Oscillations
In the absence of friction, the time to complete one oscillation remains constant and is called
the period (T). Its units are usually seconds, but may be any convenient unit of time. The word
‘period’ refers to the time for some event whether repetitive or not, but in this chapter, we shall deal
primarily in periodic motion, which is by definition repetitive.
A concept closely related to period is the frequency of an event. Frequency (f) is defined to be the
number of events per unit time. For periodic motion, frequency is the number of oscillations per unit
time. The relationship between frequency and period is
f=1T.f=1T.
The SI unit for frequency is the hertz (Hz) and is defined as one cycle per second:
1Hz=1cyclesecor1Hz=1s=1s−1.1Hz=1cyclesecor1Hz=1s=1s−1.
A cycle is one complete oscillation.
EXAMPLE
Determining the Frequency of Medical Ultrasound
Ultrasound machines are used by medical professionals to make images for examining internal organs of the
body. An ultrasound machine emits high-frequency sound waves, which reflect off the organs, and a computer
receives the waves, using them to create a picture. We can use the formulas presented in this module to
determine the frequency, based on what we know about oscillations. Consider a medical imaging device that
produces ultrasound by oscillating with a period of 0.400μs0.400μs. What is the frequency of this oscillation?
Strategy
The period (T) is given and we are asked to find frequency (f).
Solution
Substitute 0.400μs0.400μs for T in f=1Tf=1T:
f=1T=10.400×10−6s.f=1T=10.400×10−6s.
Solve to find
f=2.50×106Hz.f=2.50×106Hz.
Significance
This frequency of sound is much higher than the highest frequency that humans can hear (the range of human
hearing is 20 Hz to 20,000 Hz); therefore, it is called ultrasound. Appropriate oscillations at this frequency
generate ultrasound used for noninvasive medical diagnoses, such as observations of a fetus in the womb.
Characteristics of Simple Harmonic Motion
A very common type of periodic motion is called simple harmonic motion (SHM). A system that
oscillates with SHM is called a simple harmonic oscillator.
Simple Harmonic Motion
In simple harmonic motion, the acceleration of the system, and therefore the net force, is proportional to the
displacement and acts in the opposite direction of the displacement.
A good example of SHM is an object with mass m attached to a spring on a frictionless surface, as
shown in (Figure). The object oscillates around the equilibrium position, and the net force on the
object is equal to the force provided by the spring. This force obeys Hooke’s law Fs=−kx,Fs=−kx, as
discussed in a previous chapter.
If the net force can be described by Hooke’s law and there is no damping (slowing down due to
friction or other nonconservative forces), then a simple harmonic oscillator oscillates with equal
displacement on either side of the equilibrium position, as shown for an object on a spring in (Figure).
The maximum displacement from equilibrium is called the amplitude (A). The units for amplitude and
displacement are the same but depend on the type of oscillation. For the object on the spring, the
units of amplitude and displacement are meters.
Figure 15.3 An object attached to a spring sliding on a frictionless surface is an uncomplicated simple harmonic oscillator. In the
above set of figures, a mass is attached to a spring and placed on a frictionless table. The other end of the spring is attached to
the wall. The position of the mass, when the spring is neither stretched nor compressed, is marked as x=0x=0 and is the
equilibrium position. (a) The mass is displaced to a position x=Ax=A and released from rest. (b) The mass accelerates as it
moves in the negative x-direction, reaching a maximum negative velocity at x=0x=0. (c) The mass continues to move in the
negative x-direction, slowing until it comes to a stop at x=−Ax=−A. (d) The mass now begins to accelerate in the positive xdirection, reaching a positive maximum velocity at x=0x=0. (e) The mass then continues to move in the positive direction until it
stops at x=Ax=A. The mass continues in SHM that has an amplitude A and a period T. The object’s maximum speed occurs as
it passes through equilibrium. The stiffer the spring is, the smaller the period T. The greater the mass of the object is, the greater
the period T.
What is so significant about SHM? For one thing, the period T and frequency f of a simple harmonic
oscillator are independent of amplitude. The string of a guitar, for example, oscillates with the same
frequency whether plucked gently or hard.
Two important factors do affect the period of a simple harmonic oscillator. The period is related to
how stiff the system is. A very stiff object has a large force constant (k), which causes the system to
have a smaller period. For example, you can adjust a diving board’s stiffness—the stiffer it is, the
faster it vibrates, and the shorter its period. Period also depends on the mass of the oscillating
system. The more massive the system is, the longer the period. For example, a heavy person on a
diving board bounces up and down more slowly than a light one. In fact, the mass m and the force
constant k are the only factors that affect the period and frequency of SHM. To derive an equation for
the period and the frequency, we must first define and analyze the equations of motion. Note that the
force constant is sometimes referred to as the spring constant.
Equations of SHM
Consider a block attached to a spring on a frictionless table ((Figure)). The equilibrium position (the
position where the spring is neither stretched nor compressed) is marked as x=0x=0. At the
equilibrium position, the net force is zero.
Figure 15.4 A block is attached to a spring and placed on a frictionless table. The equilibrium position, where the spring is
neither extended nor compressed, is marked as x=0.x=0.
Work is done on the block to pull it out to a position of x=+A,x=+A, and it is then released from rest.
The maximum x-position (A) is called the amplitude of the motion. The block begins to oscillate in
SHM between x=+Ax=+A and x=−A,x=−A, where A is the amplitude of the motion and T is the
period of the oscillation. The period is the time for one oscillation. (Figure) shows the motion of the
block as it completes one and a half oscillations after release. (Figure) shows a plot of the position of
the block versus time. When the position is plotted versus time, it is clear that the data can be
modeled by a cosine function with an amplitude A and a period T. The cosine
function cosθcosθ repeats every multiple of 2π,2π, whereas the motion of the block repeats every
period T. However, the function cos(2πTt)cos(2πTt)repeats every integer multiple of the period. The
maximum of the cosine function is one, so it is necessary to multiply the cosine function by the
amplitude A.
x(t)=Acos(2πTt)=Acos(ωt).x(t)=Acos(2πTt)=Acos(ωt).
Recall from the chapter on rotation that the angular frequency equals ω=dθdtω=dθdt. In this case, the
period is constant, so the angular frequency is defined as 2π2π divided by the period, ω=2πTω=2πT.
Figure 15.5 A block is attached to one end of a spring and placed on a frictionless table. The other end of the spring is anchored
to the wall. The equilibrium position, where the net force equals zero, is marked as x=0m.x=0m. Work is done on the block,
pulling it out to x=+Ax=+A, and the block is released from rest. The block oscillates between x=+Ax=+A and x=−Ax=−A. The
force is also shown as a vector.
Figure 15.6 A graph of the position of the block shown in (Figure) as a function of time. The position can be modeled as a
periodic function, such as a cosine or sine function.
The equation for the position as a function of time x(t)=Acos(ωt)x(t)=Acos(ωt) is good for modeling
data, where the position of the block at the initial time t=0.00st=0.00s is at the amplitude A and the
initial velocity is zero. Often when taking experimental data, the position of the mass at the initial
time t=0.00st=0.00s is not equal to the amplitude and the initial velocity is not zero. Consider 10
seconds of data collected by a student in lab, shown in (Figure).
Figure 15.7 Data collected by a student in lab indicate the position of a block attached to a spring, measured with a sonic range
finder.
The
data
are
collected
starting
at
time t=0.00s,t=0.00s, but
the
initial
position
is
near
position x≈−0.80cm≠3.00cmx≈−0.80cm≠3.00cm, so the initial position does not equal the amplitude x0=+Ax0=+A. The
velocity is the time derivative of the position, which is the slope at a point on the graph of position versus time. The velocity is
not v=0.00m/sv=0.00m/s at time t=0.00st=0.00s, as evident by the slope of the graph of position versus time, which is not zero
at the initial time.
The data in (Figure) can still be modeled with a periodic function, like a cosine function, but the
function is shifted to the right. This shift is known as a phase shift and is usually represented by the
Greek letter phi (φ)(φ). The equation of the position as a function of time for a block on a spring
becomes
x(t)=Acos(ωt+φ).x(t)=Acos(ωt+φ).
This is the generalized equation for SHM where t is the time measured in seconds, ωω is the angular
frequency with units of inverse seconds, A is the amplitude measured in meters or centimeters,
and φφ is the phase shift measured in radians ((Figure)). It should be noted that because sine and
cosine functions differ only by a phase shift, this motion could be modeled using either the cosine or
sine function.
Figure 15.9 (a) A cosine function. (b) A cosine function shifted to the right by an angle φφ. The angle φφ is known as the phase
shift of the function.
The velocity of the mass on a spring, oscillating in SHM, can be found by taking the derivative of the
position equation:
v(t)=dxdt=ddt(Acos(ωt+φ))=−Aωsin(ωt+ϕ)=−vmaxsin(ωt+φ).v(t)=dxdt=ddt(Acos(ωt+φ))=−Aωsin(ωt+
ϕ)=−vmaxsin(ωt+φ).
Because the sine function oscillates between –1 and +1, the maximum velocity is the amplitude times
the angular frequency, vmax=Aωvmax=Aω. The maximum velocity occurs at the equilibrium
position (x=0)(x=0) when the mass is moving toward x=+Ax=+A. The maximum velocity in the
negative direction is attained at the equilibrium position (x=0)(x=0) when the mass is moving
toward x=−Ax=−A and is equal to −vmax−vmax.
The acceleration of the mass on the spring can be found by taking the time derivative of the velocity:
a(t)=dvdt=ddt(−Aωsin(ωt+φ))=−Aω2cos(ωt+ϕ)=−amaxcos(ωt+φ).a(t)=dvdt=ddt(−Aωsin(ωt+φ))=−Aω2
cos(ωt+ϕ)=−amaxcos(ωt+φ).
The maximum acceleration is amax=Aω2amax=Aω2. The maximum acceleration occurs at the
position(x=−A)(x=−A), and the acceleration at the position (x=−A)(x=−A) and is equal
to −amax−amax.
Summary of Equations of Motion for SHM
In summary, the oscillatory motion of a block on a spring can be modeled with the following equations
of motion:
x(t)=Acos(ωt+φ)x(t)=Acos(ωt+φ)
v(t)=−vmaxsin(ωt+φ)v(t)=−vmaxsin(ωt+φ)
a(t)=−amaxcos(ωt+φ)a(t)=−amaxcos(ωt+φ)
xmax=Axmax=A
vmax=Aωvmax=Aω
amax=Aω2.amax=Aω2.
Here, A is the amplitude of the motion, T is the period, φφ is the phase shift,
and ω=2πT=2πfω=2πT=2πf is the angular frequency of the motion of the block.
EXAMPLE
Determining the Equations of Motion for a Block and a Spring
A 2.00-kg block is placed on a frictionless surface. A spring with a force constant
of k=32.00N/mk=32.00N/m is attached to the block, and the opposite end of the spring is attached to the
wall. The spring can be compressed or extended. The equilibrium position is marked as x=0.00m.x=0.00m.
Work is done on the block, pulling it out to x=+0.02m.x=+0.02m. The block is released from rest and
oscillates between x=+0.02mx=+0.02mand x=−0.02m.x=−0.02m. The period of the motion is 1.57 s.
Determine the equations of motion.
Strategy
We first find the angular frequency. The phase shift is zero, φ=0.00rad,φ=0.00rad, because the block is
released from rest at x=A=+0.02m.x=A=+0.02m. Once the angular frequency is found, we can determine
the maximum velocity and maximum acceleration.
Solution
The angular frequency can be found and used to find the maximum velocity and maximum acceleration:
ω=2π1.57s=4.00s−1;vmax=Aω=0.02m(4.00s−1)=0.08m/s;amax=Aω2=0.02m(4.00s−1)2=0.32m/s2.ω=2π1.57s=4.00s
−1;vmax=Aω=0.02m(4.00s−1)=0.08m/s;amax=Aω2=0.02m(4.00s−1)2=0.32m/s2.
All that is left is to fill in the equations of motion:
x(t)=Acos(ωt+φ)=(0.02m)cos(4.00s−1t);v(t)=−vmaxsin(ωt+φ)=(−0.08m/s)sin(4.00s−1t);a(t)=−amaxcos(ωt+φ)=(−0.
32m/s2)cos(4.00s−1t).x(t)=Acos(ωt+φ)=(0.02m)cos(4.00s−1t);v(t)=−vmaxsin(ωt+φ)=(−0.08m/s)sin(4.00s
−1t);a(t)=−amaxcos(ωt+φ)=(−0.32m/s2)cos(4.00s−1t).
Significance
The position, velocity, and acceleration can be found for any time. It is important to remember that when using
these equations, your calculator must be in radians mode.
The Period and Frequency of a Mass on a Spring
One interesting characteristic of the SHM of an object attached to a spring is that the angular
frequency, and therefore the period and frequency of the motion, depend on only the mass and the
force constant, and not on other factors such as the amplitude of the motion. We can use the
equations of motion and Newton’s second law (→Fnet=m→a)(F→net=ma→) to find equations for the
angular frequency, frequency, and period.
Consider the block on a spring on a frictionless surface. There are three forces on the mass: the
weight, the normal force, and the force due to the spring. The only two forces that act perpendicular
to the surface are the weight and the normal force, which have equal magnitudes and opposite
directions, and thus sum to zero. The only force that acts parallel to the surface is the force due to the
spring, so the net force must be equal to the force of the spring:
Fx=−kx;ma=−kx;md2xdt2=−kx;d2xdt2=−kmx.Fx=−kx;ma=−kx;md2xdt2=−kx;d2xdt2=−kmx.
Substituting the equations of motion for x and a gives us
−Aω2cos(ωt+φ)=−kmAcos(ωt+φ).−Aω2cos(ωt+φ)=−kmAcos(ωt+φ).
Cancelling out like terms and solving for the angular frequency yields
ω=√km.ω=km.
The angular frequency depends only on the force constant and the mass, and not the amplitude. The
angular frequency is defined as ω=2π/T,ω=2π/T, which yields an equation for the period of the
motion:
T=2π√mk.T=2πmk.
The period also depends only on the mass and the force constant. The greater the mass, the longer
the period. The stiffer the spring, the shorter the period. The frequency is
f=1T=12π√km.f=1T=12πkm.
Vertical Motion and a Horizontal Spring
When a spring is hung vertically and a block is attached and set in motion, the block oscillates in
SHM. In this case, there is no normal force, and the net effect of the force of gravity is to change the
equilibrium position. Consider (Figure). Two forces act on the block: the weight and the force of the
spring. The weight is constant and the force of the spring changes as the length of the spring
changes.
Figure 15.9 A spring is hung from the ceiling. When a block is attached, the block is at the equilibrium position where the weight
of the block is equal to the force of the spring. (a) The spring is hung from the ceiling and the equilibrium position is marked
as yoyo. (b) A mass is attached to the spring and a new equilibrium position is reached ( y1=yo−Δyy1=yo−Δy) when the force
provided by the spring equals the weight of the mass. (c) The free-body diagram of the mass shows the two forces acting on the
mass: the weight and the force of the spring.
When the block reaches the equilibrium position, as seen in (Figure), the force of the spring equals
the weight of the block, Fnet=Fs−mg=0Fnet=Fs−mg=0, where
−k(−Δy)=mg.−k(−Δy)=mg.
From the figure, the change in the position is Δy=y0−y1Δy=y0−y1 and
since −k(−Δy)=mg−k(−Δy)=mg, we have
k(y0−y1)−mg=0.k(y0−y1)−mg=0.
If the block is displaced and released, it will oscillate around the new equilibrium position. As shown
in (Figure), if the position of the block is recorded as a function of time, the recording is a periodic
function.
If the block is displaced to a position y, the net force
becomes Fnet=k(y−y0)−mg=0Fnet=k(y−y0)−mg=0. But we found that at the equilibrium
position, mg=kΔy=ky0−ky1mg=kΔy=ky0−ky1. Substituting for the weight in the equation yields
Fnet=ky−ky0−(ky0−ky1)=−k(y−y1).Fnet=ky−ky0−(ky0−ky1)=−k(y−y1).
Recall that y1y1 is just the equilibrium position and any position can be set to be the
point y=0.00m.y=0.00m. So let’s set y1y1 to y=0.00m.y=0.00m. The net force then becomes
Fnet=−ky;md2ydt2=−ky.Fnet=−ky;md2ydt2=−ky.
This is just what we found previously for a horizontally sliding mass on a spring. The constant force of
gravity only served to shift the equilibrium location of the mass. Therefore, the solution should be the
same form as for a block on a horizontal spring, y(t)=Acos(ωt+φ).y(t)=Acos(ωt+φ). The equations
for the velocity and the acceleration also have the same form as for the horizontal case. Note that the
inclusion of the phase shift means that the motion can actually be modeled using either a cosine or a
sine function, since these two functions only differ by a phase shift.
Figure 15.10 Graphs of y(t), v(t), and a(t) versus t for the motion of an object on a vertical spring. The net force on the object
can be described by Hooke’s law, so the object undergoes SHM. Note that the initial position has the vertical displacement at its
maximum value A; v is initially zero and then negative as the object moves down; the initial acceleration is negative, back toward
the equilibrium position and becomes zero at that point.
SUMMARY

Periodic motion is a repeating oscillation. The time for one oscillation is the period T and the
number of oscillations per unit time is the frequency f. These quantities are related by f=1Tf=1T.

Simple harmonic motion (SHM) is oscillatory motion for a system where the restoring force is
proportional to the displacement and acts in the direction opposite to the displacement.

Maximum displacement is the amplitude A. The angular frequency ωω, period T, and
frequency f of a simple harmonic oscillator are given
by ω=√kmω=km, T=2π√mk,andf=12π√kmT=2πmk,andf=12πkm, where m is the mass of the
system and k is the force constant.
Displacement as a function of time in SHM is given
byx(t)=Acos(2πTt+φ)=Acos(ωt+φ)x(t)=Acos(2πTt+φ)=Acos(ωt+φ).
The velocity is given
by v(t)=−Aωsin(ωt+φ)=−vmaxsin(ωt+φ),wherevmax=Aω=A√kmv(t)=−Aωsin(ωt+φ)=−vmaxsin(ω
t+φ),wherevmax=Aω=Akm.
The acceleration
is a(t)=−Aω2cos(ωt+φ)=−amaxcos(ωt+φ)a(t)=−Aω2cos(ωt+φ)=−amaxcos(ωt+φ),
where amax=Aω2=Akmamax=Aω2=Akm.



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