1

CHAPTER 8

VIBRATIONS AND WAVES

8.1 HOOKE’S LAW

-The motion that repeat itself over and over is referred as periodic motion.

-The motion of an oscillating object depends on the restoring forces that make it

go back and forth.

Eg: the beating of your heart, the ticking of a clock, the movement of a child on a

swing.

-If the spring is stretched or compressed a small distance, x, from its unstretched

(equilibrium) position, and then released, it exerts a force on the mass.

The spring force ( or restoring force),

kxFS Hooke's law (spring-mass system) …(8.1)

where k = spring constant,

2

x = displacement of the mass from its unstretched (not too large)

(-) sign: the force exerted by the spring is always directed opposite the

displacement of the mass.

-The value of k is a measure of the stiffness of the spring.

-Stiff spring large k

-Soft spring small k

-A particular type of periodic motion is Simple Harmonic Motion (SHM).

-Simple harmonic motion occurs when the net force along the direction of motion is a

Hooke’s law type of force-that is, when the net force is linearly proportional to the

magnitude of the system's displacement from equilibrium and in the opposite direction.

Eg: A clock pendulum moving back & forth

SHM is described by the following parameters:

Amplitude (A) : the magnitude of the maximum displacement (+A or-A) of a mass

from its equilibrium position,

Period (T) : the time needed to complete one cycle of oscillation.

SI unit: second (s)

Frequency (f) : the number of cycles (oscillations) per second

SI unit : Hertz (Hz) or 1/s or cycle per second (cycle/s)

-How to find acceleration???

-By combining Newton's 2nd law with Hooke's law :

3

-kx = ma

xm

k

m

Fa

…(8.2)

8.2 ELASTIC POTENTIAL ENERGY

-Mechanical energy – 4 types:

1. Gravitational Potential Energy

2. Translational Kinetic Energy

3. Rotational Kinetic Energy

4. Elastic Potential Energy

-An object has potential energy by virtue of its shape or position.

-Energy is stored in a spring only when it is either stretched or compressed.

-The energy stored in a stretched or compressed spring or other elastic material is called

elastic potential energy (PE).

PE of a spring of the system :

2

2

1kxPEs …(8.3)

KE of a spring :

2

2

1mKE

…(8.4)

-The elastic potential energy is stored in a spring is zero when the spring is unstretched or

uncompressed (x=0)

-The elastic potential energy is a maximum when a spring has reached its maximum

compression or extension.

-PE is always positive because it proportional to x2.

4

Figure shows the energy transfers as the spring-mass system oscillates

-In SHM, the total energy is proportional to the square of the amplitude of the motion.

-From conservation of mechanical energy

fi EE …(8.5)

fsgisg PEPEKEPEPEKE …(8.6)

-If nonconservative forces (eg friction) are present, the final mechanical energy dos not

equal the initial mechanical energy.

From work-energy theorem:

isgfsgnc PEPEKEPEPEKEW …(8.7)

Because the spring force is conservative and since there are no external forces that can

do work on the system, the total mechanical energy of the system consisting the block and

spring remains constant.

-A x=0 x1 +A E

nerg

y

PE=1/2kx2

KE=1/2 mv2

KEmax=E KE

E=1/2 kA2

5

Example:

A 0.50-kg object is attached to a spring of spring constant 20N/m along a horizontal,

frictionless surface. The object oscillates in SHM and has a speed of 1.5m/s at the

equilibrium position.

a. What is the total energy of the system?

b. What is the amplitude

c. At what locations are the valued for the PE and KE the same?

Solution:

a. At x=0, PE=0, So total E=KE

JsmkgmKE 56.0)/5.1)(50.0(2

1

2

1 22

max

b. At x= A, v=0, so total E =PE

22

2

1

2

1kAkxPEE

6

mmN

J

k

EA 24.0

/20

)56.0(22

c. When KE = PE, E = KE+PE = PE + PE = 2 2

2

1kx = kx2

So mmN

J

k

Ex 17.0

/20

56.0

8.3 VELOCITY AS A FUNCTION OF POSITION

-Refer to Fig, as the mass moves toward the origin to some new position, x, part of this

energy is transformed into KE, and the PE stored in the spring is reduced to 1/2kx2.

Figure (a): A mass attached to a spring on a frictionless surface is released from rest with

the spring extended a distance of A. Just before released, E=1/2kx2.

Figure (b) : when the mass reaches position x, it has KE and PE (decreased)

Total energy, E=KE + PE

7

222

2

1

2

1

2

1kxmkA

Solving for v, xAm

k 2

At x=0, v=vmax

At v= A , v=0

8.4 COMPARING SIMPLE HARMONIC MOTION WITH UNIFORM CIRCULAR

MOTION

-As the turntable rotates with constant angular speed, the shadow of the ball moves back

and forth with SHM.

8

-Figure shows the ball moving with constant speed, v0, in a direction tangent to circular

path.

sin0 …(8.8)

A

xA 22

0

sin

-Equating the right-hand sides of the last two expressions.

22220 xACxAA

…(8.9)

Period and Frequency

-Period, T=time required for one complete trip back and forth and T=the time it takes the

ball to make one complete circular trip on the turntable.

-For the ball moves through a distance 2 A, the speed of the ball around the circular

path is

T

A

20

Period 0

2

AT

…(8.10)

For a mass-spring system moving with SHM

9

Period : k

mT 2

…(8.11)

Unit : second (s)

Frequency : m

k

Tf

2

11 …(8.12)

Unit : s-1 or hertz (Hz)

Angular frequency : m

kf 2 …(8.13)

8.5 POSITION, VELOCITY AND ACCELERATION AS A FUNCTION OF TIME

-See back the relationship between SHM and uniform circular motion.

)cos(cos tAAx …(8.14)

10

Displacement: tAx cos …(8.15)

Velocity : tA sin ..(8.16)

Acceleration : xtAa 22 cos …(8.17)

Where A = amplitude

= angular frequency of the motion

f = frequency of revolution/rotation

The maximum speed : Amax

The maximum acceleration: 2

max Aa

11

8.6 MOTION OF A PENDULUM

Restoring force

sinmgFt …(8.18)

For motion with a small angles

mgFt where Ls

So s

L

mgFt

g

LT 2

..(8.19)

Eg : Pendulum clocks

Comparison :

12

For a simple pendulum

g

LT 2

L

gf

2

1

L

g

where L = length of the pendulum, g = acc due to gravity.

-The period is independent of the suspended mass.

For a mass-spring system:

k

mT 2

m

kf

2

1

m

k

m = mass of the object, k = spring constant

-The period is dependent of the suspended mass

Example:

An object with a mass of 1.0kg is attached to a spring with a spring constant of 10N/m.

The object is displaced by 3.0cm from the equilibrium position and let go.

a. What is the amplitude A?

b. What is the period T?

c. What is the frequency f?

Solution:

a. The initial stretch is the max displacement

So A=3.0cm

b. smN

kg

k

mT 0.2

/10

0.122

c. HzsT

f 50.00.2

11

13

8.7 TYPES OF WAVES

-The waves that travels to the right with a definite speed is called traveling wave.

-Interfering waves of the same f and A traveling in opposite directions in a rope can

produce a standing wave.

-Waves can be divided into 2 types:

1. Transverse wave - the particle motion is perpendicular to the direction of the wave

velocity

Eg: A wave on a rope

2. Longitudinal wave - the particle motion is parallel to the direction of the wave velocity

Eg: Sound wave

Some waves are combinations of TW and LW.

Eg: water waves and seismic waves

14

(a) A longitudinal wave on a spring

(b) The crests and troughscorrespond to compressed and stretched regions of the

spring

(c) The displacement wave

8.8 FREQUENCY, AMPLITUDE, AND WAVELENGTH

Wave Characteristics:

Amplitude : the magnitude of the maximum displacement of the particles of the

material from their equilibrium positions.

Wavelength : the distance between two successive crests or troughs

Frequency : the number of wavelengths that passes by a given point in a second

Wave speed : the speed of the wave motion

15

f (8.20)

8.9 THE SPEED OF WAVES ON STRINGS

-Wave speed: the speed of the wave motion

-The relationship of the velocity, wavelength and frequency of a wave is

f

-The speed of a wave traveling on a stretched string of mass per unit length and under

tension F is

F

…(8.21)

-Speed of a mechanical wave, such as wave on a string, depends only on the properties of

the medium through which the disturbance travel.

Eg : Bass Guitar Strings

8.10 SUPERPOSITION AND INTERFERENCE OF WAVES

Interference occurs when waves meet or overlap.

Principle of Superposition : (Fig:13.14)

If two or more traveling waves are moving through a medium, the resultant wave is

found by adding the individual waves together point by point.

INTERFERENCE

CONSTRUCTIVE DESTRUCTIVE

16

-When waves meet crest to crest and trough to trough, they undergo constructive

interference.

-When crest meets trough, the wave undergo complete constructive interference.

Constructive interference Destructive interference

Total constructive interference occurs when

Two waves of the same frequency and amplitude are exactly in phase

Resultant wave – has the same frequency as the individual waves but twice their

amplitude.

Total destructive interference occurs when

Two waves of the same frequency and amplitude are completely 1800 out of

phase

Resultant wave – complete cancellation.

17

8.11 REFLECTION OF WAVES

-When a wave pulse reflects from a rigid boundary, the pulse is inverted.

-When the boundary is free, the reflected pulse is not inverted.

(a) The reflection of a traveling wave at a fixed end of a stretched string. The

reflected pulse is inverted, but its shape remains constant.

(b) The reflection of a traveling wave at the free end of a stretched string. The

reflected pulse is not inverted.