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AAST/AEDT

AP PHYSICS B:

Oscillations

This is a motion of an object that moves back and force over the same path. This motion

can be described by several physical quantities:

Period - is the time of one cycle - T

Frequency is the number of cycles in a unit of time f

Period and frequency are reciprocals T=1/ f

Amplitude is the maximum object's distance from the equilibrium point.

In physics we study two types of vibrations.

Free oscillations are the oscillations when you supply energy once and then vibrations

are continuing as the result of action of the internal forces of the system. Examples of that

kind of oscillations are - SHO and simple pendulums.

Forced oscillations are the result of the action of the external periodical force.

Example of such oscillations is the piston’s motion in the car’s engine.

2 conditions are mandatory for the existence of the free oscillations:

a) The returning force must appear when the object is moved from its equilibrium

position.

b) Low damping.

Let us discover was does change when the mass vibrates on a spring.

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We do observe that all physical quantities that describe motion do change when the

spring with the mass vibrates. All these quantities vary according to the sin or cos law.

That is why we call that motion- Simple Harmonic motion (SHM)

As you can observe from the graph, there is a certain phase shift between different

quantities. For example the phase shift between acceleration and velocity is 90°. That

means: When velocity is at maximum, acceleration is zero.

Let us derive equations for the acceleration, velocity and position of the vibrating object.

In case of the mass on the spring (SHO-Simple Harmonic Oscillator) the elastic force

provides the acceleration. Thus, the second Newton law for the system is

Fel=ma. or if we take into consideration Hooke’s law

ma = -kx, where k is the elastic coefficient and x -deformation or position if mass’s position at

equilibrium is zero. We also know that acceleration is the second derivative from the

position. Thus, we obtain the deferential equation.

The only function in math, whose second derivative is equal to the function itself

with an opposite sign, is the harmonic function sin or cos.

Thus, the solution of that equation should be

x k

mx (1)

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x= A sin (t) (2)

To prove the statement, we substitute (2) into (1). The first derivative from position

Velocity (v)= x’= Acos t = Vocos (t) (3)

Acceleration (a) = x’’ =- A 2sin (t) = -2

x (4)

As we observe (4) is equivalent to (1). Conclusions from the derivation are: velocity and

acceleration both experience harmonic motion, but there is a phase shift. From the

properties of the harmonic functions it is known that is an angular frequency. If we

compare (1) and (4), it is obvious that for the mass on the spring 2 = k/m. We also

know that

Thus, we obtain the formula for the period of the mass on the spring.

We theoretically obtained a very interesting result that the period of SHM is proportional

to the square root of the mass and is inversely proportional to the elasticity of the

oscillating object.

It is important to note, that period is independent on the amplitude and on the spring’s

dimensions.

SIMPLE PENDULUM

By definition it is a material point, suspended on a nonelastic cord with a length of L.

This is an ideal model were we ignore dimensions of the suspended object, cord’s

deformation and mass.

In a case of a SHO the elastic force of a

spring is responsible for the springs acceleration.

In a case of a simple pendulum as you can

observe on a diagram below, the resultant force

of the gravity and elastic force provides

acceleration.

Two forces are exerted on the object -the

tension of the cord -T and gravity force Fg. The

second Newton's law in vector shape for the

pendulum is

T =2

T =2

2

km

2m

k (5)

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T+ Fg = ma (6)

We pick Y-axis along the cord and X-axis along the tangent to the pendulum's trajectory.

If we project equation (6) on axis X the result is

Fgsin = ma, or mgsin = ma (7)

From the triangle ABC we can conclude that sin = BD/BC

If the angle is small we can substitute BD with the displacement from equilibrium

position AB, That displacement is similar to the spring’s deformation for the SHM. We

also can observe that AB is the length of the cord L.

Keeping that in mind we are able to rewrite (7)

Because the direction of acceleration and the direction of displacement AB are opposite,

in a vector shape the equation becomes

As you can see equation for simple pendulum is similar to that for SHO - (5). That allows

us to substitute expression k/m in the formula for the SHO period by g/L.

That is why the period of the vibrations of the simple pendulum can be expressed

by formula

where L- is the pendulum’s length.

g- acceleration due to gravity

3 laws describe the vibrations of the simple pendulum.

1. The period does not depend on the mass and amplitude

2. It depends on a length of the pendulum and its place.

3. The plane of the vibrations remains the same.

The 3-d law had been used by Focault to prove the Earth’s rotation

MECHANICAL WAVES

There are only two possible ways of transporting energy.

Example 1. If a baseball it moving, it transports energy from the pitcher to the catcher.

Energy is transported with a massive object (ball).

gsin() = a or a g

LAB (8)

a g

LAB

(9)

T = 2L

g

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Example 2: The energy is transported from the sun to the Earth without any mass

transport. We deal with the energy that is transported by waves.

There are two types of the waves in physics - mechanical waves and electromagnetic

waves. These waves have several common features and some differences. We will discus

the properties of the mechanical waves.

A mechanical wave is the disturbance which, once started, propagates on its own

through the medium.

The mandatory condition for the wave propagation is the existence of the tension

forces between the neighboring particles in the medium.

Let us assume that we have a row of particles along the straight line at the initial

instant of time

t = 0

We start to vibrate particle 1 in a vertical plane. As it moves up, the distance between the

particles 1 and 2 increases and elastic force is exerted on the second particle. Thus, it

starts to move up. Of course a little bit later than 1. Particle 2 experienced a phase shift.

Particle 3 stars to move also as the result of the 2 - 3 distance increase and so on. When

the particle 1 reaches the upper point (t= T/4) the wave has a shape as you can observe on

a diagram below. (T- is the period of the particle’s oscillations)

t= T/4

In the next moment particle 1 starts to move down. Particle two as the result of inertia

reaches the upper point and then continues to follow particle 1 and so on. At an instant t =

T/2 the shape of the wave is

t=T/2

At an instant t = 3T/4 the wave’s shape is

t =3T/4

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Finally, after point 1 completes a full oscillation at the time instant t = T, the wave’s

shape is

t=T

As we can observe in the wave the particles oscillate perpendicularly to the

direction of the wave propagation. Such wave is defined as a transverse wave.

Point A is defined as a crest. Point B is called a trough.

Thus: The transverse wave consists of crests and troughs

As you can observe, mass does not propagate with the wave. Wave transfers only

energy.

Another kind of waves occurs when a particle starts to vibrate along the direction of the

wave propagation. That wave consists of the compressions and expansions.

That type of waves is defined as longitudinal waves.

Several physics quantities are used to describe the wave.

They are:

Period (T) - the time of one full particles’ oscillation.

Frequency (f) - the number of oscillations per unit of time.

Wave length ( - lambda)- the distance between the two nearest particles that

vibrate with an equal phase.

For example it is the distance between points A and B, or C and D.

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It is easy to observe from the picture on the previous page that a wave travels a

distance that is equal to the wavelength during the period.

That allows us to derive a formula to compute the velocity of the wave.

We also know that the relationship between the period and the frequency is

Thus, the final formula is

Amplitude (A) - is the crest’s height or the trough’s depth.

Amplitude is an important characteristic that describes the wave's property to transfer

energy. It can be proved that energy transported by wave is directly proportional to

the second power of amplitude.

Waves in a medium

Different mediums have different ability to propagate waves.

In solids we can observe propagation of both transverse and longitudinal waves. The

reason of that effect is that elastic force can be created as the result of any deformation in

solids.

In liquids and gases only a longitudinal waves can propagate, because the layer shift in

liquids does not create elastic forces.

We also can observe the transverse waves on the liquids surface. That phenomenon is the

result of the existence of the liquids’ surface tension.

The greater the elasticity of the medium the greater the waves velocity. Waves in gases

have the lowest velocity. Waves in solids have the greatest velocity.

Interference

Let us assume that we have two independent sources of the waves located at opposite

ends of the cord. For the simplicity we assume that those sources produce

waves with an equal frequencies and equal amplitudes. Such sources we call coherent.

When the waves reach

the same point. There are

two possibilities.

a. If a crest of one wave

coincides with the crest

of the another one we

will observe the

amplitude increase.

(picture below).

v

T

T 1

f

v f

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b. If a trough of one wave coincides with the trough of the another one we will observe

the trough with the doubled depth. (Picture below)

c. If a trough of one wave coincides with the crest of the another one we will observe

that they destruct each other.

That phenomenon of the mutual amplification or destruction of the waves that

happened when two or more waves travel through the medium is defined as interference.

The interference in an example (a) is called constructive,

The interference in an example (c) is called destructive

Examples above are examples for two pulses. The effect becomes more interesting when

we observe the interference between two infinite waves traveling along the cord with an

equal frequency, amplitude but an opposite direction.

On the picture below you can observe the wave running right (solid line), the wave that is

running left (dashed line) and the resultant wave (bold line) in a different instants of time.

Points b,d, f,h and j are permanently at rest. That occurs because both waves arrive to

those points with opposite phases and cancel each other. Those points are called nodes.

Points a, c, e, g, i, and k vibrates with the doubled amplitude. Those points we call

antinodes.

The wave that consists of nodes and antinodes is called a standing wave.

It is important to note that the standing wave does not transport energy. The only energy

transformation that occurs in a standing wave is the transformation from the kinetic into

the potential energy in an antinodes and vice versa.

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We can observe an interference pattern from two

coherent sources in a ripple tank. The pattern as it is

shown on a diagram also consists of nodes and

antinode

Huygen’s principle

It states

Every point on a wave front can be considered as a

source of the secondary waves. The new wave

front is the result of the interference of those secondary waves.

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Examples: 1. If we deal with a two dimensional wave (wave in a cord) we can assume

that point 3 is the

source of the

waves respectively

to the point 4. The

reason is that for

the point 4 it is

indifferent whether

point 1 and 2

vibrate.

2. If we have a wave front AB, every point of

those front starts to emit the secondary waves and the

new front CD is the result of the interference of the

secondary waves.

Reflection and refraction

When wave reaches the solid boundary it reflects. If we apply Huygens principle we can

prove that the angle of incidence is equal to the angle of reflection.

When wave reaches the boundary between to mediums it refracts as it shown on the

diagram.

We define refractive indices as the ratio of the sin of angle of incidence over the sin of

angle of refraction. It can be proved that

The indices of refraction is permanent for the wave transition between the two given

mediums.

We can also prove that a refractive index is equal to the ratio of the wave’s

speed in the first medium over the

wave’s speed in the second one.

n sin i

sin r

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Mathematically that can be expressed as

SOUND

Sound is mechanical waves with the frequencies from 20 to 20,000 Hz.

Hz(Hertz) is the unit of frequency that is equal to 1 vibration per second.

1Hz = 1 1/sec

That range is chosen, because only the human ear detects that range of waves.

The waves with the frequencies that are greater then 20,000 Hz is called ultrasonic.

The sound can spread in a various mediums, i.e. solids, liquids and air.

The sound does not propagate through the vacuum. That can be easily proved if we will

place the electric bell in a vacuum.

The speed of the sound

It depends on the medium and for air it is equal approximately 330m/s.

Vibrating strings

Let us assume that we have a vibrating string with two fixed ends. L- is the length

of the string. Because on that string we always have incident wave and reflected wave the

result of their interference would be the standing wave.

Because the ends are fixed the first possible standing wave will have nodes at the fixed

ends and the antinode at the middle.

We know that the distance between the two consecutive nodes is the half of the

wavelength. That allows us to find a relationship between the string’s length and

frequency

We do know that v = f

Because L = /2, we have v = 2Lf

Thus,

n v1

v2

f1 v

2L

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f2 = vL = 2 * v

2L

where v is the wave’s speed in the string medium.

That frequency is defined as fundamental - which is the lowest possible frequency for

the given string. Subscript 1 is used for that goal.

But we also can observe another picture of the standing wave on a given string. That

picture consists of 3 nodes - two at the end and one in the middle and two antinodes in

between them

Now L=, and , that frequency we define as f2 and that

frequency is called the second harmonic.

Next possible situation you can observe on a diagram below

Now L= 3 /2, and f3= 3V/2L.

and so on

In general the string can create waves with the frequencies

As we have mentioned the wave with the lowest frequency is called fundamental. It

defines what is the note that the string creates.

Waves with the higher frequencies are called overtones.

Every string has its own set of overtones. They are responsible for tone color or timbre.

Quality of sound also depends on the quality of tuning of the different sound sources.

For example if two musical instruments produce the sounds with the equal frequencies

the sound will be perfect. But if they produce sounds that are slightly differ by

frequencies, we will be able two hear the differences in volume with a certain frequency.

That phenomenon is called beats and of great importance in acoustic engineering. The

beat frequency is equal to the difference of the frequencies of the sources. For example if

the frequency of one tuning fork is 440 Hz and the second has a frequency of 452 Hz, the

frequency of the beats will be 12 Hz.

Example of the beats you can observe below.

fn = n * v2L

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The characteristics of the sound

Sound is created by the vibrated object. If we connect the microphone with the

oscilloscope, we would observe the sine shape of the sound wave.

There are three major sound characteristics

1. Pitch is the characteristic that is proportional to the sound’s frequency.

2. Loudness is the characteristic that indicates the amplitude of the sound’s wave.

The unit of loudness is Bell. It is a large unit. You often can hear the unit decibel.

The loudness can reach the dangerous level - 120 dB. Loudness is measured in

logarithmic scale. That means that the sound level of 20 decibels is 100 times more

intense than the sound level of 10 dB.

3. Timbre is the characteristics that describe the “color” of sound. It depends on what

additional harmonics are present in sound in addition to the main one.

Doppler Effect Let us assume that the source of sound with a frequency of f is moving toward you with a

velocity of V.

As you can observe from the picture the frequency heard by the observer increases. It can

be proved that the formula to calculate that increase is

If the source of the sound is moving away from the observer, than the frequency

decreases.

f f

1 vvs

f f

1 vvs

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Doppler effect is widely used to measure the speed of the remote objects. (Speed gun)

Problem: A hollow tube of length L, open at both ends as shown on the diagram, is held

in midair. A tuning fork with a

frequency fo vibrates at one end of

the tube and causes the air in the

tube to vibrate at its fundamental

frequency. Express your answers in

terms of L and fo.

a. Determine the wavelength of the

sound?

b. Determine the speed of the sound

in the air inside the tube

c. Determine the next higher frequency at which that air column will resonate

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d. The tube is submerged in a large graduated cylinder with water. The tube is slowly

raised out of the water and the same tuning fork is vibrating with the frequency fo is held

at fixed distance from the top of the tube.

Determine the height h of the tube above the water when the air column resonates for the

first time. Express your answer in terms of L.

Home assignment:

Cutnell: Chapter 10; Conceptual questions:: Page 298 ,# 2, 4, 5, 7, 9, 10,

Problems: Page 299… #3, 9,10, 17, 21, 22 31, 33, 35, 41, 43,44

Chapter 16; Conceptual questions:: Page 485 ,# 1,3,6,10, 19

Problems: Page 486… #5, 7, 9, 11,13, 15, 31, 35, 39, 41, 75, 77