IB 12

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Oscillations and Waves

Oscillation:

Wave:

Examples of oscillations:

1. mass on spring (eg. bungee jumping) 2. pendulum (eg. swing) 3. object bobbing in water (eg. buoy, boat) 4. vibrating cantilever (eg. diving board) 5. earthquake 6. bouncing ball 7. musical instruments (eg. strings, percussion, brass, woodwinds, vocal chords)

8. heartbeat

Mean Position (Equilibrium Position) – position of object at rest

Displacement (x, meters) – distance in a particular direction of a particle from its mean position

Amplitude (A or x0, meters) – maximum displacement from the mean position

Period (T, seconds) – time taken for one complete oscillation

Frequency (f, Hertz) – number of oscillations that take place per unit time

Phase Difference – difference in phase between the particles of two oscillating systems, measured in radians

Angular Frequency -

Relationship between period

and frequency:

Symbol:

Units:

Formula:

1. A pendulum completes 10 swings in 8.0 seconds.

a) Calculate its period.

b) Calculate its frequency.

c) Calculate its angular frequency.

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Example of an Oscillating System

A mass oscillates on a horizontal spring without friction as shown below. At each

position, analyze its displacement, velocity and acceleration.

Force from the Spring:

1. When is the velocity of the mass at its maximum value?

2. When is the acceleration of the mass at its maximum value?

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The Displacement Function

A mass on a spring is allowed to oscillate up and down about its mean position without friction.

Two traces of the displacement (x) of the mass versus time (t) are shown.

Initial condition:

Function:

Initial condition:

Function:

Analyzing the Displacement Function

1. Analyze the displacement function shown at right.

2. What is the displacement of the mass when:

a) t = 1.0 s?

b) t = 2.0 s?

c) t = 2.5 s?

.

a) What is the amplitude?

x0 =

b) What is the period?

T =

c) What is the frequency?

f =

d) What is the angular frequency?

ω =

e) Write the displacement function.

x =

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Velocity and Acceleration for Simple Harmonic Motion

a) Displacement Function

b) Velocity Function

c) Acceleration Function

Simple Harmonic Motion (SHM) –

Defining Equation for SHM: Negative Sign:

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1. The graph shown at right shows the displacement

of an object in SHM. Use the graph to find the:

a) period of oscillation

b) amplitude of oscillation

c) displacement function

d) maximum velocity

e) velocity at 1.3 seconds

f) maximum acceleration

g) acceleration at 1.3 seconds

Alternate Velocity Function

2. Use the alternate form of the velocity function

to find the velocity of the object at 1.3 s.

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Example of SHM – Mass on a Horizontal Spring

A mass m oscillates horizontally on a spring

without friction, as shown. Is this SHM?

Angular frequency, period, and frequency for a mass on a spring

1. A 2.00 kg mass oscillates back and forth 0.500m from its rest position on a horizontal spring

whose constant is 40.0 N/m.

a) Calculate the angular frequency, period and frequency of this system.

b) Write the displacement, velocity and acceleration functions for this system.

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Alternate Forms of the Equations of Motion for SHM

1. Write the equations of motion for the graphs shown below.

2. Write the equations of motion for the graphs shown below.

3. What is the difference between the motions described by the two sets of equations?

4. a) Write the equations of motion for the system whose

displacement is shown on the graph at right.

b) State two times when the:

i) speed is maximum

ii) magnitude of the acceleration is maximum.

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Example of SHM – Simple Pendulum

1. A mass is allowed to swing freely from the end of a light-weight string. Show that

the motion of this simple pendulum is approximately simple harmonic motion.

2. Determine the angular frequency, period and frequency for the pendulum.

3. A 20.0 g pendulum on an 80.0 cm string is pulled back 5.0 cm and then swings. Determine its:

a) angular frequency

b) displacement function

c) velocity function

d) maximum velocity

e) maximum acceleration

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Energy and Simple Harmonic Motion

A mass oscillates back and forth on a spring. Analyze the energy in the system at each location.

When the mass is at its mean position . . .

When the mass is at any position . . .

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1. A 2.00 kg mass is oscillating on a spring and its

displacement function is shown.

a) At what time(s) does the mass have the most kinetic energy?

b) Determine the maximum kinetic energy of the mass.

c) At what time(s) does the mass have maximum potential energy? Determine this value.

d) What is the total energy of the system at 1.5 seconds?

e) Determine the kinetic and potential energy of the system at 1.5 seconds.

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Energy Graphs and SHM

Energy-Displacement Functions

Energy-Time Functions

Note that in simple harmonic motion, the energy of a system is proportional to:

1.

2.

3.

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Damping in Oscillations

Damping

Sketch the displacement function for a system without and with damping.

Without Damping With Damping

Effect of damping:

Degrees of Damping

Light damping (under-damping):

small resistive force so only a small

percentage of energy is removed each

cycle – period is not affected – can take

many cycles for oscillations to die out

eg. – car shock absorbers

Critical damping: intermediate resistive force so time

taken for object to return to mean position is minimum –

minimal or no “overshoot”

eg. – electric meters with pointers, automatic door closers

Heavy damping (over-damping): large resistive force –

can completely prevent any oscillations from taking place

– takes a long time for object to return to mean position

eg.- oscillations in viscous fluid

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Resonance

Natural Frequency of Vibration: when a system is displaced from equilibrium and allowed to oscillate

freely, it will do so at its natural frequency of vibration

Forced Oscillations – a system may be forced to oscillate at any given frequency by an outside driving

force that is applied to it

Resonance –

Amplitude vs. frequency graph

for forced oscillations

Factors that affect the frequency response and

sharpness of curve:

1)

2)

3)

4)

1. Sketch the frequency response for a lightly

damped system whose natural frequency is

20 Hz that experiences forced oscillations.

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Pulse – single oscillation or disturbance Continuous traveling wave – succession of

oscillations (series of periodic pulses)

Waves

Both pulses and traveling waves:

transfer energy though there is

no net motion of the medium

through which the wave passes.

A transverse wave is

one in which the

direction of the

oscillation of the

particles of the medium

is perpendicular to the

direction of travel of the

wave (the energy).

Examples: light, violin

and guitar strings, ropes,

earthquake S waves

Mechanical Waves: require a medium to transfer energy eg. – sound waves, water waves, waves on strings, earthquake waves

Electromagnetic Waves: do not require a medium to transfer energy eg. – light waves, all EM waves

A longitudinal wave is

one in which the direction

of the oscillation of the

particles of the medium is

parallel to the direction of

travel of the wave (the

energy).

Example: sound,

earthquake P waves

Displacement (x, meters) – distance in a particular direction of a particle

from its mean position

Amplitude (A or x0, meters) – maximum displacement from the mean

position

Period (T, seconds) – time taken for one complete oscillation

- time for one complete wave (cycle) to pass a given point

Frequency (f, Hertz) – number of oscillations that take place per unit time

Wavelength (λ, meters) – shortest distance along the wave between two points that are in phase

-the distance a complete wave (cycle) travels in one period.

Compression: region where particles of medium are close together

Rarefaction: region where particles of medium are far apart

Note that transverse mechanical waves cannot propagate (travel) through a gas – only longitudinal waves can.

Compare the motion of a single particle to

the motion of the wave as a whole (the

motion of the energy transfer).

Particle Speed: Wave Speed:

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Motion of the Wave Motion of a Particle

1.

2.

Intensity -

Control variable: in one medium - wave speed

Wave speed depends on the properties of the medium, not

how fast the medium vibrates. To change wave speed, you

must change the medium or its properties.

Control variable: across a boundary - frequency

As a wave crosses a boundary between two different media, the

frequency of a wave remains constant not the speed or

wavelength.

Light: Sound:

Waves in Two Dimensions

Wavefront – line (or arc) joining

neighboring points that have the same

phase or displacement

Ray – line indicating direction of

wave motion (direction of energy

transfer).

Rays are perpendicular to wavefronts.

At great distances,

the wavefronts are

approximately

parallel and are

known as plane

waves.

Symbol:

Units:

NOTE:

Formula:

1. 12 x 10-5

W of sound power pass

through each surface as shown. Surface

1 has area 4.0 m2 and surface 2 is twice

as far away from the source. Calculate

the sound intensity at each location.

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Reflection and Refraction

Sketch the incident and reflected rays as well as the reflected wavefront.

Law of Reflection

The angle of incidence is

equal to the angle of

reflection when both

angles are measured with

respect to the normal line

(and the incident ray,

reflected ray and normal

all lie in the same plane).

Refraction: the change in direction of a wave (due to a change in speed) when it crosses a boundary between

two different media at an angle

Glass to air:

Air to glass:

Refractive Index (Index of refraction)(n):

ratio of sine of angle of incidence to sine of angle

of refraction, for a wave incident from air

Snell’s Law: the ratio of the sine of the angle of

incidence to the sine of the angle of refraction is a

constant, for a given frequency

Mirror