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Units:

Restoring Force: A force that tends to restore the system to its equilibrium position.

Therefore, it is . . .

In general: Force from a Spring:

c) Calculate its angular frequency.

b) Calculate its frequency.

a) Calculate its period.

1. A pendulum completes 10 swings in 8.0 seconds.

Angular Frequency

Formula:

Symbol:

Oscillations and Waves

Relationship between period and frequency:

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 per unit time

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

Examples of oscillations: mass on spring (eg. bungee jumping), pendulum (eg. swing), object bobbing in water (eg. buoy, boat), vibrating cantilever (eg. diving board), earthquake, musical instruments (eg. strings, percussion, brass, woodwinds, vocal chords)

Oscillation:

Wave:

Simple Harmonic Motion

3. At which position(s) is the mass in equilibrium? Why?

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

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

| |Displacement |Velocity |Restoring Force |Acceleration |

|1 | | | | |

|2 | | | | |

|3 | | | | |

|4 | | | | |

A mass oscillates on a horizontal spring without friction. At each position, analyze the restoring force, displacement, velocity and acceleration.

1. If the mass is pulled back twice as far and then allowed to oscillate, what is the new period of the system?

2. If a second mass of the same amount is added to the spring, what is the new period of the system?

Period of SHM for

mass-on-spring system:

Initial condition:

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.

The Displacement Function

Initial condition:

Function:

Defining Equation for SHM:

Relationship

Simple Harmonic Motion (SHM) – motion that takes place when the acceleration of an object is proportional to its displacement from its equilibrium position and is always directed toward its equilibrium position

1. The graph at right shows the displacement as a function of time for an object in simple harmonic motion.

a)

d) At what time(s) is the velocity of the object at a maximum?

c) At what time(s) is the displacement of the object at a maximum?

ii) t = 2.5 s?

h) Use the function you wrote in part (g) to determine the displacement of the mass at each of the times listed below. Then, check your answer by using the graph.

i) t = 1.0 s?

g) Write a function for the displacement of the object versus time.

f) What is the acceleration of the object at 3.0 seconds?

e) At what time(s) is the acceleration of the object at a maximum?

b) What is the period, frequency, and angular frequency of the system?

a) What is the amplitude of the motion?

Alternative Equations of Motion

Equations of Motion for Simple Harmonic Motion

a) Displacement Function

c) Acceleration Function

b) Velocity Function

Maximum velocity:

Alternate Velocity Function

Defining Equation for SHM:

Maximum acceleration:

Does this motion agree with the defining equation for SHM?

When would these equations be used?

d) Determine the velocity of the object at 1.3 seconds. (Use both formulas.)

c) At what time(s) does the maximum acceleration occur?

b) At what time(s) does the maximum velocity occur?

vi) maximum acceleration

v) maximum velocity

iv) displacement function

iii) angular frequency

ii) period of oscillation

i) amplitude of oscillation

1. The graph shown at right shows the displacement of an object in SHM.

a) Use the graph to find the:

Special Note: The total energy of a system in simple harmonic motion is proportional to . . .

Derivations

| |Kinetic Energy |Potential Energy |Total Energy |

|1 | | | |

|2 | | | |

|3 | | | |

A mass oscillates back and forth on a spring between its extreme positions at 1 and 3. Analyze the energy in the system at each location and derive expressions for each type of energy. Then sketch the energies as a function of displacement from equilibrium.

Energy and Simple Harmonic Motion

ii) total energy

i) period

b) If the mass is doubled and then released from the original position, what is the effect on the mass’s:

ii) total energy

i) period

a) If the mass is pulled back twice as far and then released, what is the effect on the mass’s:

1. A mass M is put on a spring, pulled back a distance A from its equilibrium position and set in simple harmonic motion with period T.

Traveling wave (progressive wave, continuous wave) – succession of oscillations (series of periodic pulses)

or a disturbance moving from a source and transferring energy from one point to another.

Pulse – single oscillation or disturbance

Waves

Note: Both pulses and traveling waves transfer energy though there is no net motion of the medium through which the wave passes.

Motion of a particle:

Motion of energy transfer:

Compare the motion of a single particle to the motion of the wave as a whole (the motion of the energy transfer).

Period: a) time taken for one complete oscillation

(T, seconds) b) time for one complete wave (cycle) to pass a given point

Wavelength: a) shortest distance along the wave between two points that are in phase

(», meters) b) the distance traveled by a wave in one period

Which type of wave can:

be polarized?

propagate through a gas?

Light:

Sound:

Transverse Wave:

one in which the directi(λ, meters) b) the distance traveled by a wave in one period

Which type of wave can:

be polarized?

propagate through a gas?

Light:

Sound:

Transverse Wave:

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).

Energy transferred is proportional to . . .

Mechanical Wave:

requires a medium to transfer energy

Longitudinal Wave:

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).

Electromagnetic Wave:

does not require a medium to transfer energy

Energy transferred is proportional to . . .

(know the orders of magnitude

of the wavelengths of the principal

regions of the EM spectrum)

:

The Electromagnetic Spectrum

Control variable:

Light:

Sound:

What is the control variable when a single wave crosses the boundary between two media?

Therefore, in one medium . . .

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 (unless it is a dispersive medium).

What is the control variable for all waves in a single medium?

Derivation of the Wave Equation:

Motion of the Wave (Energy Transfer)

Motion of a Particle

c) What would be the intensity of the sound if they stood 6.0 meters away?

b) How much energy do they hear every minute?

a) Determine the intensity of the sound heard by this person.

1. A person stands 3.0 meters away from a 100 watt speaker.

NOTE: For a wave, its intensity is proportional . . .

Total area of energy:

As distance from source doubles . . .

Units:

Formula:

Intensity:

Power:

Formula:

Units:

At great distances, the wavefronts are approximately parallel and are known as plane waves.

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.

Waves in Two Dimensions

Where:

f =

f ' =

v =

us =

Doppler Formula (moving source)

For truck moving at constant velocity:

For truck speeding up:

Observer #2 (source moving toward):

Observer #1 (source moving away):

Moving source and stationary observers

The number of compressions reaching each observer’s ear per second is the same so each hears a sound of the same frequency. This frequency is identical to the frequency of the source so there is no Doppler shift.

Stationary source and stationary observers

Light –

Sound –

Doppler Effect: The change of frequency of a wave due to the movement of the source or the observer relative to the medium of wave transmission.

The Doppler Effect

Observer moving away from source:

Observer moving toward source:

2. The security alarm on a parked car goes off and produces a frequency of 960 Hz. The speed of sound in air is 343 m/s. What is the frequency you perceive as you drive toward this parked car at 20. m/s?

Where:

v =

uo =

Doppler Formula (moving observer)

Doppler shift for moving observer and stationary source

1. A high-speed train is traveling at a speed of 44.7 m/s (100 mi/hr) when the engineer sounds the 415-Hz warning horn. The speed of sound in air is 343 m/s. What are the frequency and wavelength of the sound, as perceived by a person standing at a crossing, when the train is approaching?

Note: This approximation is only valid when . . .

3. A star is moving away from Earth at a speed of 3.0 x 105 m/s. One of the elements in the star emits light with a frequency of 6.0 x 1014 Hz. By how much is the frequency shifted when it is received by a telescope on Earth?

Where:

Δf =

v =

Doppler Shift for EM Radiation

Doppler Formula (EM radiation)

Red shift:

Blue shift:

Exception: However, light is unique in that there is no medium of propagation so it is the relative velocity of the source and detector that is relevant.

In terms of the amount of Doppler shift, is a moving observer equivalent to a moving source, that is, is it just the motion of the source and observer relative to each other that is important? Explain.

No, it is the motion of the source and/or observer relative to the medium through which the wave travels that determines the amount of the Doppler shift. A source moving relative to the air (the medium through which the sound wave is traveling) is not equivalent to an observer moving relative to the air since the velocity of each with respect to the medium is not the same in each case. Therefore, the amount of the Doppler shift will be different depending on whether it is the source of the sound or the observer of the sound that is in motion relative to the air.

Mirror

When a wave reflects from a barrier, are there any changes in

a) direction? b) speed? c) wavelength? d) frequency? e) phase?

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).

3. Determining rotation rates

When an object such as a cyclone or a distant star rotates, one side is moving toward the observer and one side is moving away. For a cyclone, radio pulses known as radar are transmitted and reflected from each side of the rotating air mass and the difference in Doppler shift from each side can be used to calculate the rate at which it is rotating. Similarly, the atomic absorption spectrum of the light from a star will be blue-shifted from one side and red-shifted from the other so this difference can be used to determine its rotational rate.

2. Blood-flow measurements

Doctors use a ‘Doppler flow meter’ to measure the speed of blood flow. Transmitting and receiving elements are placed directly on the skin and an ultrasound signal (sound whose frequency is around 5 MHz) is emitted, reflected off moving red blood cells and then received. The difference in transmitted and received frequencies is then used to calculate how fast blood is flowing which can help doctors identify constricted arteries.

1. Measuring the speed of vehicles

Police use radar to measure the speed of moving vehicles to see if they are breaking the speed limit. A pulse of radio waves of known frequency is emitted, reflected off the moving vehicle and reflected back to the source. The change between the frequency emitted and the frequency received is used to calculate the speed of the car.

Examples of the Doppler Effect

Glass to air:

Air to glass:

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

When a wave refracts, are there any changes in

a) direction? b) speed? c) wavelength? d) frequency? e) phase?

Formulas:

Refractive Index (Index of refraction):

1) the ratio of the speed of the wave in the refracted medium to the speed of the wave in the incident medium

2) the ratio of the sine of the angle of incidence to the sine of the angle of refraction

Index of Refraction of 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

Critical Angle: the smallest possible angle of incidence for which light rays are totally reflected at a boundary between substances of different refractive index

Total Internal Reflection: the complete reflection of light at the boundary of two media, when the ray is in the medium with greater refractive index

[pic]

Rule for Refraction:

Complete the path of the light ray through the Lucite block in each diagram below.

Examine the tables of Absolute Indices of Refraction in your textbook (pg 174).

1. In which substance will light travel the fastest? What is its index of refraction?

2. In which substance will light travel the slowest? What is its index of refraction?

3. What is the relationship between the index of refraction of a substance and the speed of light in that substance?

4. Calculate the speed of light in water.

5. Will light slow down if it travels from corn oil to glycerol? Explain.

6. If light crosses a boundary between two substances with very different indices of refraction . . .

7. If light crosses a boundary between two substances with very similar indices of refraction . .

1. A beam of monochromatic yellow light whose frequency is 5.09 x 1014 Hz enters a block of lucite from air.

a) What is the frequency of the light in the Lucite?

b) What is the wavelength of the light in air?

c) What is the wavelength of the light in the Lucite?

d) What is the speed of the light in the Lucite?

Relationship:

As the wave enters a more optically dense medium . . .

Why does refraction occur

Refraction and Wavelength

2. Refer to the picture to the right. The angle between the normal and the direction of travel of the wavefronts is 200. The speed of the waves in medium M1 is 0.75 ms-1. The speed of the waves in medium M2 is

1.3 ms-1. Calculate the angle between the direction of travel of the wavefronts in medium M2 and the normal. Also, sketch the wavefronts in medium M2.

3. Calculate the angle of refraction when the angle of incidence from air at a glass surface is 550. The refractive index of this glass is 1.48

4. Calculate the critical angle for a material with an index of refraction of 1.20

5. Light of wavelength 686 nm in air enters water, making an angle of 40.40 with the normal. The refractive index of water is 1.33

Determine the:

a) Angle of refraction

b) Wavelength of light in water

Lengthening of the Day

How does refraction by Earth’s atmosphere lengthen the number of hours of daylight?

Rainbows Caused by Dispersion

Rainbows are due to sunlight from over an observer’s shoulder being refracted by water droplets in the air. Each color is refracted by a different amount with the result being the dispersion of the light into its component colors.

Explanation:

Dispersion Rule:

Violet:

Red:

Dispersion:

Apparent Depth

Mirages

Bent Objects

Optical Effects due to Refraction

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