Helpful animation

https://phet.colorado.edu/sims/html/wave-on-a-string/latest/wave-on-a-string_en.html

http://www.acs.psu.edu/drussell/Demos/waves/wavemotion.html

Doppler!

The Doppler Effect

You have no doubt heard about the Doppler Effect - what is it exactly?  The key in the Doppler effect is that motion makes the "detected" or "perceived" frequencies higher or lower.  We will consider this first for sound and then generalize to light.

Let's play around with this:  

http://www.lon-capa.org/~mmp/applist/doppler/d.htm

How how the number of waves you receive per second will be the same regardless of where you stand, UNLESS the source is moving.  And then:

If the source is moving toward you, you detect/measure a higher frequency - this is called a BLUE SHIFT.

If the source is moving away from you, you detect/measure a lower frequency - this is called a RED SHIFT. 

It's worth noting that the effect also works in reverse. If you (the detector) move toward a sound-emitter, you'll detect a higher frequency. If you (the detector) move away from a sound-emitter, you'll detect a lower frequency.

Mind you, these Doppler effects only happen WHILE there is relative motion between source and detector (you).

And they also work for light. In fact, the terms red shift and blue shift refer mainly to light (or other electromagnetic) phenomena.

If your computer runs Java:

http://falstad.com/mathphysics.html

Run the Ripple tank applet -

http://falstad.com/ripple/

Distant galaxies in the universe are moving away from us, as determined by their red shifts. This indicates that the universe is indeed expanding (first shown by E. Hubble). The 2011 Nobel Prize in Physics went to local physicist Adam Riess (and 2 others) for the discovery of the accelerating expansion of the universe. Awesome stuff!

http://www.nobelprize.org/nobel_prizes/physics/laureates/2011/

Sound problems related to music

Wave questions II

Consider the musical note G, 392 Hz.  Find the following:

1.  The frequencies of the next two G's, one and two octaves above.

2.  The frequency of the G one octave lower than 392 Hz.

3.  The frequency of G#, one semi-tone (piano key or guitar fret) above this G.

4.  The frequency of A#, 3 semi-tones above G.

5.  The wavelength of the 392 Hz sound wave, assuming that the speed of sound is 340 m/s.

answers:

1.  392 x 2 = 784 Hz; 392 x 4 = 1568 Hz

2.  392/2 = 196 Hz

3.  392 x 1.0594 = 415 Hz

4.  392 x 1.0594 x 1.0594 x 1.0594  (or 392 x 1.0594^3) = 466 Hz

5.  wavelength = speed / frequency = 340/392 = 0.87 m

Sound sites from class tonight

https://www.youtube.com/watch?v=cPALfz-6pnQ

Shattering glass with sound



http://www.szynalski.com/tone-generator/

Tone generator



Fairly amazing guy to check out:

https://www.youtube.com/watch?v=XKRj-T4l-e8

https://www.youtube.com/watch?v=7aLlvkxfUL4

The sound of music!

Music 1 - Notes, Harmonics

In western music, we use an "equal tempered (or well tempered) scale."  It has a few noteworthy characteristics;

The octave is defined as a doubling (or halving) of a frequency.

You may have seen a keyboard before.  The notes are, beginning with C (the note immediately before the pair of black keys):

C

C#

D

D#

E

F

F#

G

G#

A

A#

B

C

(Yes, I could also say D-flat instead of C#, but I don't have a flat symbol on the keyboard.  And I don't want to split hairs over sharps and flats - it's not that important at the moment.)

There are 13 notes here, but only 12 "jumps" to go from C to the next C above it (one octave higher).  Here's the problem.  If there are 12 jumps to get to a factor of 2 (in frequency), making an octave, how do you get from one note to the next note on the piano?  (This is called a "half-step" or "semi-tone".)

The well-tempered scale says that each note has a frequency equal to a particular number multiplied by the frequency that comes before it.  In other words, to go from C to C#, multiply the frequency of the C by a particular number.

So, what is this number?  Well, it's the number that, when multiplied by itself 12 times, will give 2.  In other words, it's the 12th root of 2 - or 2 to the 1/12 power.  That is around 1.0594.

So to go from one note to the next note on the piano or fretboard, multiply the first note by 1.0594.  To go TWO semi-tones up, multiply by 1.0594 again - or multiply the first note by 1.0594^2.  Got it?

>

Let us examine "harmonics", visible on a string (as demonstrated in class).  Harmonics are wave shapes produced that have a maximum amplitude under given conditions (tension in string, length of string, composition of string, etc.).  Every stretched string has a particular lowest frequency at which it will naturally resonate or vibrate.  However, there are also higher frequencies that will also give "harmonics" - basically, pretty wave shapes (also known as "standing waves").  These higher frequencies are integer multiples of the lowest frequency.

So, if the frequency of the lowest frequency is 10 Hz (for an N = 1 harmonic), the next harmonic (N = 2) occurs at 20 Hz.  N = 3 is at 30 Hz, and so on.

For those of you who play guitar, you know that you get harmonics on certain frets.  In the exact center of the neck (12th fret) you get a harmonic (the 2nd one) and the frequency is twice that of the open string - one octave above, as we will discuss.

Monday class 10/24 canceled

Folks - I have a last-minute family crisis to tend to and must cancel class for tomorrow evening.  (Nothing too terrible, but my presence is needed.)

I'm so sorry about this!  

Practice problems and answers are posted. I will add more notes in the next few hours.

Please also review the notes on waves, particularly the sites linking to E/M spectrum pages.

See you Wednesday - email me if you have any questions.

Sincerely,

Sean Lally

Wave practice problems

Wave questions I

1.  Differentiate between mechanical and electromagnetic waves.  Give examples.

2.  Draw a wave and identify the primary parts (wavelength, crest, trough, amplitude).

3.  Find the speed of a 500 Hz wave with a wavelength of 0.25 m.

4.  What is the frequency of a wave that travels at 24 m/s, if 3 full waves fit in a 12-m space?  (Hint:  find the wavelength first.)

5.  Approximately how much greater is the speed of light than the speed of sound?

6.  Show how to compute the wavelength of WTMD's signal (89.7 MHz).  Note that MHz means 'million Hz."  Recall that radio waves travel at the speed of light.

7.  Middle C vibrates at 262 Hz (approximately).  Find the frequencies of the next 2 C's (1 and 2 octaves above this one).

Answers:

1/2.  See notes

3.  v = f l = 500(0.25) = 125 m/s

4.  wavelength (l) = 12/3 = 4 m

v = f l

24 = f (4)

f = 6 Hz

5.  300,000,000 / 340 = approx one million times greater

6.  v = f l

300,000,000 = (89,700,000) l

l = 300,000,000/89,700,000

7.  In music, octaves are found by doubling the frequency of the first note.

524 Hz, 1048 Hz

Introduction to Waves

So - Waves.....  

We spoke about energy.  Energy can, as it turns out, travel in waves.  In fact, you can think of a wave as a traveling disturbance, capable of carrying energy with it.  For example, light "waves" can have energy - like solar energy.  Ocean waves can certainly carry energy.  

There are several wave characteristics (applicable to most conventional waves) that are useful to know:

amplitude - the "height" of the wave, from equilibrium (or direction axis of travel) to maximum position above or below

crest - peak (or highest point) of a wave

trough - valley (or lowest point) of a wave

wavelength (lambda - see picture 2 above) - the length of a complete wave, measured from crest to crest or trough to trough (or distance between any two points that are in phase - see picture 2 above).  Measured in meters (or any units of length).

frequency (f) - literally, the number of complete waves per second.  The unit is the cycle per second, usually called:  hertz (Hz)

wave speed (v) -  the rate at which the wave travels.  Same as regular speed/velocity, and measured in units of m/s (or any unit of velocity).  It can be calculated using a simple expression:

There are 2 primary categories of waves:

Mechanical – these require a medium (e.g., sound, guitar strings, water, etc.)

Electromagnetic – these do NOT require a medium and, in fact, travel fastest where is there is nothing in the way (a vacuum). All e/m waves travel at the same speed in a vacuum (c, the speed of light):

c = 3 x 10^8 m/s

First, the electromagnetic (e/m) waves:

General breakdown of e/m waves from low frequency (and long wavelength) to high frequency (and short wavelength):

Radio

Microwave

IR (infrared)

Visible (ROYGBV)

UV (ultraviolet)

X-rays

Gamma rays

In detail, particularly the last image:

http://hyperphysics.phy-astr.gsu.edu/hbase/ems1.html

http://csep10.phys.utk.edu/astr162/lect/light/spectrum.html

http://www.unihedron.com/projects/spectrum/downloads/full_spectrum.jpg

Mechanical waves include:  sound, water, earthquakes, strings (guitar, piano, etc.)....

Again, don't forget that the primary wave variables are related by the expression:

v = f l

speed = frequency x wavelength

(Note that 'l' should be the Greek symbol 'lambda', if it does not already show up as such.)

For e/m waves, the speed is the speed of light, so the expression becomes:

c = f l

Note that for a given medium (constant speed), as the frequency increases, the wavelength decreases.

What is energy?

I stole my energy story from the famous American physicist Richard Feynman. Here is a version adapted from his original energy story. He used the character, "Dennis the Menace." The story below is paraphrased from the original Feynman lecture on physics (in the early 1960s).

Dennis the Menace

Adapted from Richard Feynman

Imagine Dennis has 28 blocks, which are all the same. They are absolutely indestructible and cannot be divided into pieces.

His mother puts him and his 28 blocks into a room at the beginning of the day. At the end of each day, being curious, she counts them and discovers a phenomenal law. No matter what he does with the blocks, there are always 28 remaining.

This continues for some time until one day she only counts 27, but with a little searching she discovers one under a rug. She realizes she must be careful to look everywhere.

One day later she can only find 26. She looks everywhere in the room, but cannot find them. Then she realises the window is open and two blocks are found outside in the garden.

Another day, she discovers 30 blocks. This causes considerable dismay until she realizes that Bruce has visited that day, and left a few of his own blocks behind.

Dennis' mother removes the extra blocks, gives the remaining ones back to Bruce, and all returns to normal.

We can think about energy in this way (except there are no blocks!). We can use this idea to track energy transfers during changes. We need to be careful to look everywhere to ensure that we can account for all of the energy.

Some ideas about energy

• Energy is stored in fuels (chemicals).
• Energy can be stored by lifting objects (potential energy).
• Moving objects carry energy (kinetic energy).
• Electric current carries energy.
• Light (and other forms of radiation) carries energy.
• Heat carries energy.
• Sound carries energy.

But is energy a real thing?  No, not exactly.  It is a mathematical concept, completely consistent with Newton's laws and the equations of motion.  It allows us to see that some number (calculated according to other manifest changes - speed, mass, temperature, position, etc.) remains constant before and after some "event" occurs.

How things fly

How things fly!

The amazing science of flight is largely governed by Newton's laws.

Consider a wing cross-section:

Air hits it at a certain speed.  However, the shape of the wing forces air to rush over it and under it at different rates.  The top curve creates a partial vacuum - a region "missing" a bit of air.  So, the pressure (force/area) on top of the wing can become less than the pressure below.  If the numbers are right, and the resulting force below the wing is greater than the weight of the plane, the plane can lift.

This is often expressed as the Bernoulli Principle:

Pressure in a moving stream of fluid (such as air) is less than the pressure of the surrounding fluid.

The image above shows another way to think of flight - imagine the wing first shown, but slightly inclined upward (to exacerbate the effect).  There is a downward deflection of air.  The reaction force from the air below provides lift and the lift is proportional to the force on the wing.

In practice, it works out (in general) to be:

Lift = 0.3 p v^2 A

where p is the density of air, v (squared) is the speed of the plane, and A is the effective area.  Note that the lift is proportional to the speed squared - so, the faster the plane goes, the (much) easier it is to take flight.

Some related animation:

http://physics.stackexchange.com/questions/13030/why-does-the-air-flow-faster-over-the-top-of-an-airfoil

Exam 1 review

General topics for exam 1.  Be sure to review all assigned homework, blog posts and your notes.

You are permitted to have a sheet of notes for this test.  I will NOT give equations.

SI units (m, kg, s) - meanings, definitions

velocity

average vs. instantaneous velocity

acceleration

related motion problems using the formulas

speed of light (c) - approx 300,000,000 m/s

gravitational acceleration (g)

freefall problems

Newton's 3 laws - applications and problems

Kepler's 3 laws - applications and problems

epicycles

Galileo and his telescope

Newton's law of universal gravitation (inverse square law)

weight vs. mass

Weightlessness

center of mass/gravity and torque

Useful equations

Average v = d/t

Vf = a t

d = 0.5 a t^2

F = m a

W = m g

a^3 = P^2

T = F L

Answer to recent Newton questions, and more questions

Answers to recent Newton problems:

1.  See notes.

2.  40 / 0.5 = 80 m/s/s

3.  lower acceleration

4.  See notes.  Tablecloth pull, etc.

5.  firearm recoil, etc.

6.  Principia Mathematica, 1687.

7.  They helped explain retrograde motion - the apparent backwards motion of planets.  Really, planets are orbiting the Sun and there are times that some bodies are "behind" -- it's like when you pass someone on the highway and they "appear" to be moving backwards at that time.

8.  mass - the amount of stuff (in kg); weight - the gravitational pull on this stuff (in newtons).  The weight depends on where you are (in terms of how the gravitational acceleration changes).  For example, your weight on the Moon is 1/6 that of Earth.

9.  newton; pound

10.  W = m g.  Weight is depending on the local value of g.


11.  Letting g = 10 m/s/s -->  40 m/s, 80 m

>

New questions (and answers):

1.  Explain the meaning of "inverse square law".

2.  Discuss each of Kepler's 3 laws.

3.  At what point in its orbit is the Earth closest to the Sun?

4.  At what point in its orbit is the Earth moving fastest?

5.  What causes seasons?

6.  What is a semi-major axis of orbit (a)?

7.  What is an Astronomical Unit (AU)?

8.  Consider Jupiter.  It's orbit is 5 AU in size (roughly).  How long should it take Jupiter to orbit the Sun once?  Show how this calculation would be done.

9.  What is the period of Earth's orbit around the Sun?

10.  What is the size of Earth's orbit (in AU)?

11.  When you stand on the Earth's surface, you experience your "normal" Earth weight.  What would happen to your Earth weight if you were one Earth radius above the surface?  (That's twice as far from the center as simply standing on the surface.)

12.  What does gravitational force between 2 objects depend on?

Some questions from Newton's laws:

13.  A 10-kg object is pushed on by a 200-N force.  What will be the acceleration?

14.  What is the weight of a 100-kg man?

15.  Would the answer to 3 be different if he was on the moon?  How so?

16.  Consider yourself standing on a scale in an elevator.  The scale reads your weight.  Compared to being at rest, how would the scale reading change (if at all) if the elevator were:

A.  Moving with constant velocity upward

B.  moving with constant velocity downward

C.  Moving with constant acceleration upward

D.  Moving with constant acceleration downward

E.  If the cable snapped (yikes!) and the elevator were falling

17.  A 50 lb child sits on a see-saw, 2 feet from the fulcrum.  Where should her 150 lb brother sit, so that they balance?

>

1.  Explain the meaning of "inverse square law".

The force (of gravity, in this case) gets progressively weaker by the factor 1 over the distance squared.  Double the distance --> force is 1/4 as great as it was.  Triple the distance --> force is 1/9 the original.

2.  Discuss each of Kepler's 3 laws.

See notes.

3.  At what point in its orbit is the Earth closest to the Sun?

Perihelion, which is approximately January 3-4 each year.

4.  At what point in its orbit is the Earth moving fastest?

Same point as 3 above.

5.  What causes seasons?

Tilt of Earth's axis.

6.  What is a semi-major axis of orbit (a)?

Half the longest distance across the orbital path (ellipse).

7.  What is an Astronomical Unit (AU)?

Defined as the semi-major axis of Earth's orbit - roughly 93,000,000 miles - or  half the longest width across Earth's orbit.

8.  Consider Jupiter.  It's orbit is 5 AU in size (roughly).  How long should it take Jupiter to orbit the Sun once?  Show how this calculation would be done.

5^3 = T^2

So, T = the square root of 125, or around 11 years.

9.  What is the period of Earth's orbit around the Sun?

1 year, or approximately 365.25 days.

10.  What is the size of Earth's orbit (in AU)?

Defined as 1 AU.

11.  When you stand on the Earth's surface, you experience your "normal" Earth weight.  What would happen to your Earth weight if you were one Earth radius above the surface?  (That's twice as far from the center as simply standing on the surface.)

1/4 your surface weight.

12.  What does gravitational force between 2 objects depend on?

mass of the objects; distance between; a universal (unchanging) constant (G)

13.  F = m a

200 = 10 a

a = 20 m/s/s

14.  W = m g

W = 100 g = 980 newtons

15.  Yes.  The weight would be smaller (1/6 as much, since Moon surface gravity is 1/6 that of Earth).

16.  a.   your regular weight

b.  your regular weight

c.  greater than your regular weight

d.  less than your regular weight

e.  zero!  (meaning that you are "weightless")

17.  Consider 'center of gravity' and torque.  

50(2) = 150 L

L = 100/150 = 0.67 feet from the fulcrum

How things balance

A very useful concept in physics is Center of Gravity (AKA CM, Center of Mass - they are usually the same point).  

Recall the demo with the mass on a stick.  Same mass, held at a further distance from the "fulcrum", is harder to support.  It twists your wrist more - it requires a greater "torque".

So, what is torque?

Torque - a "rotating" force

T = F L

For an object to be "in equilibrium," not only must the forces be balanced, but the torques must also be balanced.

Consider a basic see-saw, initially balanced at the fulcrum:  See image below.

You can have two people of different weight balanced, if their distances are adjusted accordingly:  the heavier person is closer to the fulcrum.  

Mathematically, this requires that the torques be equal on both sides.

Consider two people, 100 lb and 200 lb.  The 100 lb person is 3 feet from the fulcrum.  How far from the fulcrum must the 200 lb person sit, to maintain equilibrium?

Torque on left = Torque on right

100 (3) = 200 (x)

x = 1.5 feet

NOTE:  The weights are NOT equal on both sides of the balance point.  But the torques ARE EQUAL.

We call the "balance point" the center of mass (or center of gravity).  

It is the point about which the object best rotates.

It is the average weighted location of mass points on the object.

It does not HAVE to be physically on the object - think of a doughnut.

The principle is believed to originate with Archimedes (287 - 212 BC).  He is believed to have said, "Give me a place to stand on, and I will move the Earth."

FYI:  http://en.wikipedia.org/wiki/Archimedes