Harmonics revisited.

The above picture represents "harmonics" on a string - waves that form and give the maximum possible amplitude.  A string can vibrate at any frequency, but certain frequencies will make the wave BEST vibrate - with the greatest height/amplitude.  These waves are called harmonics and they occur at frequencies which are integer multiples of the lowest possible "resonant" frequency.  For example, if the frequency will produce a simple 1 "hump" pattern (n = 1) at 10 Hz, the next harmonic (2 "humps" or n = 2) will be at 20 Hz.  The next harmonic (n = 3) is at 30 Hz, and so on.

Another way to put it:

Harmonics are a specific case of waves "interfering" with each other – making “standing waves” or “harmonics.” Here we see that certain frequencies produce larger amplitudes than other frequencies.  There is a lowest possible frequency (the resonant frequency) that gives a “half wave” or “single hump”. Every other harmonic has a frequency that is an integer multiple of the resonant frequency. So, if the lowest frequency is 25 Hz, the next harmonic will be found at 50 Hz – note that that is 1 octave higher than 25 Hz. Guitar players find this by hitting the 12^th fret on the neck of the guitar. The next harmonics in this series are at 75 Hz, 100 Hz and so on.

Sound waves are mechanical - that is, they require a medium.  There can be NO sound where there is no medium through which to travel.  That makes them very different from light (or any e/m waves).

Also note that it is easy to calculate the wavelength of a harmonic.  The formula is:

wavelength = 2 L / n

That means, it is twice the length of the string divided by the harmonic number.


Useful terms in music:

In music, the concept of “octave” is defined as doubling the frequency. For example, a concert A is defined as 440 Hz. The next A on the piano would have a frequency of 880 Hz. The A after that? 1760 Hz. The A below concert A? 220 Hz. Finding the other notes that exist is trickier and we’ll get to that later.

Interference:

Waves can “interfere” with each other – run into each other. This is true for both mechanical and e/m waves, but it is easiest to visualize with mechanical waves. When this happens, they instantaneously “add”, producing a new wave. This new wave may be bigger, smaller or simply the mathematical sum of the 2 (or more) waves. For example, 2 identical sine waves add to produce a new sine wave that is twice as tall as one alone. Most cases are more complicated.

In music, waves can add nicely to produce chords, as long as the frequencies are in particular ratios. For example, a major chord is produced when a note is played simultaneously with 2 other notes of ratios 5/4 and 3/2. (In a C chord, that requires the C, E and G to be played simultaneously.) Of course, there are many types of chords (major, minor, 7ths, 6ths,…..) but all have similar rules. In general, musicians don’t remember the ratios, but remember that a major chord is made from the 1 (DO), the 3 (MI) and the 5 (SO). It gets complicated pretty quickly.

Harmonics in tubes - also, transverse and longitudinal waves:

I will demonstrate this in class.  Here are relevant sites which show longitudinal waves (as well as transverse waves).  You know the difference between mechanical and EM waves - mechanical waves need a medium, but EM waves do not (though they can travel in many different mediums, but at a speed less than the speed of light).  Waves can also be classified according to shape:  transverse (standard-looking waves, like sine waves, which move perpendicular to the direction of wave travel.  Longitudinal waves move parallel to the direction of wave travel.  The animations below show this better than words can express:

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

http://www.animatedscience.co.uk/blog/wp-content/uploads/focus_waves/tl-wave.html

Note that light is a transverse wave, but sound is a longitudinal (or compression) wave.

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.

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.

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.  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.

So, 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!

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 topics

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)
gravitational acceleration (g)
freefall problems
Newton's 3 laws - applications and problems
Kepler's 3 laws - applications and problems
epicycles
Newton's law of universal gravitation (inverse square law)
weight vs. mass
center of mass/gravity

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

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 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

New practice problems and questions 2 (Kepler)

Gravitation and flight questions and problems:

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

(Answers below.)

>


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 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")