Light 2 - Reflection

Reflection - light "bouncing" off a reflective surface. This obeys a simple law, the law of reflection!

The incident (incoming) angle equals the reflected angle. Angles are generally measured with respect to a "normal" line (line perpendicular to the surface).

Note that this works for curved mirrors as well, though we must think of a the surface as a series of flat surfaces - in this way, we can see that the light can reflect in a different direction, depending on where it hits the surface of the curved mirror.

So - light reflects from mirrors, according to the law of reflection.  However, if the mirrors is curved, light still obeys this rule - it just looks a bit different.  You have to visualize the curved mirror as a series of little flat mirrors.

A convex mirror (top) acts reflects light rays "outward" - the light rays seem as though they are coming from inside the convex mirror, so it seems as though there is an image inside.  We call this a VIRTUAL IMAGE.  Think of convenience store mirrors or side view mirrors.

 A concave mirror (bottom) acts sort of the opposite way.  The parallel light rays bend "inward" - so the light rays converge at a FOCAL POINT.  Where they meet, an image is formed - we call this a REAL IMAGE.

Note however that this happens in this case because the light rays were initially parallel (which is what happens if the object the light rays reflect from is far away).  If they are NOT initially parallel - in other words, if the object is reasonably close to the mirror, the rays may converge at some other point.  Examples of concave mirrors are found in makeup/shaving mirrors and reflecting telescope mirrors.  But again - the light rays ONLY meet at the focal point IF they were initially parallel.  If not, they meet elsewhere (or maybe not at all).  More about this next class when we talk about refraction and lenses.

Light 1 - review of the basics of EM radiation

Recall that waves can be categorized into two major divisions:

Mechanical waves, which require a medium. These include sound, water and waves on a (guitar, etc.) string

Electromagnetic waves, which travel best where there is NO medium (vacuum), though they can typically travel through a medium as well. All electromagnetic waves can be represented on a chart, usually going from low frequency (radio waves) to high frequency (gamma rays). This translates to: long wavelength to short wavelength.

All of these EM waves travel at the same speed in a vacuum: the speed of light (c). Thus, the standard wave velocity equation becomes:

c = f l

where c is the speed of light (3 x 10^8 m/s), f is frequency (in Hz) and l (which should actually be the Greek letter, lambda) is wavelength (in m).

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

Don't forget - electromagnetic waves should be distinguished from mechanical waves (sound, water, earthquakes, strings on a guitar/piano/etc.). 

ALL E/M waves (in a vacuum) travel at the SPEED OF LIGHT (c).

Wave problems 1 and 2, with answers

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.  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.

6.  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.  v = f l

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

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

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

524 Hz, 1048 Hz

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

Doppler!!!

The Doppler Effect

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

http://falstad.com/mathphysics.html

Run the Ripple tank applet -

http://falstad.com/ripple/

The key in the Doppler effect is that motion makes the "detected" or "perceived" frequencies higher or lower.

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. 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/

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 move away from a 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.

The sound of music!

Music

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?

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.

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.

Newton's take on gravitation

Newton's take on gravity and orbits - which is the genesis of our modern conception of it, is based on:

Universal Gravitation (1687, Principia)

Newton's take on orbits was quite different. For him, Kepler's laws were a manifestation of the bigger "truth" of universal gravitation. That is:

All bodies have gravity unto them. Not just the Earth and Sun and planets, but ALL bodies (including YOU). Of course, the gravity for all of these is not equal. Far from it. The force of gravity can be summarized in an equation:

or.... the force of gravitation is equal to a constant ("big G") times the product of the masses, divided by the distance between them (between their centers, to be precise) squared.

Big G = 6.67 x 10^-11, which is a tiny number - therefore, you need BIG masses to see appreciable gravitational forces.

This is an INVERSE SQUARE law, meaning that:

- if the distance between the bodies is doubled, the force becomes 1/4 of its original value
- if the distance is tripled, the force becomes 1/9 the original amount
- etc.

Weight

Weight is a result of local gravitation. Since F = G m1 m2 / d^2, and the force of gravity (weight) is equal to m g, we can come up with a simple expression for local gravity (g):

g = G m(planet) / d^2

Likewise, this is an inverse square law. The further you are from the surface of the Earth, the weaker the gravitational acceleration. With normal altitudes, the value for g goes down only slightly, but it's enough for the air to become thinner (and for you to notice it immediately!).

Note that d is the distance from the CENTER of the Earth - this is the Earth's radius, if you're standing on the surface.

If you were above the surface of the earth an amount equal to the radius of the Earth, thereby doubling your distance from the center of the Earth, the value of g would be 1/4 of 9.8 m/s/s. If you were 2 Earth radii above the surface, the value of g would be 1/9 of 9.8 m/s/s.

The value of g also depends on the mass of the planet. The Moon is 1/4 the diameter of the Earth and about 1/81 its mass. You can check this but, this gives the Moon a g value of around 1.7 m/s/s. For Jupiter, it's around 2.5 m/s/s.

>

The current view of gravitation is much more subtle, and is largely the work on Einstein.  Gravity is a function of the geometry of space - a curvature caused by the presence of mass, affecting space and time around it.  If that seems complex, well, it is.  The theory is known as the General Theory of Relativity.

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

Frames of reference - video links inside

http://www.iflscience.com/physics/what-happens-when-you-fire-ball-out-cannon-travelling-opposite-way/

Links from class - thanks, Brianna!

https://www.youtube.com/watch?v=RvD7mFT1eT4

https://www.youtube.com/watch?v=AO-Nbpbe8Sk

https://www.youtube.com/watch?v=7GRZQUN1ADQ

https://www.youtube.com/watch?v=6KL9jBCnsMc

https://www.youtube.com/watch?v=p-klYOgcYHg



Things to remember from tonight's class demonstrations.

- the ball shot from the cart lands in the cart, because it was moving at the same speed as the cart.  This is similar to Galileo's ship problem, or the idea of apples falling near the base of the apple tree.

- the ball released from the top of the hook (on the cart) still lands in the cart itself, for the same reason:  ball is moving at the same speed as the cart.

Physics - Yay!!!