Light problems pt. 2

1.  Define "focal point."

2.  Do images always form at the focal point?  If not, what determines if/where an image is formed?

3.  How are convex and concave lenses different?

4.  How are convex and concave mirrors different?

5.  Revisit the final problem from the last problemset.  Draw what happens to parallel rays of light as they pass through:
a.  convex lens
b.  concave lens
c.  convex mirror
d.  concave mirror

6.  Play around with the lens applet used in class to see how image formation depends on the location of the object.

7.  Describe the difference between nearsightedness and farsightedness, and the lenses used to correct each.

8.  Describe the operation of a basic reflecting telescope.

Doppler and Light etc. problems

1.  Doppler effect.  A police car approaches you.  According to the siren's manufacturer, the frequency is 1000 Hz.  How would the frequency change if you were located:
a.  behind the car as it passed you
b.  in front of the car as it approached you
c.  in the car
d.  running toward the car, if it were at rest
e.  running away from the car, if it were at rest

2.  What does the red shift of distant galaxies suggest?

3.  What do red shift and blue shift mean?

4.  Explain the Doppler effect.

5.  Consider a 320 Hz note (E, approximately).  What is the frequency of:
a.  the next E (one octave above)
b.  the E one octave below
c.  an E that is 3 octaves above
d.  an F, one semi-tone above
e.  a G, three semi-tones above

(You don't have to calculate these, but show how it would be done.)

6.  Explain the idea of reflection (of light).  What is the 'law of reflection'?

7.  What happens during refraction?  What changes exactly?  (Hint:  more than one thing could change during refraction.)

8.  Light goes from air into a rectangular piece of glass.  If the light hits the glass at an angle of 45 degrees (with respect to the normal line), how is it refracted inside the glass?  Draw this situation.  Would the refraction angle be greater than, less than or the same as 45 degrees?

9.  Light passes from water into air - basically the opposite of the previous problem.  Light hits the surface of the water (from below) at an angle of 45 degrees.  Revisit the previous problem and answer/draw what is expected.

* Problem 9 can refer to "total internal reflection", or what happens when light hits the surface INSIDE the substance (water, plexiglas, etc.) at angle too great to leave.  This phenomenon allows for fiber optics, as well as strange effects seen when you look into an aquarium.

10.  Draw what is expected when parallel rays of light hit:
a.  convex lens
b.  concave lens
c.  convex mirror
d.  concave mirror
e.  flat mirror

Lenses

Lenses

As shown and discussed in class, light refracts TOWARD a normal line (dotted line, perpendicular to surface of lens) when entering a more dense medium.

Note, however, that this direction of bend changes from down (with the top ray) to up with the bottom ray. This is due to the geometry of the lens. Look at the picture to make sure that this makes sense.


The FOCAL LENGTH (f) of a lens (or curved mirror) where the light rays would intersect, but ONLY IF THEY WERE INITIALLY PARALLEL to each other. Otherwise, they intersect at some other point, or maybe not at all!

The location of images can be predicted by a powerful equation:

1/f = 1/di + 1/do

In this equation, f is the theoretical focal length (determined by the geometry of the lens or mirror), do is the distance between the object and lens (or mirror) and di is the distance from lens (or mirror) to the formed image.

We find several things to be true when experimenting with lenses. If the object distance (do) is:

greater than 2f -- the image is smaller
equal to 2f -- the image is the same size as the object (and is located at a di equal to 2f)
between f and 2f -- the images is larger
at f -- there is NO image
within f -- the image is VIRTUAL (meaning that it can not be projected onto a screen) and it appears to be within the lens (or mirror) itself

If an image CAN be projected onto a screen, the image is REAL. Convex lenses (fatter in the middle) and concave mirrors (like the inside of a spoon) CAN create real images - the only cases where there are no images for convex lenses or concave mirrors are when do = f, or when do < f. In the first case, there is NO image at all. In the second case, there is a magnified upright virtual image within the lens.

Concave lenses (thinner in the middle) NEVER create real images and ONLY/ALWAYS create virtual images. This is also true for convex mirrors (like the outside of a spoon, or a convenience store mirror).

Play around with this applet:

http://www.physics.metu.edu.tr/~bucurgat/ntnujava/Lens/lens_e.html

Convex lenses (which are defined to have a positive focal length) are similar to concave mirrors.

Concave lenses (which are defined to have a negative focal length) are similar to convex mirrors.


This is a bit more complicated, but here are some images and information for mirrors:

http://www.physicstutorials.org/home/optics/reflection-of-light/curved-mirrors/concave-mirrors


>

http://www.physics.metu.edu.tr/~bucurgat/ntnujava/Lens/lens_e.html

The key thing to note is that whether or not an image forms, and what characteristics that image has, depends on:

- type of lens or mirror
- how far from the lens or mirror the object is

In general, convex lenses and concave mirrors CAN form "real" images.  In fact, they always form real images (images that can be projected onto screens) if the object is further away from the lens/mirror than the focal length.

If the object is AT the focal point, NO image will form.

If the object is WITHIN the focal point, only virtual images (larger ones) will form "inside" the mirror or lens.

Concave lenses and convex mirrors ONLY form virtual images; they NEVER form real images.  Think of convenience store mirrors and glasses for people who are nearsighted.

http://www.physics.metu.edu.tr/~bucurgat/ntnujava/Lens/lens_e.html

Problems in light - 1

1.  All electromagnetic waves have the same _______.  Discuss.

2.  Explain the law of reflection.  What does it mean and what does it have to do with mirrors?

3.  When and why does refraction occur?

4.  Draw a picture that depicts a single light ray entering and exiting a glass rectangular block.

5.  Draw a picture that depicts a single light ray entering and exiting a convex lens.

6.  Repeat for a concave lens.

7.  Repeat for an equilateral prism.

8.  What is total internal reflection and when does it occur?

9.  What is an index of refraction?  Related to this, what is the point of high-index lenses?
10.  What is the difference between nearsightedness and farsightedness?

Refraction notes

Light 2 - Refraction

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. More to come here.

Refraction:

Refraction is much different. In refraction, light enters a NEW medium. In the new medium, the speed changes. We define the extent to which this new medium changes the speed by a simple ratio, the index of refraction:

n = c/v

In this equation, n is the index of refraction (a number always 1 or greater), c is the speed of light (in a vacuum) and v is the speed of light in the new medium.

The index of refraction for some familiar substances:

vacuum, defined as 1

air, approximately 1

water, 1.33

glass, 1.5

polycarbonate ("high index" lenses), 1.67

diamond, 2.2

The index of refraction is a way of expressing how optically dense a medium is. The actual index of refraction (other than in a vacuum) depends on the incoming wavelength. Different wavelengths have slightly different speeds in (non-vacuum) mediums. For example, red slows down by a certain amount, but violet slows down by a slightly lower amount - meaning that red light goes through a material (glass, for example) a bit faster than violet light. Red light exits first.

In addition, different wavelengths of light are "bent" by slightly different amounts. This is trickier to see. We will explore it soon.

Refraction, in gross gory detail

Consider a wave hitting a new medium - one in which is travels more slowly. This would be like light going from air into water. The light has a certain frequency (which is unchangeable, since its set by whatever atomic process causes it to be emitted). The wavelength has a certain amount set by the equation, c = f l, where l is the wavelength (Greek symbol, lambda).

When the wave enters the new medium it is slowed - the speed becomes lower, but the frequency is fixed. Therefore, the wavelength becomes smaller (in a more dense medium).

Note also that the wave becomes "bent." Look at the image above: in order for the wave front to stay together, part of the wave front is slowed before the remaining part of it hits the surface. This necessarily results in a bend.

The general rule - if a wave is going from a lower density medium to one of higher density, the wave is refracted TOWARD the normal (perpendicular to surface) line. See picture above.

http://stwww.weizmann.ac.il/lasers/laserweb/java/twoangles2.htm

http://lectureonline.cl.msu.edu/~mmp/kap25/Snell/app.htm

http://www.physics.uoguelph.ca/applets/Intro_physics/refraction/LightRefract.html

Light 1 - Electromagnetic Spectrum revisited

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 be the Greek letter, lambda) is wavelength (in m).

Sites

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

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

https://www.youtube.com/watch?v=aWAjLOwOHU0

http://www.acs.psu.edu/drussell/Demos/TuningFork/fork-modes.html

Doppler:

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

http://falstad.com/ripple/

Additional music questions

Consider concert A - 440 Hz.  Find the following:

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

2.  The frequency of the A below concert A.

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

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

5.  The wavelength of the 440 Hz sound wave, assuming that the speed of sound is 340 m/s.

The Doppler Effect

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.

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.)

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?

Wave problems

Problems

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 0.4 m.

3.  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.)

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

5.  Draw the first 3 harmonics for a wave on a string.  If the length of the string is 1-m, find the wavelengths of these harmonics.

6.  Show how to compute the wavelength of WTMD's signal (89.7 MHz).  Note that MHz means 'million Hz."

7.  Explain the Chladni plate seen in class.

8.  A C-note vibrates at 262 Hz (approximately).  Find the frequencies of the next 2 C's (1 and 2 octaves above this one).

Harmonics on a string

Introduction to Waves

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)

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

Waves have several characteristics associated with them, most notably: wavelength, frequency, speed. These 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.

Note the units:

Frequency is in hertz (Hz), also known as a cycle per second.

Wavelength is in meters or some unit of length.

Speed is typically in meters/second (m/s).

Sound waves

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.

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.

We looked at specific cases of waves interfering with each other – the case of “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.

Answers to Gravitation questions

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.

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)