Einstein notes

Worth a read (from an Einstein scholar and former prof of mine):

http://www.pitt.edu/~jdnorton/Goodies/Einstein_and_S/index.html

Notes from class:

And then there was Einstein…

Albert Einstein 1879-1955

http://www.aip.org/history/einstein/index.html

What’s happening around the turn of the 20th century? Physics was set to explode with 30 brilliant years of excitement and unprecedented activity

X-rays – Roentgen

Radioactivity – Becquerel, Marie & Pierre Curie

Blackbody radiation (and the quantum discontinuity) – Planck

1905/6 – Einstein publishes 6 major papers:

a) “On the electrodynamics of moving bodies”

b) “Does the inertia of a body depend upon its energy content?”

c) “On a heuristic point of view about the creation and conversion of light”

d) “On the theory of the Brownian movement”

e) “On the movement of small particles suspended in stationary liquid demanded by the molecular-kinetic theory of heat”

f) “A new determination of molecular dimensions”

What are these about anyway?

a. Special relativity (SR)

b. E = m c2 (actually, L = m c2)

c. Photoelectric effect, light quanta, fluorescence

d. Same as title

e. Brownian motion agan

f. Avagadro’s number, etc.

Now, these are interesting (and very different fields of study), but is this why we revere Uncle Al? Not necessarily. Others (Poincare, Lorentz) were working on what would become SR. Planck had introduced the quantum discontinuity (E = h f) and quantum mechanics (QM) would have many contributors. The photoelectric effect had also several investigators (Lenard, et.al.).

Mostly, Einstein’s legend grows because of General Relativity (GR), which appears 1912-1915 and later and on which he worked largely alone with pad and pen. He forced us to re-examine how we see ourselves in the universe; indeed, how we think of gravitation. All of this around the time his marriage was falling apart (he married young after fathering 1 illegitimate child) and he began an affair with his cousin (whom he would later marry). Also, between 1902 and 1909, Einstein held a modest post in Bern, Switzerland as a Patent Clerk. By 1914, he would be director of the Kaiser Wilhelm Institute (later Max Planck Institute).

Special Relativity

Spaceship – Inside an inertial reference frame (constant velocity), you can’t tell whether or not you’re moving (“Principle of Relativity”)

Biographical notes

1879 – born in Ulm, Germany

1884 – receives first compass

1895 – attempts to gain entrance to Swiss Polytechnic (and finish high school early), but is rejected

1896 – begins Federal Polytechnic (ETH) in Zurich, Switzerland

1898 – meets Mileva Maric

1900 – graduates from ETH

1901 – Einstein becomes Swiss citizen and moves to Bern; Mileva becomes pregnant

1902 – Lieserl born (put up for adoption); Hermann dies

1903 – Albert and Mileva marry

1904 – Hans Albert born

1905 – Einstein’s “Annus Mirabilus”, his miracle year; Ph.D. (Zurich)

1919 – divorces Mileva (having lived apart for 5 years); marries Elsa; GR verified

1921 – awarded the Nobel Prize in Physics "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect".

1933 – settles in Princeton, NJ

1936 – Elsa dies

1939 – E. writes FDR

1940 – E. becomes American citizen

1949 – Mileva dies

1955 – E. dies

http://www.aip.org/history/einstein/index.html

http://www.albert-einstein.org/

http://einstein.stanford.edu/

http://en.wikipedia.org/wiki/Einstein

General Relativity

By 1907, E. wanted to advance the SR theory to include non-inertial (accelerated) frames of reference. Around this time, E. has the “happiest thought of my life”. In a uniformly accelerated spaceship, a stationary thing (ball, etc.) would appear to be falling (accelerating) down – it would be indistinguishable from normally accelerated motion. Light, too, follows this idea. Two clocks at different ends of an accelerated spaceship would be out of sync. Gravity is the result of the curvature of space and time?

Why does the Earth follow the Sun? Gravity – the very presence of the Sun causes Earth to veer from its otherwise straight (Newtonian inertial) path. With the Sun, it takes an elliptical path as its natural motion. Getting to this point, and showing that mass alters space and time is the real genius.

There is a breakdown of the observed geometry (Euclidean). E. must use non-Euclidean geometry (Gauss, surfaces, infinitesimal geometry) to consider the behavior of things (rods, etc.) on surfaces. He also considers the shortest distance between 2 points on a sphere (geodesic, great circle). He obtains the mathematical advice of his friend Marcel Grossman and studies (at great length) tensor calculus, differential geometry, Riemann and Minkowski math … it’s all puzzle solving. Soon, the principle of equivalence emerges. Eventually, the gravitational field equations appear (to show how matter “produces” gravity. GR was mostly worked out by 1913

Skepticism 101 FYI

Related to our brief foray yesterday into all things skeptical.

Good books, sites, etc.

by Michael Shermer:

Why people believe weird things

The believing brain

The science of good and evil

Science friction

Why Darwin matters

Skeptic Magazine

http://www.skeptic.com/

http://www.michaelshermer.com/

James "The Amazing" Randi

Flim Flam

Conjuring

The Faith Healers

An encyclopedia of claims, frauds and hoaxes of the occult and supernatural

http://www.randi.org/site/

Skeptical Inquirer

http://www.csicop.org/si/

Skeptic's Dictionary

http://skepdic.com/

Richard Feynman - "Cargo Cult Science" essay

http://www.lhup.edu/~DSIMANEK/cargocul.htm

Martin Gardner

Fads and fallacies in the name of science

Carl Sagan

The demon-haunted world

Richard Dawkins

Climbing mount improbable

Schick/Vaughn

How to think about weird things

Other good essays and sites:

http://www.lhup.edu/~dsimanek/pseudo/scipseud.htm

http://www.quackwatch.org/01QuackeryRelatedTopics/pseudo.html

http://www.rationallyspeaking.org/

http://skepticblog.org/

More problems, in prep for final.

Final topics:

circuit basics (series, parallel, power, current, voltage, resistance)

combination circuits

magnetism, electromagnetism, electromagnetic induction

special theory of relativity

>

1. Find the charge that "flows" past a given point, if a 12-V battery is in series with a 3-ohm resistor.

2. Consider 4 identical resistors, connected to a power supply. Two are in series, and these are connected to two that are in parallel. If each resistor is a light bulb, which one(s) are brightest? (Hint: Consider the equation for power, particularly P = i^2 R)

3. In the above problem, if one of the series "bulbs" are removed, what happens to the brightness of all of them? If one of the parallel bulbs are removed, what happens to the brightness of all of them?

4. Review the meaning and subtleties of magnetic fields, especially what compasses do in a magnetic field.

5. Consider a coil (100 turns) of wire, with a radius of 0.1-m. A magnetic field increases from 0-20 T in 5 seconds, directly onto the coil. What voltage is induced?

6. Distinguish between motors and generators.

7. Distinguish between speakers and microphones.

8. Imagine two twins: Earthy and Spacey. Earthy stays home, while Spacey travels at 0.5c to a star system 5 light-years away. How long will this trip last according to each twin? There are two answers for this question. Also, leave the distance (5 light-years) in light-years (or c-years). This makes the time work out to years, and in general, makes the math much easier. No conversions to do.

Problems a-plenty

Consider 4 resistors, each 10 ohms. Two are in series - these are followed by the other two resistors which are in parallel. They are connected to a 20 volt battery. Find all currents and voltages.

Draw the magnetic field around: a bar magnet, a current carrying wire, and a coil with current.

A magnetic field is directed into the page. A 0.3 meter long wire carries 2 amperes of current to the left. How strong is the magnetic field if the force on the wire is 5 newtons? Also, which way is the force?

Consider the previous problem. If a proton had been shot to the right in this field, what would the path look like? Draw.

Define and give units for all recently defined quantities.

Circuit info

Some information about circuits, in general.

voltage (V) = energy(work)/charge

V = W/q

The unit is the volt (joule/coulomb).


current (I) = charge/time

I = q/t

The unit is the ampere (coulomb/second).


Resistance = voltage/current

R = V/I

The unit is the ohm (volt/ampere).


The last relationship is often referred to as Ohm's Law, typically written as:

V = I R

Furthermore, power (used or radiated) in a circuit can be expressed by:

P = I V = I^2 R = V^2 / R

The unit is the joule/second, also called a watt (W).



Series circuit reminders

In a simple series circuit:

The current is the same in each resistor
The voltages ("over" each resistor) add to the total voltage (battery) available
The total resistance of the circuit is equal to the sum of the individual resistances -
Rs = R1 + R2 + R3 + .....


Parallel circuit reminders

In a simple parallel circuit:

The voltage is the same over each resistor
The currents ("through" each resistor) add to the total current (battery) available
The total resistance of the circuit is equal to the inverse of the sum of the inverted individual resistances. That is -

1/Rp = 1/R1 + 1/R2 + 1/R3 + ......


Combination circuits

Solving combination circuits (series and parallel together) is not too difficult, IF you break the circuit down to a simpler one first.

Determine what resistors are in series (and add them appropriately) and what resistors are in parallel (and add them appropriately). You'll have a simpler circuit that should be able to be simplified even further. We will examine this in class with several examples. Here is one to consider:

Imagine having 2 resistors (10 ohms and 20 ohms) in series with each other. These two are in series with a pair of resistor (3 ohms and 6 ohms) in parallel with each other. The combinaton is powered by a 12-V battery. What is the total resistance of this circuit?

The 2 series resistors make 30 ohms. The two parallel resistors make 2 ohms (do the math). The total combination makes 32 ohms.

The next step would be to find the total current. Take the total voltage (12-V) and divide it by the total resistance (32 ohms). This will give you the battery current. And since the battery is directly in series with the 10 and 20 ohm resistors, THEY TOO have that same current.

More to see in class this evening.

Exam 2 Practice

Test 2 is tomorrow (Wednesday, 8/3). Play with these problems for some practice:

1. Determine the charge due to a cluster of one million electrons.

2. If the one million electron cluster (from above) were 0.5-m from a half-million cluster of protons, what force would exist between them? (Calculate this.)

3. Consider a convex mirror (f = -0.25 m). A 10-cm tall object is 0.5-m in front of it. Find:
a. the location, type (R or V), magnification, and orientation (up or down) of the image

4. Consider a red laser (632nm) hitting a diffraction grating (1000 lines/mm). Find the separation (on a wall, 1-m away) between the central image and the n=1 image.

Emergency

Sorry folks - I had a family emergency come up on the way to class this evening - I tried calling the campus offices, but was redirected to people who could not help me. I hope at least that you were able to work on labs with Mr. Scott.

For Monday, work on posted blog problems and be prepared to learn about voltage, current and resistance. In preparation, please investigate the meanings of these terms. You can also look up series circuits, parallel circuits and combination circuits.

Wednesday's test will cover lenses/mirrors, diffraction and interference and electrostatics (Coulomb's law and electric fields). Do take some time to play with the e-field applets.

See you Monday.

SL

Problems in Diffraction / Electrostatics

1. Consider a diffraction grating, marked at 100 lines (slits) per mm. A 632nm laser hits it. A screen is 0.75-m away from the grating. Find the following:

a. the distance between slits (in mm)
b. the distance between slits (in m)
c. the diffraction angle for n=1
d. the distance between n=0 and n=1 on the wall (which is 0.75-m away)
e. the highest order (n) that you can get from this grating and laser combination

2. Repeat the above problem for a 450nm laser.

3. Explain superposition of waves.

4. Explain diffraction.

5. Two identical charges are 0.25-m apart. If the force between them is 25-N, what is the magnitude of each charge? Is this force attractive or repulsive? Can you tell the sign of the charges?

6. Consider a 100 uC (10^-6) charge, 0.01-m away from a -300 uC charge. Find the following:

a. the force between them
b. whether or not this force is attractive or repulsive
c. the new force, if the distance is doubled
d. draw the electric field between the charges

In preparation for next class:

7. Define voltage (electric potential), current, resistance and power. Also, give the units for each.

8. What is Ohm's Law?

9. If a 9-V battery is in series with a 25-ohm resistor, what current is drawn from the battery? How much charge "flows" during one minute? What is the power radiated (in heat) by the resistor?

Electric Fields Applets

In 2-d:

http://falstad.com/vector2de/

In 3-d (much cooler):

http://falstad.com/vector3de/

Also worth a look:

http://www.cco.caltech.edu/~phys1/java/phys1/EField/EField.html

http://www.cbu.edu/~jvarrian/applets/efield1/elefi_z.htm

Interference and Diffraction

http://www.falstad.com/ripple/index.html

This is the "ripple tank" applet I showed in class. Play around with 2-source interference and note locations of constructive and destructive interference. This also happens with light waves.

The mathematical relationship:

n lambda = d sin(theta)

n is the "image order number," going from 0 (central image, directly in line with the light source) to n=1 (first order image, the same on either side of the central image), to n=2, etc.

lambda is the wavelength of light

d is the distance of separation between "slits"

theta is the angle of diffraction

Lens and Mirror problems

1. Compare and contrast convex and concave lenses.

2. Compare and contrast convex and concave mirrors.

3. Consider a lens, f = +12cm, with an object located 20cm in front of it. Find the following:
a. type of lens
b. di
c. type of image (real or virtual)
d. magnification of image
e. whether or not image is upside-down or right-side up
f. Where could you place object so that you get NO image?
g. Where could you place object so that you only get virtual images?

4. Repeat question 3 for a lens with f = -12cm.

5. Repeat question 3 for a mirror with f = +20cm.

6. Give a practical use for a convex lens, concave lens, convex mirror and concave mirror. (One for each.)

7. What is, in general, the effect of covering a lens or mirror in half?

Lens / Mirror Applet

http://www.phys.hawaii.edu/~teb/java/ntnujava/Lens/lens_e.html

Recall:

Lenses:
+f, convex lens (can form both real and virtual images, depending on do)
-f, concave lens (forms ONLY virtual images, since light rays always diverge)

Mirrors
+f, concave mirror (can form both real and virtual images, depending on do)
-f, convex mirror (forms ONLY virtual images, since light rays always diverge)

1/f = 1/di + 1/do

f = (theoretical) focal length
di = image distance (where image forms)
do = object distance (where object is located, relative to lens or mirror)

-di indicates a virtual image
+di indicates a real image

mag = -di/do

-mag indicates upside-down image
+mag indicates right-side up image

If absolute value of mag is > 1, image is larger than object.
If absolute value of mag is < 1, image is smaller than object.

Chladni Art

http://www.wasserklangbilder.de/

Reflection and Refraction

REFLECTION

You may recall from class the simple, elegant Law of Reflection:

angle of incidence equals angle of reflection.

Think about pool balls hitting the side of a billiards table - angle in equals angle out.

The only tricky part is the way we measure angles - they are measured with respect to a "normal line", a line that is perpendicular to the surface where they hit.


REFRACTION

Refraction refers to a wave changing mediums - going from air to glass, air to water, water to air, glass to air, etc. We first begin by defining a new quantity, the index of refraction (n):

n = c/v

where c is the speed of light and v is the speed of light in the NEW medium. Indices of refraction are always greater than (or approximately equal to) one, and have NO units.

For example, if a substance (say, glass) slows down light to 2/3 of the speed of light (in a vacuum), its index is:

n = c/(2/3 c) = 1.5

A convenient relationship can be derived that relates the angle of incidence and the angle of refraction, along with the indices of refraction of the two mediums. It is called Snell's Law:

n1 sin(theta 1) = n2 sin(theta 2)

As before, the angles are measured with respect to a normal (perpendicular) line. It may be helpful to remember:

- When light goes from a lower density medium (n1) to a higher density medium (n2 > n1), the light ray is refracted TOWARD the normal line. And vice versa.


CRITICAL ANGLE

There is an angle, above which light can not leave the medium. Imagine a light ray trying to go from water into air. Clearly, the light ray will refract AWAY from the normal line. If you gradually increase the angle of incidence (theta 1), eventually the refracted angle (theta 2) will become 90 degrees - light "skating" across the surface.

Any theta 1 greater than this angle will result in "total internal reflection", wherein the light simply cannot leave the substance - it is reflected back inside the original medium. This is the secret of fiber optics. The mathematics come from Snell's Law:

n1 sin(theta 1) = n2 sin(90)

n1 sin(critical angle, ic) = 1 (1)

sin ic = 1/n

That is, the sine of the critical angle equals 1 over the index of refraction for that particular medium (assuming that medium 2 is air, so that n2 = 1).

Got it? Good!

Waves and Sound notes

Here is a skeleton outline of the notes from the past few classes.

Waves - a periodic disturbance, typically oscillating with sinusoidal behavior. Imagine a simple harmonic oscillator moving through space.

Types of waves (in general):

mechanical - require medium
electromagnetic - do not require a medium, and travel at the speed of light (in a vacuum)
c = 3 x 10^8 m/s

Types of wave (by geometry):

transverse - disturbance is perpendicular to wave speed
longitudinal - disturbance is parallel to wave speed

Wave speed, v = frequency x wavelength

Standing waves on a string:

There are harmonics - "standing" waves which occur at points where the energy maximizes the displacement of the string. A string can typically vibrate at any frequency, but some are dramatically better than others. These frequencies are called harmonics.

wavelength = 2L / n

This is an expression for the wavelength of given harmonic (with harmonic number n)

To find the frequency (or speed), use the wave speed expression above.


For organ pipes open at both ends, the mathematical treatment is very similar. The waves, however, are NOT transverse - they are longitudinal. That is, the sound jiggles back and forth, NOT up and down. However, we can represent the motion of the particles (in terms of particle density inside the tube) as a sine function with anti-nodes on both ends. This gives rise to the pictures we saw in class (and on the applets posted earlier).

If the tube is capped at one end, the wavelengths are given by 4L/n. Furthermore, due to this geometry, the tube can only generate ODD numbered harmonics.


The Doppler Effect

The changed in perceived/detected frequency, due to relative motion between source and observer. See previously posted applets.

f' = f [(v +/- vd) / (v -/+ vs)]

where v is the speed of sound, vd is the speed of the detector, and vs is the speed of the source.

Problems in Snell's Law

1. Define index of refraction.

2. Is it possible for an index of refraction to be less than one? Why or why not?

3. Consider a light beam hitting a block of plexiglas (index = 1.6). It hits at an angle of 25 degrees, with respect to a normal line at the surface. Find the angle of refraction inside the material.

4. A light ray hits the side of an equilateral prism on its left side, striking it at an angle of 30 degrees with respect to a normal line. If it refracts at an angle of 20 degrees inside the prism, what is the index of refraction of the prism? Also, draw an diagram that represents this problem.

5. What is a critical angle of refraction?

Funny, via XKCD.com

Applets and sites worth your time

http://www.physicsclassroom.com/

A good source for general help.

Organ Pipe

http://www.walter-fendt.de/ph14e/stlwaves.htm

Doppler Effect

http://www.astro.ubc.ca/~scharein/a311/Sim/doppler/Doppler.html

(Enable "source motion" to make the source move.)

http://www.fisica.uniud.it/~deangeli/applets/Multimedia/Waves_java/Doppler/doppler.htm

http://www.falstad.com/ripple/ex-doppler.html

Waves on a string

http://zonalandeducation.com/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html

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

http://hep.physics.indiana.edu/~rickv/Standing_Waves_on_String.html

Wave addition

http://www.smaphysics.ca/phys30s/waves30s/waveadd1.html

Interference

http://www.walter-fendt.de/ph14e/interference.htm

http://paws.kettering.edu/~drussell/Demos/superposition/superposition.html

Sound and Wave problems

1. Consider a string vibrating on a guitar at 320 Hz (E note). If the string length is 0.6-m, find the following:

a. wavelength of first 3 harmonics

b. frequency of first 3 harmonics

c. speed of first 3 harmonics

2. Now imagine an organ pipe, open on both ends. The speed of sound is 340 m/s. Find the following:

a. length of tube needed to create a resonant frequency of 150 Hz.

b. frequencies and wavelengths of first 3 harmonics

c. wave shapes of first 3 harmonics

d. the effect of capping the tube at one end

3. Most instruments in western music are tuned to a 440 Hz (A) standard. That is, the note middle A is defined as 440 Hz. Find the following frequencies:

a. one octave below

b. two octaves below

c. 1 semi-tone (piano key) above

d. 3 semi-tones above

4. (In advance of next class) Consider a police car, traveling at 35 m/s, with a siren at 1000 Hz. Find the frequencies when the car does the following:

a. approaches you (with you at rest)

b. passes you (with you at rest)

c. approaches you (with you traveling toward the car at 15 m/s)

d. passes you (with you traveling away from the car at 15 m/s)

5. Distinguish between mechanical and electromagnetic waves, giving examples.

6. Distinguish between transverse and longitudinal waves, giving examples.

7. What is the frequency of a 120 x 10^-11 m X-ray?

Harmonics on a string

Simple harmonic motion:

http://positron.ps.uci.edu/~dkirkby/music/html/demos/SimpleHarmonicMotion/index.html


Harmonics on a stretched string/spring - "standing waves"

http://zonalandeducation.com/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html


Superposition (adding) of waves:

http://www.phy.ntnu.edu.tw/ntnujava/index.php?PHPSESSID=83b90d00c3a8b37a065ec5abe61acb7f&topic=19.msg124#msg124

Adding harmonics with instruments:

http://www13.fisica.ufmg.br/~wag/transf/TEACHING/ONDAS/Library_thinkquest/Harmonics.html

Einstein references

John Norton's "Einstein for Everyone" e-book:

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/index.html

Supplementary readings:

J. Schwartz and M. McGuinness, Einstein for Beginners. New York: Pantheon.

J. P. McEvoy and O. Zarate, Introducing Stephen Hawking. Totem.

J. P. McEvoy, Introducing Quantum Theory. Totem.

Pre-Final Problems

1. Consider a 7-m long pendulum (a Foucault pendulum, typically used to demonstrate the revolution of the Earth). Find its period on Earth, and on the Moon.

2. How long must a pendulum be such that its period is 0.5 seconds?

3. A spring-mass oscillator has a period of 0.8 seconds. What are the first 3 times where the oscillator will have its maximum speed?

4. What is the acceleration due to gravity at a point above the surface of the Earth equal to the radius of the Earth?

5. Mercury orbits the Sun once every 88 days. What is the size of its orbit (semi-major axis) and what is its average speed around the Sun (in km/hr)?

6. What is the angular velocity of a 33 1/3 album? If it takes 0.5 seconds to accelerate up to this speed, what is the angular acceleration and how many turns does it take to get up to this speed?

*7. Two masses (2 kg and 8 kg) are 5-m apart. Where is the center of mass located (as measured from the 2 kg mass)? Hint: consider one distance as x and the other distance as (5-x), then solve for x.

8. A meter stick is set up such that the fulcrum is located at the 25-cm mark. If a 100-g mass is at the 15-cm mark, what is the mass of the meter stick (assuming that it is at the 50-cm mark).

Simple Harmonic Motion

Practice Problems

Gravitation

1. Consider Jupiter, which has an orbital size (a) of 5 AU.

- How long does it take to orbit the Sun once?

- What exactly is 5 AU, in this problems?

2. If an asteroid were discovered that took 2.5 years to orbit the Sun once, how far away from the Sun must it be (on average)?

3. Consider the planet Mars, with mass 1/10 that of Earth and a radius 1/2 as much. What is its acceleration due to gravity? Also, if it is 1.8 AU from the Sun, how long does it take to orbit the Sun? Finally, what is its average speed (in km/sec) around the Sun? To do this, you'll need to convert AU to km first.

Torque and Center of Mass

4. On a see-saw, a 40-kg child is located 1.5-m away from the fulcrum. Where must a 75-kg adult be located, to balance with the child?

5. In the above problem, the 40-kg child now moves twice as far away from the fulcrum as she originally was. A third child (25-kg) wanders in. If the adult remains in the same location as above, where can the third child sit to balance the see-saw?

Rotation

6. If a cd can go from rest to 400 revolutions per minute in 4 seconds, find the following:

a. the final angular velocity (in radians/sec) - this is a conversion

b. the angular acceleration required to get to this angular velocity

c. the linear speed of a point at the edge of the cd (radius = 0.06 m)

13 - Angular Motion / Rotation

12 - Center of Mass and Torque

11 - Gravitation -- Kepler and Newton

As discussed in class, Kepler's laws were based on Tycho Brahe's massive amount of data. The laws can be summarized as follows:

1. Planetary orbits are elliptical with the Sun at one focus.

2. Planets sweep out equal areas in equal amounts of time.

3. The square of the period of orbit is proportional to the cube of the semi-major axis. If the units are years and AUs, this is an equality:

T^2 = a^3

e.g, Consider an asteroid with a 4 AU semi-major axis of orbit. How long does it take to orbit once?

Answer: 8 years

>

Several decades after Kepler, Newton derived (from geometry) his law of universal gravitation. The derivation is prohibitive to discuss here, but it can be found in Principia (prop. 71). I don't recommend that you read it - many key points are omitted by Newton. In modern language:

F = G m1 m2 / r^2

That is, the force of gravitational attraction is equal to a constant (6.67 x 10^-11 Nm^2/kg^2) times the product of the masses, divided by the distance between the masses squared.

Setting this equal to the local force of gravity (weight) yields a simple expression for local gravitation:

g = G m(planet) / r^2

Finally, we saw in class how Newton's law of gravitation, along with the expression for centripetal acceleration (v^2 / r) can yield Kepler's third law. That is, Newton's law was powerful enough to predict anything known before it, as well as make predictions about the future.

Practice Problems

1. Consider a 0.1-kg coconut, falling from a tree 4-m above the ground. Find the following:

a. type (and amount) of energy before it falls
b. type (and amount) of energy right before it hits the ground
c. velocity right before it hits the ground
d. velocity at a point 1-m above the ground

2. Two toy cars collide on a track. A red car (0.2-kg) is traveling at 4 m/s and it smacks into the back end of a blue car (0.4 kg) traveling at 2 m/s in the same direction. They stick together. Find:

a. the type of collision that this is
b. the velocity of the red/blue cars (stuck together) immediately after the collision
c. what happens to the kinetic energy of the system before and after the collision. Calculate it, if you're not sure

3. Explain the following:

a. conservation of energy
b. conservation of momentum
c. elastic vs. inelastic collisions
d. the difference between mass and weight (and how to calculate it)

4. Two cars spring (explode) apart. If the first (1-kg) moves to the left at a final speed of 3 m/s, what is the speed of the second car (0.5-kg)?

5. Revisit the Newton's 2nd Law track problem: A 0.5-kg cart is pulled across a track by a string. The string is connected over a pulley to a 0.1-kg mass.

a. Find the acceleration of the cars
b. Find the acceleration of the cars if there is 0.5-N of friction between the car and track

6. How fast would a rollercoaster car have to travel so that you would just lose contact with the seat over a 5-m (radius) hill?

Session 10 - Energy

Session 9 - Newton's Second Law

As we have seen, Newton's second law is easy to express mathematically:

F = m a

Strictly speaking, the force (F) is the NET FORCE. The mass (m) refers to the mass of the system, and the acceleration (a) also refers to the system.

The unit is the kg m / s^2

This is defined as a newton (N).

We consider a few cases in detail.

1. Weight

First, the force due to gravity - weight, W

Since F = m a, and the acceleration is due to gravity,

W = m g

Mass (m) is the amount of matter (or stuff). The weight is a way of quantifying how much gravity pulls on the mass. This means, if g is different, the weight is also different. You weigh less on the Moon, more on Jupiter, less at high altitudes, etc.

2. Inclined Planes

Objects resting on inclined planes are not free to fall directly down. Rather, they are constrained by the geometry of the plane. Part of the weight (a parallel component) can be thought of as acting DOWN the plane. Part of the weight (a perpendicular component) can be thought of as acting ONTO the plane. By trig:

W(parallel) = mg sin(theta)

W(perpendicular) = mg cos(theta)

Without any resistance whatsoever, all objects slide down a plane with the same acceleration:

F = m a

W(parallel) = m a

mg sin(theta) = m a

a = g sin(theta)

3. Friction

Friction is a catch-all term for any resistive force (really due to electromagnetic interactions between surfaces).

The frictional force (f) is the amount of force that resists motion - that is, it acts in the direction opposite the motion. We can quantify friction by introducing a coefficient of friction (u).

u = f / mg

Meaning: the ratio of frictional force that exists to the weight of the object is defined as the coefficient of friction. Typically, this is a (unitless) number much less than 1.

4. Circular Motion

As Newton's 1st law would predict, any object that moves in a circular path has a REASON to do that - some force is causing it to happen. Recall that acceleration is a change in velocity - since velocity refers to a magnitude (speed) AND a direction, if the direction of a body is changing, it MUST be accelerating even if the speed does not change.

Consider a ball spinning on a string at a constant speed. Even if the speed remains constant, we know that the ball is accelerating - its direction is constantly changing. Some force must be causing that to happen. We call such a force - a center-directed force - centripetal. The center-directed acceleration that results is called centripetal acceleration (ac).

ac = v^2 / r

Or, to compute the magnitude of the centripetal acceleration, we take the speed squared and divide it by the radius of orbit.

The units of acceleration are still m/s^2.

Test practice problems, without answers

1. How high will a ball travel into the air, if thrown straight up at 25 m/s?

2. If this ball were thrown at a 40-degree angle (also at 25 m/s), find the following: time in air, horizontal displacement, max vertical displacement

3. Add the following vectors together, which are acting concurrently at a right angle to each other: 40 m/s and 60 m/s. Find the angle from the 40 m/s vector.

Session 7 part 3 - some historical info FYI

http://en.wikipedia.org/wiki/Isaac_Newton

This is really exhaustive - only for the truly interested.

This one is a bit easier to digest:

http://galileoandeinstein.physics.virginia.edu/lectures/newton.html

We'll return to Newton's gravitation (along with Kepler) later in the course.

For Galileo:

http://galileo.rice.edu/

http://galileo.rice.edu/bio/index.html

I also recommend "Galileo's Daughter" by Dava Sobel. Actually, anything she writes is pretty great historical reading. See also her "Longitude."

It is also worth reading about Copernicus and the Scientific Revolution.

For those of you interested in ancient science, David Lindberg's "Beginnings of Western Science" is amazing.

In general, John Gribbin's "The Scientists" is a good intro book about the history of science, in general. I recommend this for all bio and chem majors.

As a science major, you owe it to yourself to find out the history of your discipline. I think it will give you new perspective and respect.

Test 1 review book pages

Session 7 - Newton and his laws.

Newton, Philosophiae naturalis principia mathematica (1687) Translated by Andrew Motte (1729)

Lex. I. Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.


Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.


Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.


Lex. II. Mutationem motus proportionalem esse vi motrici impressae, & fieri secundum lineam rectam qua vis illa imprimitur.


The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.


If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.


Lex. III. Actioni contrariam semper & aequalem esse reactionent: sive corporum duorum actiones in se mutuo semper esse aequales & in partes contrarias dirigi.


To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.


Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other.

Session 7 - some biographical details

Some background details will be discussed in class. Here are some dates of note:

Nicolaus Copernicus
1473 - 1543
De Revolutionibus Orbium Celestium


Tycho Brahe
1546 - 1601


Johannes Kepler
1571 - 1630
Astronomia Nova

Galileo Galilei
1564 - 1642
Siderius Nuncius
Dialogue on Two Chief World Systems
Discourse on Two New Sciences


Isaac Newton
1642 - 1727
Philosophiae Naturalis Principia Mathematica (1687)

Session 6 - more details on Projectiles

The Range Equation is very useful expression for determine the maximum horizontal displacement, when a projectile lands on the same plane from which it was launched.

R = (vo^2 / g) sin (2 theta)

This relationship (derived below) has an interesting prediction (since there is a sine of 2 times the angle). Since the sine relationship is symmetrical from 0 to 180 degrees (about the 90 degree point), the sine of any angle less than 90 degrees is the same as the sine of its supplementary angle. That is, the sine of 80 = the sine of 100.

Since there is a double angle, you get equal sine values for (twice) a launch angle and (twice) the compliment of that angle. For example, the sine of 40 = the sine of 50.

You could also think of this as: you get equal ranges for angle 45 + or - the same amount. The range is the same for 30 and 60, 10 and 80, 35 and 55, etc.

Pretty neat, eh?

Session 6 - Projectile Motion in general

Problem to try:

Consider a baseball hit at 40 m/s at an angle of 35 degrees. Find the following:

a. initial components of velocity

b. time to apogee

c. max height attained

d. max horizontal displacement

e. total time in air

f. the OTHER angle that would give the same horizontal displacement*

Session 5 - Projectile Motion

Projectile motion is a beautiful, but slightly tricky application of vectors. It's vector math mixed with one-dimensional motion.

First, let us consider vector components. Just like 2 vectors can be added together to give a resultant vector, a single vector can be "deconstructed" into its "perpendicular components."

If the angle is measured with respect to horizontal, a vector V at angle theta (T) has components:

Vh = V cos (T)

Vv = V sin (T)

Horizontal (h)

Vertical (v)

And now, projectiles:

Consider first a projectile launched with nothing but horizontal velocity (vox), from a vertical height (xy). There is NO horizontal acceleration, ONLY vertical acceleration. So, the ball lands in the same exact time as if it had been simply dropped. True, it goes further horizontally, but it doesn't take any longer to do that. This will be demonstrated in class.

To solve projectile motion problems, you still use equations of motion, but you use them with horizontal and vertical variables - and you NEVER mix the horizontals with verticals. Time (t) is neither horizontal nor vertical - it is the same (t) for both dimensions.

Given an initial horizontal velocity (vox) of 12 m/s and vertical height (xy) of 5 m, find:

a. time to fall (t)

b. horizontal displacement (xx)

Practice problems from the text

Hello,

By popular demand, I present you some suggested textbook problems from College Physics, 9th Edition, by Serway/Vuille.

* tricky problems

Trig review:

Chapter 1: 41-49 odd

One-dimensional Motion:

Chapter 2: 7, 29, 33, 37, 45, 53*

Vectors:

Chapter 3: 8, 10, 11

Session 4, part 3 -- Vector Addition yet again!

Adding vectors which are at NON-right angles is a bit trickier. Draw a parallelogram and determine the diagonal. You will need to use the law of cosines, but must first find the angle opposite the diagonal. Details furnished in class.

c^2 = a^2 + b^2 - 2ab cos (C)

where C is the angle across from c (the side). In practice, this angle is (again) the supplement of the angle between the vectors.

Session 4, part 2 -- Vector Addition

Vector quantities can be visualized as arrows. Adding two arrows in the same direction gives a larger arrow. Likewise, adding two (similar) vector quantities (say, 2 velocities) yields a larger velocity.

Subtracting vector quantities is simply adding one vector to a negative vector, resulting in a smaller vector.

Imagine walking on a "people mover". If you walk in the direction of the mover's velocity, you ultimately have a greater velocity than the mover itself (with respect to the people observing on the side). If you walk in the OPPOSITE direction to the mover's velocity, which is a pretty lame thing to do, you will have a velocity that is less than that of the mover itself.

How about adding vectors at angles? This is a bit trickier. See the second slide above.

If two vectors, A and B, are at a right angle to each other, the "resultant vector" C is given by the Pythagorean Theorem:

A^2 + B^2 = C^2

and the angle between C and the horizontal is given by the tangent relationship.

tan (angle) = OPP / ADJ = A/B

so that the angle itself is given by the inverse (or arc) tangent of A/B:

angle = arctan (A/B)

If trigonometry is not something you remember that well, check out:

http://howdoweknowthat.blogspot.com/2009/07/how-far-away-is-that.html


Some problems to try:

1. A flagpole makes an angle of 55 degrees with respect to horizontal, as measured from a point 10-m away. How tall is the flagpole?

2. Add 2 vectors together: 10 and 20, at a right angle to each other. Find the resultant vector and the angle as measured from the 20 vector.

3. Imagine trying to cross a river, 100-m wide with a current of 2 m/s. If you head out in your motorboat at 10 m/s, find the following:

a. how long it will take you to cross
b. how far downstream you will be
c. the angle you take with respect to your original path

In this problem, there is NO acceleration - rather, the velocities are added simultaneously (or "concurrently").

Session 4, part 1 -- Problems to try

Hiya. First a comment from yesterday's class. I remarked that that the "area under the curve" for a v vs. t graph gives displacement. This is true - see the image from yesterday's class (in the previous blog post). The units of (v times t) gives units of displacement. For those of you who speak calculus, we refer to this area as an integral, a process made easier if you have a mathematical function that describes the velocity as a function of time.

If this makes no sense to you - fret not, physics friends. Simply know that the area under the curve gives you displacement.

And now, some problems to play with:

Woo Hoo – it’s physics problems with motion! OH YEAH!!!

1. Determine the average velocity of your own trip to school: in m/s and miles per hour.

2. Consider an echo-y canyon. You stand 200-m from the canyon wall. How long does it take the echo of your scream (“Arghhhh! Curse you Physics!!!”), if the speed of sound is 340 m/s? (Sound travels at a constant speed.)

3. What is the difference between traveling at an average speed of 65 mph for one hour and a constant speed of 65 mph for one hour? Will you go further in either case?

4. What is the meaning of instantaneous velocity? How can we measure it?

5. What is the acceleration of a toy car, moving from rest to 6 m/s in 4 seconds?

6. How far will a light pulse (say, a cell phone radio wave) travel in 10 minutes?

7. What does a negative acceleration indicate? How about a negative velocity? Negative displacement?

8. Consider an automobile starting from rest. It attains a speed of 30 m/s in 8 seconds. What is the car’s acceleration during this period, and how far has it traveled?

9. You are driving down a dark road when out pops a deer, 40 m away. Your speed is 20 m/s and you instantly slam on the brakes providing your car with a -2 m/s2 acceleration. Will you hit the deer? (Hint: Find out how far your car will travel given this acceleration, assuming that you come to a complete stop.)

10. What is the acceleration due to gravity? What does this value mean?

11. How does the acceleration due to gravity vary on the Moon? On Jupiter?

12. If you are “pulling 5 g’s”, what acceleration do you experience?

13. How long will it take a rock falling from rest to drop from a 100-m cliff?

14. If you drop a pebble from a bridge into a river below, and it takes 2.5 seconds to hit water, how high is the bridge?

15. Drop a bowling ball from atop a high platform. How fast will it be traveling after 3 seconds of freefall?

16. In order to pass another car on the highway, you increase your velocity from 25 m/s to 30 m/s. This occurs in 10 seconds. What is your acceleration? How far have you moved during this time?

17. You kick a soccer ball straight up into the air with an initial velocity of 22 m/s. How long will it take to reach apogee? To return to the Earth? How high will it rise? What will be its final velocity before hitting Earth?

18. Draw two (approximate) graphs of motion (displacement vs. time, velocity vs. time) for each of the following scenarios:
a. A car accelerates from rest for 5 seconds, travels at a constant velocity for 5 seconds, and then slows down for 5 seconds, finally stopping for 5 seconds.
b. A person walks at a constant velocity for 10 seconds, stops for 5 seconds, then walks back to where they started (faster) for 5 seconds.
c. You toss a ball up in the air and it lands back in your hand.

Session 3 (one more time!) - Graphs of Motion

An object's motion can be represented on graphs:

x vs. t

v vs. t

a vs. t

For a non-moving object, all 3 graphs resemble horizontal lines.

For an object moving with constant velocity:

the x vs. t graph is linear - that is, the displacement increases by the same amount each time interval.

For an object moving with constant acceleration, the displacement increases exponentially with respect to time.

See image above.

The slope of an x vs. t graph will give the velocity. Of course, if the graph is a curve, things are a bit complicated. You could take a tangent line to a particular point on the curve - then take the slope of the tangent line to get the instantaneous velocity at that point.

The slope of a v vs. t graph will give the acceleration.

Session 3 redux - The acceleration due to gravity

Tonight we're discussing the acceleration due to the ubiquitous force of gravity.

Contrary to popular belief, all objects do NOT experience the same force of gravity. An object experiences a gravitation force directly related to its mass. That is, greater mass equals greater gravitational force. All objects undergo the same acceleration (in the absence of air resistance) for a quite subtle reason. (Short answer: there is a greater attractive force of gravity for more massive objects; however, the greater force of attraction is offset by the greater inertia and resistance to motion. The two effects exactly cancel each other.)

Near the surface of the earth, the acceleration due to gravity (g) is:

g = 9.8 m/s^2

Which is to say, an object will accelerate (in free-fall) at a rate of 9.8 m/s^2 - increasing its speed by 9.8 m/s with each second of free-fall. Or, if it is rising, losing 9.8 m/s with each second of upward motion.

So, after 1 second of free-fall, the object will attain a speed of 9.8 m/s.

After 2 seconds, 19.6 m/s.

After 3 seconds, 29.4 m/s.

And so on.

Galileo's odd numbers rule was also discussed in class. Galileo determined that accelerating objects fall (or roll) through total displacements that are proportional to the amount of time squared. If we approximate g as 10 m/s^2, then:

x(1 sec) = 5 m

x(2 sec) = 20 m

x(3 sec) = 45 m

x(4 sec) = 80 m

Note that these displacements are proportional to 1, 4, 9, 16 - perfect squares. And interestingly, the differences between these "base" displacements are 1, 3, 5, 7...... odd numbers. Neat, eh?

Sessions 2 and 3

Hi again, and welcome back to Physics 101! So nice to see you again. How've you been?

First, let us chat about order of magnitude questions, so-called "Fermi Questions" (as they were made legendary by Enrico Fermi). Here are a few we will discuss in class:

Number of days to walk around the world
Number of times your blink in your lifetime
Number of revolutions a car's tire will make
Number of dollar bills to stack to the Moon
Number of hairs on your head
Area needed for all people on Earth to gather for a party

We will discuss strategies for solving these questions today. Now clearly these are not physics questions, per se, but the way of thinking about them can be informative to our process. Physics is a way of thinking about problems, and often applying the language of mathematics to weird situations.

OK? OK!!!

And now......

THE EQUATIONS OF MOTION!

First, let's look at some definitions.

Vector vs. Scalar:

Vector quantities - quantities that must be expressed with a magnitude AND direction. Examples: velocity, displacement, force, momentum, magnetic field, electric field

Scalar quantities - quantities that only requires a magnitude (mass, time, energy, speed)

>

Average velocity

v = x / t

That is, displacement divided by time.

Another way to compute average velocity:

v = (v0 + v) / 2

where vo ("v naught") is the original velocity, and v is the final (or current) velocity.

Average velocity should be distinguished from instantaneous velocity:

v(inst) = x / t, where t is a very, very, very tiny time interval.

[In calculus speak, this is: v(inst) = dx/dt, a derivative.]

>

Acceleration, a

a = (change in velocity) / time

a = (v - vo) / t

The units here are m/s^2.

>

Today we will chat about the equations of motion. There are 5 useful expressions that relate the variables in questions:

vo - "v naught", or original velocity. Note that the 0 is a subscript.
v - velocity at a specific point (usually at time t)
a - acceleration
t - time
x - displacement

Now these equations are a little tricky to come up with - we will derive them in class. (Remember, never drink and derive. But anyway....)

We start with 3 definitions, two of which are for average velocity:

v (avg) = x / t

v (avg) = (vo + v) / 2

and the definition of acceleration:

a = (change in v) / t or

a = (v - vo) / t

Through the miracle of algebra, these can be manipulated (details shown in class) to come up with:

v = v0 + at

x = 0.5 (vo + v) t

x = vo t + 0.5at^2

v^2 = vo^2 + 2ax

x = v t - 0.5at^2

Note that in each of the 5 equations, one main variable is absent. Each equation is true - indeed, they are the logical result of our definitions - however, each is not always helpful or relevant. The expression you use will depend on the situation.

Let's look at a sample problem:

Consider a car, starting from rest. It accelerates uniformly (meaning that the acceleration remains a constant value) at 1.5 m/s^2 for 7 seconds. Find the following:

- the speed of the car after 7 seconds
- how far the car has traveled after 7 seconds

Then, the driver applies the brakes and brings the car to a halt in 3 seconds. Find:

- the acceleration of the car in this time
- the distance that the car travels during this time

Finally, find:

- the total distance the car travels during this "experiment"
- graphs of motion for: displacement vs. time, velocity vs. time, acceleration vs. time

Got it? Hurray!

Physics - YEAH!!!

Session 1

Howdy, and welcome to Physics 101!

Some comments on the first class. I speak about SI units at some length. To remind you:

Mass is measured based on a kilogram (kg) standard.
Length (or displacement or position) is based on a meter (m) standard.
Time is based on a second (s) standard.

How do we get these standards?

Length - meter (m)

- originally 1 ten-millionth the distance from north pole (of Earth) to equator
- then a distance between two fine lines engraved on a platinum-iridium bar
- (1960): 1,650,763.73 wavelengths of a particular orange-red light emitted by atoms of Kr-86 in a gas discharge tube
- (1983, current standard): the length of path traveled by light during a time interval of 1/299,792,458 seconds

That is, the speed of light is 299,792,458 m/s. This is the fastest speed that exists. Why this is is quite a subtle thing. Short answer: the only things that can travel that fast aren't "things" at all, but rather massless electromagnetic radiation. Low-mass things (particles) can travel in excess of 99% the speed of light.

Long answer: See relativity.

Time - second (s)

- Originally, the time for a pendulum (1-m long) to swing from one side of path to other
- Later, a fraction of mean solar day
- (1967): the time taken by 9,192,631,770 vibrations of a specific wavelength of light emitted by a cesium-133 atom

Mass - kilogram (kg)

- originally based on the mass of a cubic decimeter of water
- standard of mass is now the platinum-iridium cylinder kept at the International Bureau of Weights and Measures near Paris
- secondary standards are based on this
- 1 u (atomic mass unit, or AMU) = 1.6605402 x 10^-27 kg
- so, the Carbon-12 atom is 12 u in mass

Volume - liter (l)

- volume occupied by a mass of 1 kg of pure water at certain conditions
- 1.000028 decimeters cubed
- ml is approximately 1 cc

Temperature - kelvin (K)

- 1/273.16 of the thermodynamic temperature of the triple point of water (1 K = 1 degree C)
- degrees C + 273.15
- 0 K = absolute zero

For further reading:

http://en.wikipedia.org/wiki/SI_units

http://en.wikipedia.org/wiki/Metric_system#History

>

In addition, we spoke about the spherocity of the Earth and how we know its size. I've written about this previously. Please see the blog entries below:

http://howdoweknowthat.blogspot.com/2009/07/how-do-we-know-that-earth-is-spherical.html

http://howdoweknowthat.blogspot.com/2009/07/so-how-big-is-earth.html


Physics - Yeah!!!