http://www.fourmilab.ch/earthview/
http://www.fourmilab.ch/solar/
http://www.fourmilab.ch/yoursky/
http://www.fourmilab.ch/cgi-bin/Solar/action?sys=-Sf
http://www.arachnoid.com/gravitation/small.html
Thursday, September 29, 2011
Wednesday, September 28, 2011
Kepler and Newton
First, the applets:
http://www.physics.sjsu.edu/tomley/kepler.html
http://www.physics.sjsu.edu/tomley/Kepler12.html
for Kepler's laws, primarily the 2nd law
http://www.astro.utoronto.ca/~zhu/ast210/geocentric.html
for our discussion on geocentrism and how retrograde motion appears within this conceptual framework
Cool:
http://galileo.phys.virginia.edu/classes/109N/more_stuff/flashlets/kepler6.htm
http://physics.unl.edu/~klee/applets/moonphase/moonphase.html
>
Now, the notes.
Johannes Kepler, 1571-1630
Kepler's laws of planetary motion - of course, these apply equally well to all orbiting bodies
1. Planets take elliptical orbits, with the Sun at one focus. (If we were talking about satellites, the central gravitating body, such as the Earth, would be at one focus.) Nothing is at the other focus. Recall that a circle is the special case of the ellipse, wherein the two focal points are coincident. Some bodies, such as the Moon, take nearly circular orbits - that is, the eccentricity is very small.
2. The Area Law. Planets "sweep out" equal areas in equal times. See the applets for pictorial clarification. This means that in any 30 day period, a planet will sweep out a sector of space - the area of this sector is the same, regardless of the 30 day period. A major result of this is that the planet travels fastest when near the Sun.
3. The Harmonic Law. Consider the semi-major axis of a planet's orbit around the Sun - that's half the longest diameter of its orbit. This distance (a) is proportional to the amount of time to go around the Sun in a very peculiar fashion:
a^3 = T^2
That is to say, the semi-major axis CUBED (to the third power) is equal to the period (time) SQUARED. This assumes that we choose convenient units:
- the unit of a is the Astronomical Unit (AU), equal to the semi-major axis of Earth's orbit (approximately the average distance between Earth and Sun). This is around 150 million km or around 93 million miles
- the unit of time is the (Earth) year
e.g. Consider an asteroid with a semi-major axis of orbit of 4 AU. We can quickly calculate that its period of orbit is 8 years.
Likewise for Pluto: a = 40 AU. T works out to be around 250 years.
>
Newton's take on this 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:
F = G m1 m2 / d^2
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.
http://www.physics.sjsu.edu/tomley/kepler.html
http://www.physics.sjsu.edu/tomley/Kepler12.html
for Kepler's laws, primarily the 2nd law
http://www.astro.utoronto.ca/~zhu/ast210/geocentric.html
for our discussion on geocentrism and how retrograde motion appears within this conceptual framework
Cool:
http://galileo.phys.virginia.edu/classes/109N/more_stuff/flashlets/kepler6.htm
http://physics.unl.edu/~klee/applets/moonphase/moonphase.html
>
Now, the notes.
Johannes Kepler, 1571-1630
Kepler's laws of planetary motion - of course, these apply equally well to all orbiting bodies
1. Planets take elliptical orbits, with the Sun at one focus. (If we were talking about satellites, the central gravitating body, such as the Earth, would be at one focus.) Nothing is at the other focus. Recall that a circle is the special case of the ellipse, wherein the two focal points are coincident. Some bodies, such as the Moon, take nearly circular orbits - that is, the eccentricity is very small.
2. The Area Law. Planets "sweep out" equal areas in equal times. See the applets for pictorial clarification. This means that in any 30 day period, a planet will sweep out a sector of space - the area of this sector is the same, regardless of the 30 day period. A major result of this is that the planet travels fastest when near the Sun.
3. The Harmonic Law. Consider the semi-major axis of a planet's orbit around the Sun - that's half the longest diameter of its orbit. This distance (a) is proportional to the amount of time to go around the Sun in a very peculiar fashion:
a^3 = T^2
That is to say, the semi-major axis CUBED (to the third power) is equal to the period (time) SQUARED. This assumes that we choose convenient units:
- the unit of a is the Astronomical Unit (AU), equal to the semi-major axis of Earth's orbit (approximately the average distance between Earth and Sun). This is around 150 million km or around 93 million miles
- the unit of time is the (Earth) year
e.g. Consider an asteroid with a semi-major axis of orbit of 4 AU. We can quickly calculate that its period of orbit is 8 years.
Likewise for Pluto: a = 40 AU. T works out to be around 250 years.
>
Newton's take on this 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:
F = G m1 m2 / d^2
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.
Monday, September 26, 2011
Newton homework
1. What is the SI unit of force and what other units is it made from?
2. What is Newton's major book and in what year what it published?
3. Consider a 100-N force acting on a 20 kg cart. Whist acceleration does it experience? What would the acceleration be if there were 40-N of friction resisting the motion?
4. You've seen a little fan cart demonstrated in class. Explain why it moves as it does, in terms of Newton's laws.
5. If you placed a sail on the cart (above), would it still move? Explain.
6. Explain each of Newton's laws.
7. What is your weight in newtons?
8. What would happen to your mass on the Moon? How about your weight? How would your answers change on Jupiter?
9. What is weightlessness and how does one experience it?
2. What is Newton's major book and in what year what it published?
3. Consider a 100-N force acting on a 20 kg cart. Whist acceleration does it experience? What would the acceleration be if there were 40-N of friction resisting the motion?
4. You've seen a little fan cart demonstrated in class. Explain why it moves as it does, in terms of Newton's laws.
5. If you placed a sail on the cart (above), would it still move? Explain.
6. Explain each of Newton's laws.
7. What is your weight in newtons?
8. What would happen to your mass on the Moon? How about your weight? How would your answers change on Jupiter?
9. What is weightlessness and how does one experience it?
Wednesday, September 21, 2011
Answers to motion problems
1. Use v = d/t for your values.
2. 1.2 sec
3. No difference
4. Speed at a particular "instant" - essentially, what a speedometer tells you
5. 1.5 m/s/s
6. 3 x 10^8 m, (60) 3 x 10^8 m, (31,557,600) 3 x 10^8 m
7. Slowing down,relative to the forward direction
8. 3.75 m/s/s, 120 m
9. 9.8 m/s/s, the amount by which an object's speed changes with each second of freefall (whether you're moving up OR down)
10. 1/6 of an Earth g for the Moon; Jupiter, about 2.5 Earth g's
11. 5 x 9.8 m/s/s
12. 31 m
13. 29.4 m/s
14. 4.5 s
15. 2.2 sec
2. 1.2 sec
3. No difference
4. Speed at a particular "instant" - essentially, what a speedometer tells you
5. 1.5 m/s/s
6. 3 x 10^8 m, (60) 3 x 10^8 m, (31,557,600) 3 x 10^8 m
7. Slowing down,relative to the forward direction
8. 3.75 m/s/s, 120 m
9. 9.8 m/s/s, the amount by which an object's speed changes with each second of freefall (whether you're moving up OR down)
10. 1/6 of an Earth g for the Moon; Jupiter, about 2.5 Earth g's
11. 5 x 9.8 m/s/s
12. 31 m
13. 29.4 m/s
14. 4.5 s
15. 2.2 sec
Text problems worth a look
For those of you with the textbook:
chapter 3
exercises: 2,3,6,8,9
problems: 1,5,8
chapter 4
ex. 3,4,5,25,26
prob. 1,2
These are problems that you might do after the (blog) posted problems - that is to say, the posted (non-text) problems are more of a direct reflection of what to expect from test problems.
Shortly, I'll post answers to the earlier blog problems.
chapter 3
exercises: 2,3,6,8,9
problems: 1,5,8
chapter 4
ex. 3,4,5,25,26
prob. 1,2
These are problems that you might do after the (blog) posted problems - that is to say, the posted (non-text) problems are more of a direct reflection of what to expect from test problems.
Shortly, I'll post answers to the earlier blog problems.
Tuesday, September 20, 2011
Newton and his laws of motion.
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.
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.
Newton!
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)
>
More historical information regarding Newton:
http://en.wikipedia.org/wiki/Isaac_Newton
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)
>
More historical information regarding Newton:
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 interested in the history of intellectual pursuits.
Wednesday, September 14, 2011
The acceleration due to gravity!
Friends....
We discussed the acceleration due to gravity in class. It is a value (g), and it is approximately equal to 9.8 m/s/s, near the surface of the Earth. At higher altitudes, it becomes lower - a related phenomenon is that the air pressure becomes less (since the air molecules are less tightly constrained), and it becomes harder to breathe at higher altitudes (unless you're used to it). Also, the boiling point of water becomes lower - if you've ever read the "high altitude" directions for cooking Mac n Cheese, you might remember that you have to cook the noodles longer (since the temperature of the boiling water is lower).
On the Moon, which is a smaller body (1/4 Earth radius, 1/81 Earth mass), the acceleration at the Moon's surface is roughly 1/6 of a g (or around 1.7 m/s/s). On Jupiter, which is substantially bigger than Earth, the acceleration due to gravity is around 2.2 times that of Earth. All of these things can be calculated without ever having to visit those bodies - isn't that neat?
Consider the meaning of g = 9.8 m/s/s. After 1 second of freefall, a ball would achieve a speed of .....
9.8 m/s
After 2 seconds....
19.6 m/s
After 3 seconds....
29.4 m/s
We can calculate the speed by using the simple equation:
vf = vi + at
In this case, vf is the speed at some time, a is 9.8 m/s/s, and t is the time in question. Note that the initial velocity is 0 m/s.
Got it?
The distance is a bit trickier to figure. This formula is useful:
d = vi t + 0.5 at^2
Since the initial velocity is 0, this formula becomes a bit easier:
d = 0.5 at^2
Or....
d = 0.5 gt^2
Or.....
d = 4.9 t^2
(if you're near the surface of the Earth, where g = 9.8 m/s/s)
This is close enough to 5 to approximate.
So, after 1 second, a freely falling body has fallen:
d = 5 m
After 2 seconds....
d = 20 m
After 3 seconds....
d = 45 m
After 4 seconds...
d = 80 m
This relationship is worth exploring. Look at the numbers for successive seconds of freefall:
0 m
5 m
20 m
45 m
80 m
125 m
180 m
If an object is accelerating down an inclined plane, the distances will follow a similar pattern - they will still be proportional to the time squared. Galileo noticed this. Being a musician, he placed bells at specific distances on an inclined plane - a ball would hit the bells. If the bells were equally spaced, he (and you) would hear successively quickly "dings" by the bells. However, if the bells were located at distances that were progressively greater (as predicted by the above equation, wherein the distance is proportional to the time squared), one would hear equally spaced 'dings."
Check this out:
Equally spaced bells:
http://www.youtube.com/watch?v=06hdPR1lfKg&feature=related
Bells spaced according to the distance formula:
http://www.youtube.com/watch?v=totpfvtbzi0
Furthermore, look at the numbers again:
0 m
5 m
20 m
45 m
80 m
125 m
180 m
Each number is divisible by 5:
0
1
4
9
16
25
36
All perfect squares, which Galileo noticed - this holds true on an inclined plane as well, and its easier to see with the naked eye (and time with a "water clock.")
Look at the differences between successive numbers:
1
3
5
7
9
All odd numbers. Neat, eh?
FYI:
http://www.mcm.edu/academic/galileo/ars/arshtml/mathofmotion1.html
We discussed the acceleration due to gravity in class. It is a value (g), and it is approximately equal to 9.8 m/s/s, near the surface of the Earth. At higher altitudes, it becomes lower - a related phenomenon is that the air pressure becomes less (since the air molecules are less tightly constrained), and it becomes harder to breathe at higher altitudes (unless you're used to it). Also, the boiling point of water becomes lower - if you've ever read the "high altitude" directions for cooking Mac n Cheese, you might remember that you have to cook the noodles longer (since the temperature of the boiling water is lower).
On the Moon, which is a smaller body (1/4 Earth radius, 1/81 Earth mass), the acceleration at the Moon's surface is roughly 1/6 of a g (or around 1.7 m/s/s). On Jupiter, which is substantially bigger than Earth, the acceleration due to gravity is around 2.2 times that of Earth. All of these things can be calculated without ever having to visit those bodies - isn't that neat?
Consider the meaning of g = 9.8 m/s/s. After 1 second of freefall, a ball would achieve a speed of .....
9.8 m/s
After 2 seconds....
19.6 m/s
After 3 seconds....
29.4 m/s
We can calculate the speed by using the simple equation:
vf = vi + at
In this case, vf is the speed at some time, a is 9.8 m/s/s, and t is the time in question. Note that the initial velocity is 0 m/s.
Got it?
The distance is a bit trickier to figure. This formula is useful:
d = vi t + 0.5 at^2
Since the initial velocity is 0, this formula becomes a bit easier:
d = 0.5 at^2
Or....
d = 0.5 gt^2
Or.....
d = 4.9 t^2
(if you're near the surface of the Earth, where g = 9.8 m/s/s)
This is close enough to 5 to approximate.
So, after 1 second, a freely falling body has fallen:
d = 5 m
After 2 seconds....
d = 20 m
After 3 seconds....
d = 45 m
After 4 seconds...
d = 80 m
This relationship is worth exploring. Look at the numbers for successive seconds of freefall:
0 m
5 m
20 m
45 m
80 m
125 m
180 m
If an object is accelerating down an inclined plane, the distances will follow a similar pattern - they will still be proportional to the time squared. Galileo noticed this. Being a musician, he placed bells at specific distances on an inclined plane - a ball would hit the bells. If the bells were equally spaced, he (and you) would hear successively quickly "dings" by the bells. However, if the bells were located at distances that were progressively greater (as predicted by the above equation, wherein the distance is proportional to the time squared), one would hear equally spaced 'dings."
Check this out:
Equally spaced bells:
http://www.youtube.com/watch?v=06hdPR1lfKg&feature=related
Bells spaced according to the distance formula:
http://www.youtube.com/watch?v=totpfvtbzi0
Furthermore, look at the numbers again:
0 m
5 m
20 m
45 m
80 m
125 m
180 m
Each number is divisible by 5:
0
1
4
9
16
25
36
All perfect squares, which Galileo noticed - this holds true on an inclined plane as well, and its easier to see with the naked eye (and time with a "water clock.")
Look at the differences between successive numbers:
1
3
5
7
9
All odd numbers. Neat, eh?
FYI:
http://www.mcm.edu/academic/galileo/ars/arshtml/mathofmotion1.html
Woo Hoo – it’s physics problems with motion! OH YEAH!!!
You will likely be able to do many of these problems, but possibly not all. Fret not, physics phriends! Try them all.
1. Determine the average velocity of your own trip to school: in miles per hour. Use GoogleMaps or something similar to get the distance, and try to recall the time from your last trip. Use your trip from home to Towson, or something that makes sense to you.
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!!!”) to return to your ears, 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 1 second? In one minute? In one year?
7. What does a negative acceleration indicate?
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. What is the acceleration due to gravity? What does this value mean?
10. How does the acceleration due to gravity vary on the Moon? On Jupiter?
11. If you are “pulling 5 g’s”, what acceleration do you experience?
12. 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?
13. Drop a bowling ball from atop a high platform. How fast will it be traveling after 3 seconds of freefall?
14. How long will it take a rock falling from rest to drop from a 100-m cliff?
15. You throw a baseball straight up into the air with an initial velocity of 22 m/s. How long will it take to reach apogee? (Hint - consider the acceleration to be -9.8 m/s/s.)
1. Determine the average velocity of your own trip to school: in miles per hour. Use GoogleMaps or something similar to get the distance, and try to recall the time from your last trip. Use your trip from home to Towson, or something that makes sense to you.
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!!!”) to return to your ears, 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 1 second? In one minute? In one year?
7. What does a negative acceleration indicate?
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. What is the acceleration due to gravity? What does this value mean?
10. How does the acceleration due to gravity vary on the Moon? On Jupiter?
11. If you are “pulling 5 g’s”, what acceleration do you experience?
12. 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?
13. Drop a bowling ball from atop a high platform. How fast will it be traveling after 3 seconds of freefall?
14. How long will it take a rock falling from rest to drop from a 100-m cliff?
15. You throw a baseball straight up into the air with an initial velocity of 22 m/s. How long will it take to reach apogee? (Hint - consider the acceleration to be -9.8 m/s/s.)
Thursday, September 8, 2011
Some velocities to ponder....
Approximately....
Keep in mind that 1 m/s is approximately 2 miles/hour.
Your walking speed to class - 1-2 m/s
Running speed - 5-7 m/s
Car speed (highway) - 30 m/s
Professional baseball throwing speed - 45 m/s
Terminal velocity of skydiver - 55 m/s
Speed skiing - 60 m/s
Speed of sound (in air) - 340 m/s
Bullet speed (typical) - 900 m/s
Satellite speed (in orbit) - 6200 m/s
Escape velocity of Earth - 11,200 m/s
(That's around 7 miles per second, or 11.2 km/s)
Speed of light (in a vacuum) -
This number is a physical constant, believed to be true everywhere in the universe. The letter c is used to represent the value being of constant celerity (speed).
Keep in mind that 1 m/s is approximately 2 miles/hour.
Your walking speed to class - 1-2 m/s
Running speed - 5-7 m/s
Car speed (highway) - 30 m/s
Professional baseball throwing speed - 45 m/s
Terminal velocity of skydiver - 55 m/s
Speed skiing - 60 m/s
Speed of sound (in air) - 340 m/s
Bullet speed (typical) - 900 m/s
Satellite speed (in orbit) - 6200 m/s
Escape velocity of Earth - 11,200 m/s
(That's around 7 miles per second, or 11.2 km/s)
Speed of light (in a vacuum) -
c = 299,792,458 m/s
This number is a physical constant, believed to be true everywhere in the universe. The letter c is used to represent the value being of constant celerity (speed).
Motion!
THE EQUATIONS OF MOTION!
First, let's look at some definitions.
Average velocity
v = d / t
That is, displacement divided by time.
Another way to compute average velocity:
v = (vi + vf) / 2
where vi is the initial velocity, and vf is the final (or current) velocity.
Average velocity should be distinguished from instantaneous velocity (what you get from a speedometer):
v(inst) = d / t, where t is a very, very, very tiny time interval. There's more to be said about this sort of thing, and that's where calculus begins.
Now this idea (velocity) is pretty useful if you care about the velocity at a specific time OR the average velocity for a trip. However, if you care about the details of velocity, if and when it changes, then we need to introduce a new concept: acceleration.
Now this idea (velocity) is pretty useful if you care about the velocity at a specific time OR the average velocity for a trip. However, if you care about the details of velocity, if and when it changes, then we need to introduce a new concept: acceleration.
>
Acceleration, a
a = (change in velocity) / time
a = (vf - vi) / t
The units here are m/s^2, or m/s/s.
Acceleration is a measure of how quickly you change your speed - that is, it's a measure of 'change in speed' per time. Imagine if you got in a car and floored it, then could watch your speedometer. Imagine now that you get up to 10 miles/hr (MPH) after 1 second, 20 MPH by the 2nd second, 30 MPH by the 3rd second, and so on. This would give you an acceleration of:
10 MPH per second. That's not a super convenient unit, but you get the idea (I hope!).
Acceleration is a measure of how quickly you change your speed - that is, it's a measure of 'change in speed' per time. Imagine if you got in a car and floored it, then could watch your speedometer. Imagine now that you get up to 10 miles/hr (MPH) after 1 second, 20 MPH by the 2nd second, 30 MPH by the 3rd second, and so on. This would give you an acceleration of:
10 MPH per second. That's not a super convenient unit, but you get the idea (I hope!).
>
Today we will chat about the equations of motion. There are 5 useful expressions that relate the variables in questions:
vi - initial velocity. Note that the i is a subscript.
vf - velocity after some period of time
a - acceleration
t - time
d - displacement
Now these equations are a little tricky to come up with - we can derive them in class, if you like. (Remember, never drink and derive. But anyway....)
We start with 3 definitions, two of which are for average velocity:
v (avg) = d / t
v (avg) = (vi + vf) / 2
and the definition of acceleration:
a = (change in v) / t or
a = (vf - vi) / t
Through the miracle of algebra, these can be manipulated (details shown, if you like) to come up with:
vf = vi + at
d = 0.5 (vi + vf) t
d = vi t + 0.5 at^2
vf^2 = vi^2 + 2ad
d = vf t - 0.5 at^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.
In general, I find these most useful:
vf = vi + at
d = 0.5 (vi + vf) t
d = vi t + 0.5 at^2
d = 0.5 (vi + vf) t
d = vi t + 0.5 at^2
By the way, note that the 2nd equation above is the SAME THING as saying distance equals average velocity [0.5 (vi + vf)] multiplied by time.
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
Got it? Hurray!
There is another way to think about motion - graphically. That is, looking (pictorially) at how the position or velocity changes with time. We'll talk about this in class, and use a motion detector to "see" the motion a little better.
Physics - YAY!
Tuesday, September 6, 2011
Schrodinger poem, as promised.
http://www.straightdope.com/columns/read/113/the-story-of-schroedingers-cat-an-epic-poem
From "The Straight Dope"
From "The Straight Dope"
The story of Schroedinger's cat (an epic poem)
May 7, 1982
Dear Cecil:
Cecil, you're my final hope
Of finding out the true Straight Dope
For I have been reading of Schroedinger's cat
But none of my cats are at all like that.
This unusual animal (so it is said)
Is simultaneously live and dead!
What I don't understand is just why he
Can't be one or other, unquestionably.
My future now hangs in between eigenstates.
In one I'm enlightened, the other I ain't.
If you understand, Cecil, then show me the way
And rescue my psyche from quantum decay.
But if this queer thing has perplexed even you,
Then I will and won't see you in Schroedinger's zoo.
Cecil replies:
Schroedinger, Erwin! Professor of physics!
Wrote daring equations! Confounded his critics!
(Not bad, eh? Don't worry. This part of the verse
Starts off pretty good, but it gets a lot worse.)
Win saw that the theory that Newton'd invented
By Einstein's discov'ries had been badly dented.
What now? wailed his colleagues. Said Erwin, "Don't panic,
No grease monkey I, but a quantum mechanic.
Consider electrons. Now, these teeny articles
Are sometimes like waves, and then sometimes like particles.
If that's not confusing, the nuclear dance
Of electrons and suchlike is governed by chance!
No sweat, though — my theory permits us to judge
Where some of 'em is and the rest of 'em was."
Not everyone bought this. It threatened to wreck
The comforting linkage of cause and effect.
E'en Einstein had doubts, and so Schroedinger tried
To tell him what quantum mechanics implied.
Said Win to Al, "Brother, suppose we've a cat,
And inside a tube we have put that cat at —
Along with a solitaire deck and some Fritos,
A bottle of Night Train, a couple mosquitoes
(Or something else rhyming) and, oh, if you got 'em,
One vial prussic acid, one decaying ottom
Or atom — whatever — but when it emits,
A trigger device blasts the vial into bits
Which snuffs our poor kitty. The odds of this crime
Are 50 to 50 per hour each time.
The cylinder's sealed. The hour's passed away. Is
Our pussy still purring — or pushing up daisies?
Now, you'd say the cat either lives or it don't
But quantum mechanics is stubborn and won't.
Statistically speaking, the cat (goes the joke),
Is half a cat breathing and half a cat croaked.
To some this may seem a ridiculous split,
But quantum mechanics must answer, "Tough shit.
We may not know much, but one thing's fo' sho':
There's things in the cosmos that we cannot know.
Shine light on electrons — you'll cause them to swerve.
The act of observing disturbs the observed —
Which ruins your test. But then if there's no testing
To see if a particle's moving or resting
Why try to conjecture? Pure useless endeavor!
We know probability — certainty, never.'
The effect of this notion? I very much fear
'Twill make doubtful all things that were formerly clear.
Till soon the cat doctors will say in reports,
"We've just flipped a coin and we've learned he's a corpse."'
So saith Herr Erwin. Quoth Albert, "You're nuts.
God doesn't play dice with the universe, putz.
I'll prove it!" he said, and the Lord knows he tried —
In vain — until fin'ly he more or less died.
Win spoke at the funeral: "Listen, dear friends,
Sweet Al was my buddy. I must make amends.
Though he doubted my theory, I'll say of this saint:
Ten-to-one he's in heaven — but five bucks says he ain't."
Syllabus!
Physics 100
Understanding Physics
with your host, Sean Lally!
T/Th: 5:30 - 6:45, SM0326
seanplally@gmail.com
slally@towson.edu
412-965-0805
Greeting physics phriends! Welcome to Physics, your new favorite class. We are here to have a look at the broad and beautiful world of physics - the science that seeks to explain the tiniest of the tiny and the hugest of the huge. Physics is a magnificent way of explaining the strangest things imaginable, as well as the most ordinary and mundane. It's a philosophical approach to physical problems - an experimentally oriented way for seeking "truth" in the universe, or at least a close model of the truth.
This semester, I will introduce you to some of Physics' greatest hits - those things in physics that (I hope) capture the imagination, explain everyday things around us, or surprise us with their inner beauty. And really, is there a better way to spend 2.5 hours a week than that? Clearly not.
Some of the ideas will be new to you. Some may seem scary - physics has a bad reputation, I fear, as a vicious unforgiving science, destroying students in its wake. Fret not, physics phriends - we are all here together to learn about our universe. Some of the ideas will be easy to you; some will be quite challenging - honestly, I wouldn't be doing my job as teacher if I didn't challenge you once in a while. Stick with me, ask questions, seek help when you need it, do homework and invest time in the class outside of the normal lecture - these are the keys to success. Above all, don't be afraid to ask questions - in class, before class, after class, by email or by a casual note left expressing something you'd like further clarification about. Got it? Awesome!
My plan is to look at ideas from the following topics: motion, gravitation (Newton, Kepler, Einstein), relativity, sound, light and optics, forces, and more. I'm open to your ideas, so don't be afraid to ask if there are topics you'd like to explore. If I can make it happen, I will.
Stuff:
Text - Paul Hewitt's Conceptual Physics
Tests - 3, equally-weighted and non-cumulative (other than the extent to which physics is naturally cumulative)
My course blog:
http://towsonphysics.blogspot.com/
Also useful, especially if you aren't fond of textbooks:
http://www.physicsclassroom.com/
And just worth watching (though I didn't quite check all the facts):
http://www.youtube.com/watch?v=RmTxr7OsPj0
So, are we ready for Physics? Yeah!!!
Tentative outline:
9/1 Intro; SI units 1
9/6 SI units 2; Pseudoscience
9/8 Motion 1 - velocity
9/13 Motion 2 - acceleration
9/15 Gravitation
9/20 Force and Newton's Laws
9/22 Newton again
9/27 Gravitation again - Kepler's Laws, universal gravitation
9/29 Center of mass
10/4 Energy
10/6 Exam 1
10/11 Simple harmonic motion
10/13 Waves
10/18 Sound
10/20 More Sound
10/25 Doppler Effect
10/27 Light
11/1 Optics
11/3 More Optics
11/8 Exam 2
11/10 Interference, Diffraction and Holography
11/15 Electrical charge
11/17 Electrical circuits
11/22 Electricity and magnetism
11/29 Magnetism 2
12/1 Magnetism 3
12/6 Einstein!
12/8 Special theory of relativity
12/13 More relativity!
All topics subject to change.
Understanding Physics
with your host, Sean Lally!
T/Th: 5:30 - 6:45, SM0326
seanplally@gmail.com
slally@towson.edu
412-965-0805
Greeting physics phriends! Welcome to Physics, your new favorite class. We are here to have a look at the broad and beautiful world of physics - the science that seeks to explain the tiniest of the tiny and the hugest of the huge. Physics is a magnificent way of explaining the strangest things imaginable, as well as the most ordinary and mundane. It's a philosophical approach to physical problems - an experimentally oriented way for seeking "truth" in the universe, or at least a close model of the truth.
This semester, I will introduce you to some of Physics' greatest hits - those things in physics that (I hope) capture the imagination, explain everyday things around us, or surprise us with their inner beauty. And really, is there a better way to spend 2.5 hours a week than that? Clearly not.
Some of the ideas will be new to you. Some may seem scary - physics has a bad reputation, I fear, as a vicious unforgiving science, destroying students in its wake. Fret not, physics phriends - we are all here together to learn about our universe. Some of the ideas will be easy to you; some will be quite challenging - honestly, I wouldn't be doing my job as teacher if I didn't challenge you once in a while. Stick with me, ask questions, seek help when you need it, do homework and invest time in the class outside of the normal lecture - these are the keys to success. Above all, don't be afraid to ask questions - in class, before class, after class, by email or by a casual note left expressing something you'd like further clarification about. Got it? Awesome!
My plan is to look at ideas from the following topics: motion, gravitation (Newton, Kepler, Einstein), relativity, sound, light and optics, forces, and more. I'm open to your ideas, so don't be afraid to ask if there are topics you'd like to explore. If I can make it happen, I will.
Stuff:
Text - Paul Hewitt's Conceptual Physics
Tests - 3, equally-weighted and non-cumulative (other than the extent to which physics is naturally cumulative)
My course blog:
http://towsonphysics.blogspot.com/
Also useful, especially if you aren't fond of textbooks:
http://www.physicsclassroom.com/
And just worth watching (though I didn't quite check all the facts):
http://www.youtube.com/watch?v=RmTxr7OsPj0
So, are we ready for Physics? Yeah!!!
Tentative outline:
9/1 Intro; SI units 1
9/6 SI units 2; Pseudoscience
9/8 Motion 1 - velocity
9/13 Motion 2 - acceleration
9/15 Gravitation
9/20 Force and Newton's Laws
9/22 Newton again
9/27 Gravitation again - Kepler's Laws, universal gravitation
9/29 Center of mass
10/4 Energy
10/6 Exam 1
10/11 Simple harmonic motion
10/13 Waves
10/18 Sound
10/20 More Sound
10/25 Doppler Effect
10/27 Light
11/1 Optics
11/3 More Optics
11/8 Exam 2
11/10 Interference, Diffraction and Holography
11/15 Electrical charge
11/17 Electrical circuits
11/22 Electricity and magnetism
11/29 Magnetism 2
12/1 Magnetism 3
12/6 Einstein!
12/8 Special theory of relativity
12/13 More relativity!
All topics subject to change.
Monday, September 5, 2011
Skepticism 101
Related to our brief foray 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
James "The Amazing" Randi
Flim Flam
Conjuring
The Faith Healers
An encyclopedia of claims, frauds and hoaxes of the occult and supernatural
Skeptical Inquirer
Skeptic's Dictionary
Richard Feynman - "Cargo Cult Science" essay
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:
Thursday, September 1, 2011
Session 1 - Physics.... YAY!
Session 1
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!!!
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