1. Explain each of Kepler's 3 laws.
2. In the (2nd) Area law, what does Kepler have to say about the speed of a planet (or asteroid, etc.) as it orbits the Sun - that is, what happens to the speed of a planet as its distance from the Sun changes?
3. What does "semi-major axis of orbit" mean?
4. What is an AU (astronomical unit)?
5. (3rd Law) If an asteroid is 6 AU from the Sun, how long does it take to orbit the Sun?
6. Consider two bodies in space. If the distance between them is doubled, what happens to the gravitational force between them? What if the distance becomes 5 times the original distance? Half the original distance?
7. The acceleration due to gravity behaves the same was as the universal gravitation law - the same as in problem 6. So that said, what will happen to your weight if you are above the surface of the Earth at a distance exactly equal to the radius of the Earth (thereby doubling your distance from the center of the Earth)?
8. What is your weight on the Moon? (g = 1.7 m/s/s)
9. What was Newton's remarkable book titled, and when was it published?
Wednesday, February 29, 2012
Tuesday, February 28, 2012
NEW EXAM DATE
Folks
I am moving the first exam to March 8, to give us a little more time to explore the new ideas, specifically gravitation.
Sorry for any inconvenience.
SL
I am moving the first exam to March 8, to give us a little more time to explore the new ideas, specifically gravitation.
Sorry for any inconvenience.
SL
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 (since 4 cubed equals 8 squared).
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 (since 4 cubed equals 8 squared).
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.
Friday, February 24, 2012
More motion problems - sorry for delay
1. A car, starting from rest, gets up to a speed of 30 m/s in 8 seconds. Find:
A. The cars acceleration
B. The distance traveled by the car in this time
2. You drop a ball from a height of 28 m. Find the time for it to hit the ground and the speed it has immediately before impact.
3. If you were to throw a ball straight up at 15 m/s, how long would it take to reach apogee? How high would it travel?
A. The cars acceleration
B. The distance traveled by the car in this time
2. You drop a ball from a height of 28 m. Find the time for it to hit the ground and the speed it has immediately before impact.
3. If you were to throw a ball straight up at 15 m/s, how long would it take to reach apogee? How high would it travel?
Thursday, February 23, 2012
Newton homework
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?
answers:
1. newton (N) = kg m/2^2
2. Principia Mathematica, 1687
3. 5 m/s/s; 3 m/s/s
4. Third law: fan blades push air; air returns the favor
5. You think about this one.
6. see notes
7. W = m(in kg) x 9.8 m/s/s. This will give you a weight in newtons.
8. mass does NOT change, but weight WILL change. On the Moon, your weight will be less (1/6 the original). On Jupiter, greater - around 2.5 times
9. You and your surroundings are accelerating together. Consider the 'vomit comet' plane.
Newton's Laws redux.
1. Newton's First Law (Inertia)
An object will keep doing what it is doing, unless there is a reason for it to do otherwise.
That means, it will stay at rest OR it will keep moving (at a constant velocity) unless acted on by an unbalanced force.
2. Newton's Second Law
An unbalanced force (F) causes an object to accelerate (a).
That means, if you apply a force to an object (and the force is unbalanced - greater than any resisting forces), the object will accelerate.
Symbolically:
The Force (F) on a mass (m) produces acceleration (a), predicted by the above equation. In detail:
Greater F means greater a
If the Force is kept constant, but the mass is increased, the acceleration will be smaller:
a = F/m
That's an inverse relationship.
There is a new unit for Force - since Force = mass times acceleration, the units are:
kg m/s^2
We give this a new name, the newton (N). It's about 0.22 lb.
Weight:
There is a special force, the force due to gravity. It's called weight (W).
W = mass x acceleration (due to gravity)
or
W = m g
This is worth noting - there is a BIG difference between mass (m) and weight (W). Mass is the amount of stuff there is and weight is the extent to which it is pulled to the Earth (or wherever).
Since g on the Moon is around 1/6 that of Earth, your weight on the Moon would be around 1/6 of your Earth weight.
3. Newton's 3rd Law
To every action there is opposed an equal reaction. Forces always exist in pairs. Examples:
You move forward by pushing backward on the Earth - the Earth pushes YOU forward.
A rocket engine pushes hot gases out of one end - the gases push the rocket forward.
If you fire a rifle or pistol, the firearm "kicks" back on you.
Since the two objects experience the same force:
m A = M a
That's a little tricky to convey in letters but, the larger object (M) will experience the smaller acceleration (a) and the smaller object (m) will have a larger acceleration (A).
An object will keep doing what it is doing, unless there is a reason for it to do otherwise.
That means, it will stay at rest OR it will keep moving (at a constant velocity) unless acted on by an unbalanced force.
2. Newton's Second Law
An unbalanced force (F) causes an object to accelerate (a).
That means, if you apply a force to an object (and the force is unbalanced - greater than any resisting forces), the object will accelerate.
Symbolically:
F = m a
The Force (F) on a mass (m) produces acceleration (a), predicted by the above equation. In detail:
Greater F means greater a
If the Force is kept constant, but the mass is increased, the acceleration will be smaller:
a = F/m
That's an inverse relationship.
There is a new unit for Force - since Force = mass times acceleration, the units are:
kg m/s^2
We give this a new name, the newton (N). It's about 0.22 lb.
Weight:
There is a special force, the force due to gravity. It's called weight (W).
W = mass x acceleration (due to gravity)
or
W = m g
This is worth noting - there is a BIG difference between mass (m) and weight (W). Mass is the amount of stuff there is and weight is the extent to which it is pulled to the Earth (or wherever).
Since g on the Moon is around 1/6 that of Earth, your weight on the Moon would be around 1/6 of your Earth weight.
3. Newton's 3rd Law
To every action there is opposed an equal reaction. Forces always exist in pairs. Examples:
You move forward by pushing backward on the Earth - the Earth pushes YOU forward.
A rocket engine pushes hot gases out of one end - the gases push the rocket forward.
If you fire a rifle or pistol, the firearm "kicks" back on you.
Since the two objects experience the same force:
m A = M a
That's a little tricky to convey in letters but, the larger object (M) will experience the smaller acceleration (a) and the smaller object (m) will have a larger acceleration (A).
Tuesday, February 21, 2012
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
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.
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.
Tuesday, February 14, 2012
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
Friday, February 10, 2012
Celestial Sphere Revisited
The CS is an ancient concept - it's a crystal sphere that was imagined
to rotate around the earth. All of the stars were imagined to be fixed
it's surface. In this model, which is of course wrong, the sun travels
around - it traces an apparent path (the ecliptic). The path of the
sun goes higher after the winter solstice, following a direct
East-West path only on the day of the spring equinox. After that, it
keeps rising higher and higher until the summer solstice - longest day
of the year. And then, the path starts to become lower.
Mainly, the CS idea is useful because it allows us to visualize the
universe as a sphere - it's convenient, even though it's not correct.
We can imagine a "celestial equator," a circle that mimics the Earth's
equator. Plus, we can imagine the ecliptic - the path that the Sun
appears to take. These 2 circles intersect at two points, the
equinoxes. The ecliptic circle/path has its highest point at the
summer solstice and lowest point at the winter solstice.
So, even though the CS idea is false, it's super useful for
visualizing how weird the motion of the Sun (and planets, which move
along the ecliptic path) is, when viewed from around the Earth.
Thursday, February 9, 2012
Physics Problems 1
Woo Hoo – it’s physics problems with motion! OH YEAH!!!
Questions from the first 2 classes:
1. What are the SI standards for mass, length and time?
2. What are these standards CURRENTLY based on?
3. What is evidence for a spherical Earth?
4. How long have we known that the Earth was spherical?
5. Roughly, how was the circumference of the Earth initially determined?
6. What is the celestial sphere concept all about?
MOTION PROBLEMS (classes 3-5)
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. How far will a light pulse (say, a cell phone radio wave) travel in 1 second? In one minute? In one year?
6. What is the acceleration of a toy car, moving from rest to 6 m/s in 4 seconds?
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?
QUESTIONS SPECIFICALLY RELATED TO THE ACCELERATION DUE TO GRAVITY (class 5):
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.)
Questions from the first 2 classes:
1. What are the SI standards for mass, length and time?
2. What are these standards CURRENTLY based on?
3. What is evidence for a spherical Earth?
4. How long have we known that the Earth was spherical?
5. Roughly, how was the circumference of the Earth initially determined?
6. What is the celestial sphere concept all about?
MOTION PROBLEMS (classes 3-5)
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. How far will a light pulse (say, a cell phone radio wave) travel in 1 second? In one minute? In one year?
6. What is the acceleration of a toy car, moving from rest to 6 m/s in 4 seconds?
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?
QUESTIONS SPECIFICALLY RELATED TO THE ACCELERATION DUE TO GRAVITY (class 5):
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.)
Syllabus!
Tentative outline, recognizing that I hope to add a couple of classes devoted to flight and engines, et. al.:
1.31 Intro; SI units 1
2.2 SI units 2; Pseudoscience
2.7 Motion 1 - velocity
2.9 Motion 2 - acceleration
2.14 Gravitation
2.16 Force and Newton's Laws
2.21 Newton again
2.23 Gravitation again - Kepler's Laws, universal gravitation
2.28 Center of mass
3.1 Energy
3.6 Exam 1
3.8 Simple harmonic motion
3.13 Waves
3.15 Sound
3.27 More Sound
3.29 Doppler Effect
4.3 Light
4.5 Optics
4.10 More Optics
4.12 Exam 2
4.17 Interference, Diffraction and Holography
4.19 Electrical charge
4.24 Electrical circuits
4.26 Electricity and magnetism
5.1 Magnetism 2
5.3 Magnetism 3
5.8 Einstein!
5.10 Special theory of relativity
5.15 More relativity!
5.17 Final Exam
1.31 Intro; SI units 1
2.2 SI units 2; Pseudoscience
2.7 Motion 1 - velocity
2.9 Motion 2 - acceleration
2.14 Gravitation
2.16 Force and Newton's Laws
2.21 Newton again
2.23 Gravitation again - Kepler's Laws, universal gravitation
2.28 Center of mass
3.1 Energy
3.6 Exam 1
3.8 Simple harmonic motion
3.13 Waves
3.15 Sound
3.27 More Sound
3.29 Doppler Effect
4.3 Light
4.5 Optics
4.10 More Optics
4.12 Exam 2
4.17 Interference, Diffraction and Holography
4.19 Electrical charge
4.24 Electrical circuits
4.26 Electricity and magnetism
5.1 Magnetism 2
5.3 Magnetism 3
5.8 Einstein!
5.10 Special theory of relativity
5.15 More relativity!
5.17 Final Exam
Tuesday, February 7, 2012
Celestial Sphere Applet
This is what I was playing around with in class this evening:
http://astro.unl.edu/naap/motion3/animations/sunmotions.swf
http://astro.unl.edu/naap/motion3/animations/sunmotions.swf
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!
Velocity!
Average velocity
v = d / t
That is, displacement divided by time.
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) -
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).
This speed is SO fast that:
it's around 7 times around the Earth's equator in one second
it's up to the Moon and back in around 2.5 seconds
it's out to the Sun in around 8 minutes
it's around 186,000 miles per second
LIGHT UNITS (for distance):
Related to this, we call the distance that light can travel (unimpeded) in one year a "light year". Other useful distinctions:
light-second -- around 300,000,000 m (or 186,000 miles)
light-minute -- 60 times the above number
light-year -- 60 x 60 x 24 x 365.35 times the above number
A light year is a big distance to us, but astronomically speaking, it's tiny. The nearest star (our Sun) is around 8 light MINUTES, but the nearest star other than the Sun is around 4.2 light YEARS away! It's Proxima Centauri, by the way, but it's not visible to the naked eye. Alpha Centauri (a binary system system) IS visible (Southern Hemisphere), and it's around 4.3 light years away.
FYI:
http://en.wikipedia.org/wiki/List_of_nearest_stars
http://en.wikipedia.org/wiki/Alpha_Centauri
v = d / t
That is, displacement divided by time.
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) -
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).
This speed is SO fast that:
it's around 7 times around the Earth's equator in one second
it's up to the Moon and back in around 2.5 seconds
it's out to the Sun in around 8 minutes
it's around 186,000 miles per second
LIGHT UNITS (for distance):
Related to this, we call the distance that light can travel (unimpeded) in one year a "light year". Other useful distinctions:
light-second -- around 300,000,000 m (or 186,000 miles)
light-minute -- 60 times the above number
light-year -- 60 x 60 x 24 x 365.35 times the above number
A light year is a big distance to us, but astronomically speaking, it's tiny. The nearest star (our Sun) is around 8 light MINUTES, but the nearest star other than the Sun is around 4.2 light YEARS away! It's Proxima Centauri, by the way, but it's not visible to the naked eye. Alpha Centauri (a binary system system) IS visible (Southern Hemisphere), and it's around 4.3 light years away.
FYI:
http://en.wikipedia.org/wiki/List_of_nearest_stars
http://en.wikipedia.org/wiki/Alpha_Centauri
Thursday, February 2, 2012
On the Earth's spherocity....
So how do we know that the Earth is "spherical?"
http://howdoweknowthat.blogspot.com/2009/07/how-do-we-know-that-earth-is-spherical.html
And for how we measured its size.....
http://howdoweknowthat.blogspot.com/2009/07/so-how-big-is-earth.html
And for how we got the distances to lots of things we can't visit....
http://howdoweknowthat.blogspot.com/2009/07/how-far-away-is-that.html
http://howdoweknowthat.blogspot.com/2009/07/how-do-we-know-that-earth-is-spherical.html
And for how we measured its size.....
http://howdoweknowthat.blogspot.com/2009/07/so-how-big-is-earth.html
And for how we got the distances to lots of things we can't visit....
http://howdoweknowthat.blogspot.com/2009/07/how-far-away-is-that.html
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