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.
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