How things balance

A very useful concept in physics is Center of Gravity (AKA CM, Center of Mass - they are usually the same point).  

Recall the demo with the mass on a stick.  Same mass, held at a further distance from the "fulcrum", is harder to support.  It twists your wrist more - it requires a greater "torque".

So, what is torque?

Torque - a "rotating" force

T = F L

For an object to be "in equilibrium," not only must the forces be balanced, but the torques must also be balanced.

Consider a basic see-saw, initially balanced at the fulcrum:  See image below.

You can have two people of different weight balanced, if their distances are adjusted accordingly:  the heavier person is closer to the fulcrum.  

Mathematically, this requires that the torques be equal on both sides.

Consider two people, 100 lb and 200 lb.  The 100 lb person is 3 feet from the fulcrum.  How far from the fulcrum must the 200 lb person sit, to maintain equilibrium?

Torque on left = Torque on right

100 (3) = 200 (x)

x = 1.5 feet

We call the "balance point" the center of mass (or center of gravity).  

It is the point about which the object best rotates.

It is the average location of mass points on the object.

It does not HAVE to be physically on the object - think of a doughnut.

The principle is believed to originate with Archimedes (287 - 212 BC).  He is believed to have said, "Give me a place to stand on, and I will move the Earth."

FYI:  http://en.wikipedia.org/wiki/Archimedes

New practice problems and questions 2 (Kepler)

Gravitation and flight questions and problems:

1.  Explain the meaning of "inverse square law".

2.  Discuss each of Kepler's 3 laws.

3.  At what point in its orbit is the Earth closest to the Sun?

4.  At what point in its orbit is the Earth moving fastest?

5.  What causes seasons?

6.  What is a semi-major axis of orbit (a)?

7.  What is an Astronomical Unit (AU)?

8.  Consider Jupiter.  It's orbit is 5 AU in size (roughly).  How long should it take Jupiter to orbit the Sun once?  Show how this calculation would be done.

9.  What is the period of Earth's orbit around the Sun?

10.  What is the size of Earth's orbit (in AU)?

11.  When you stand on the Earth's surface, you experience your "normal" Earth weight.  What would happen to your Earth weight if you were one Earth radius above the surface?  (That's twice as far from the center as simply standing on the surface.)

12.  What does gravitational force between 2 objects depend on?

Some questions from Newton's laws:

13.  A 10-kg object is pushed on by a 200-N force.  What will be the acceleration?

14.  What is the weight of a 100-kg man?

15.  Would the answer to 3 be different if he was on the moon?  How so?

16.  Consider yourself standing on a scale in an elevator.  The scale reads your weight.  Compared to being at rest, how would the scale reading change (if at all) if the elevator were:

A.  Moving with constant velocity upward

B.  moving with constant velocity downward

C.  Moving with constant acceleration upward

D.  Moving with constant acceleration downward

E.  If the cable snapped (yikes!) and the elevator were falling

(Answers below.)

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1.  Explain the meaning of "inverse square law".

The force (of gravity, in this case) gets progressively weaker by the factor 1 over the distance squared.  Double the distance --> force is 1/4 as great as it was.  Triple the distance --> force is 1/9 the original.

2.  Discuss each of Kepler's 3 laws.

See notes.

3.  At what point in its orbit is the Earth closest to the Sun?

Perihelion, which is approximately January 3 each year.

4.  At what point in its orbit is the Earth moving fastest?

Same point as 3 above.

5.  What causes seasons?

Tilt of Earth's axis.

6.  What is a semi-major axis of orbit (a)?

Half the longest distance across the orbital path (ellipse).

7.  What is an Astronomical Unit (AU)?

Defined as the semi-major axis of Earth's orbit - roughly 93,000,000 miles - or  half the longest width across Earth's orbit.

8.  Consider Jupiter.  It's orbit is 5 AU in size (roughly).  How long should it take Jupiter to orbit the Sun once?  Show how this calculation would be done.

5^3 = T^2

So, T = the square root of 125, or around 11 years.

9.  What is the period of Earth's orbit around the Sun?

1 year, or approximately 365.25 days.

10.  What is the size of Earth's orbit (in AU)?

Defined as 1 AU.

11.  When you stand on the Earth's surface, you experience your "normal" Earth weight.  What would happen to your Earth weight if you were one Earth radius above the surface?  (That's twice as far from the center as simply standing on the surface.)

1/4 your surface weight.

12.  What does gravitational force between 2 objects depend on?

mass of the objects; distance between; a universal (unchanging) constant (G)

13.  F = m a

200 = 10 a

a = 20 m/s/s

14.  W = m g

W = 100 g = 980 newtons

15.  Yes.  The weight would be smaller (1/6 as much, since Moon surface gravity is 1/6 that of Earth).

16.  a.   your regular weight
b.  your regular weight
c.  greater than your regular weight
d.  less than your regular weight
e.  zero!  (meaning that you are "weightless")

New practice problems and questions - 1 (Newton)

homework for Newton's laws, etc.

1. Describe each of Newton's 3 laws.

2. A 0.5 kg toy car is pushed with a 40 newton force. What is the car's acceleration?

3. Without calculating anything, what would be the effect (in problem 2) of increasing the mass of the car?

4. Give an example of Newton's 1st law in action.

5. Give an example of Newton's 3rd law in action.

6. Newton's "big book", what I claim is the most important non-religious book of all time is _____ and was published in _____.

7.  What are epicycles and why are they important in the history of science?

8. Distinguish between weight and mass.

9.   What is the SI unit of force?  What is the English unit of force?

10.  How does weight depend on gravitational acceleration?

11.   Freefall review.  Consider a ball falling from rest.  How fast would it be moving after 4 seconds?  How far would it fall in this time?

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1.  See notes.

2.  40 / 0.5 = 80 m/s/s

3.  lower acceleration

4.  See notes.  Tablecloth pull, etc.

5.  firearm recoil, etc.

6.  Principia Mathematica, 1687.

7.  They helped explain retrograde motion - the apparent backwards motion of planets.  Really, planets are orbiting the Sun and there are times that some bodies are "behind" -- it's like when you pass someone on the highway and they "appear" to be moving backwards at that time.

8.  mass - the amount of stuff (in kg); weight - the gravitational pull on this stuff (in newtons).  The weight depends on where you are (in terms of how the gravitational acceleration changes).  For example, your weight on the Moon is 1/6 that of Earth.

9.  newton; pound

10.  W = m g.  Weight is depending on the local value of g.


11.  Letting g = 10 m/s/s -->  40 m/s, 80 m

Sunday evening's lunar eclipse FYI

Friends,

There is a lunar eclipse this  Sunday, September 27, at around 8 PM.  While lunar eclipses are not that uncommon, this eclipse has the distinction of occurring during a so-called “super Moon”, wherein the full Moon occurs at perigee (when the Moon, in its elliptical orbit, is closest to Earth).  Expect a slightly brighter full Moon than usual!  You will have to wait until 2033 to see that again!

Moonrise is approximately 6:48 PM and the Moon will be full at approximately 10:52 PM.  However, the eclipse itself will start close to 9 PM.  It should reach totality (where the Moon turns “blood red”) around an hour later. 

So, go outside between 9 PM and midnight and enjoy the greatest show from Earth this year.  And as my hero Jack Horkheimer used to say, Keep looking up! 

Clear skies!

Further information:

http://www.skyandtelescope.com/astronomy-news/observing-news/get-ready-for-septembers-total-lunar-eclipse-091420155/

https://www.rt.com/news/316005-supermoon-lunar-eclipse-coincide/

What's all this about gravity?

Universal Gravitation (1687, Principia)

Newton's take on orbits 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:

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.

How things orbit

Kepler's laws of planetary motion

Johannes Kepler, 1571-1630

Note that these laws apply equally well to all orbiting bodies (moons, satellites, comets, etc.)

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

The image below calls period P (rather than T), but the meaning is the same:

Example problem:  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.


The applet I used::

http://astro.unl.edu/classaction/animations/renaissance/kepler.html




Cool:
http://galileo.phys.virginia.edu/classes/109N/more_stuff/flashlets/kepler6.htm

Some thoughts on size and space

http://astro.unl.edu/naap/motion3/animations/sunmotions.swf

On the size of things:

http://htwins.net/scale2/

http://scaleofuniverse.com/

http://xkcd.com/482/

http://xkcd.com/1331/
This is just cool.

http://workshop.chromeexperiments.com/stars/

http://www.youtube.com/watch?v=8yCzzTkDSMo

Jack Horkheimer, FYI.