Donate to JF Business Ventures

If You Love The Content Provided On The Rational Theorist and You Want To Contribute, then feel free to donate money, which will help get my company "JF Business Ventures" up and running and thereby the goal of bringing future innovation and technology to the masses. Not only will your donations be greatly appreciated, they also will go to the excellent cause of making the world a better and more hi-tech place for all humanity one project at a time. However, regardless if you donate or not, I must say thanks for taking the time to read/listen to my blog and hopefully you’ll learn many valuable things from it which will stimulate your thoughts and ideas about the world. $-]
Note: Click banner for my Tutor profile on WyzAnt

Saturday, June 28, 2008

Orbits

This shows various kinds of orbits. All orbits have something in common, they always orbit about the common center of mass between both masses and the center of mass (red cross hairs) doesn't ever move unless there is an external influence on the orbit system.

(1a) Shows a planetary orbit around a star.

(1b) shows international space station orbit around earth which is orbiting the sun.(1c) shows geostationary satellite orbit....notice how the satellite orbits the Earth at the same rate that the Earth is rotating on its axis.

(2a) shows the formation of the moon when a large mass called Theia impacted the Earth some 4 billion years ago when the Earth was still forming.


(2b) shows a moon orbit around a planet

(2c) shows the phases of our moon


(3) This shows a slight binary orbit, as you'll notice the center of mass lies outside of the more massive body

(4a) shows a binary orbit that is typically seen in binary stars
(4b) shows a black hole and supergiant binary star system (4c) shows the effect that gravity waves emminating from a black hole - binary star system should have on a satellite array many light years away.

(5a) shows a binary orbit that is often seen in stars that are captured by other stars

(5b) shows a chaotic 3 body orbit system, where two satellites are given the same exact tragectory but are initially offset from each other by an extremely small distance and hence the butterfly effect occurs.

(Note: All images are a courtessy of Wikipedia

Thursday, June 26, 2008

Airplane Axis of Rotation

(1) The control stick "B" is used by the pilot for controlling the aerodynamic surfaces of the airplane. In general, these 3 axis of rotation are perpendicular to each other and lie somewhere near the center of mass of the airplane.


(1) Pitch is a rotation about the front wing axis as controlled by the elevators on the tail wings "C".

(2) Roll is rotation about the axis of the fuselage as controlled by the ailerons "A".



(3) Yaw is rotation about the axis that is perpindicular to the plane of those other two axis, and is controlled by the tail rudder "D".

Wednesday, June 25, 2008

Doppler Shifts and Bow/Shock Waves

The following particles are emitting sound waves which travel at the same velocity in all directions (in a way, these animations can represent other types of waves too such as the water-wake from a boat or jet ski.... although, to try to compare this to siesmic waves would require a moving epicenter which doesn't quite happen in nature to my knowledge, and the comparison to moving charged particles emitting EM-fields could be made too, at least for non-relativistic velocities where the energy isn't being transferred into the mass of the particle). Anyhow....

(1) The particle at rest is emitting a wave that travels outward in all directions at a constant speed.




(2) This next particle is moving to the right at a speed which is less than the speed of sound (or the corresponding wave speed). This creates creates a bow wave whereby the frequency in front is greater than the frequency behind, and hence a doppler shift.


(3) This, the fastest particle, is traveling at the speed of sound, or Mach 1, which creates a shock wave or a shock-cone.


(4) Here is a shadowgraph of a scale model rocket-plane doing a pitch maneuver at Mach speeds in a wind tunnel, whereby the denser air at the high pressure wavefronts change the index of light refraction which shows up as blackened lines on the exposed film (courtessy NASA, I think).

Monday, June 23, 2008

Simple Harmonic Motion

(1) The middle picture is what a mass on a spring would do if it were sliding on a completely frictionless surface.

(2) Notice how the mass is always directly below the red circle. That mass on the spring represents the X-component (i-vector) of the circular trajectory of the top picture.

(3) Notice how the graph on the bottom shows how the kinetic energy "K" plus the potential spring energy "U" is always constant for the system (the sum of the energies is always equal to E-total). That is representative of the conservation of energy for the spring-mass system, the middle picture.

(4) Simple Harmonic Motion, or systems that oscillate in a sinousoidal fashion, can be used to model/represent a lot of different things in physics .... orbits, springs, the magnetic core in stereo speakers, pendulums, electron resonances, sound waves, light waves, antennas, or anything else that oscillates at a specified frequency.

The parametric equation for the top animation is:
{x(t),y(t)}= [Ao*cos(kt)] i + [Ao*sin(kt)] j

where Ao => radius; k=> angular velocity



Sunday, June 22, 2008

Hydraulic Press

(1A) Pressure applied to a confined fluid is the same throughout that fluid. If you apply 10 PSI at point 1, there will be 10 PSI at point 2. What this means, is that if you apply 10 PSI over a 1 square inch area on a fluid, then you apply 10 PSI over everything that fluid is in contact with, and hence you've also applied 10 PSI on the second area of the fluid that is maybe 5 square inches.

(1B) What this means is that if you apply a force of 10 pounds over 1 square inch on side 1, that means you are applying a force of 50 pounds over an area of 5 square inches on side two .... meaning you can lift your heavy car by applying a relatively small force on the car jack.

(1C) What gives? This pressure difference will cause the fluid to flow because the fluid in your car jack is incompressible. The energy that you exert on side one, say you push side one down 5 inches, is the same energy that you give to side two.

Thus, you've exerted 50 inch-pounds of energy on side 1 (Force one, 10 lbs, times the distance you've exerted that force over, 5 inches gives you 50 inch-pounds), well, then you've also exerted 50 inch-pounds on side two resulting in side two rising by 1 inch (50 pounds on side two, times 1 inch on side two gives you the balance of 50 inch-pounds).

(1D) The volume of displacement on both sides is the same too.

Side 1 displacement .... (1 square inch)x(5 inches) = 5 cubic inches

Side 2 displacement .... (5 square inches)x(1 inch) = 5 cubic inches

(2) Then, a hydraulic jack draws fluid from the fluid "reservoir" to replace the fluid that you've jacked up, as it acts like a positive displacement piston pump. Due to energy inefficiencies, you never quite displace the amount of fluid as in this perfect example because some of the fluid flows back into the reservoir on the recharge stroke.

Radio Waves

(1) Electrons in antenna oscillate back and forth.

(2) This sends out an EM-wave* which is in the plane perpendicular to the line of oscillation (antenna).
(3) This signal moves out in all directions at the speed of light.
(4,5) This is showing how the radio waves bounce off of convex and flat surfaces



(6) A satellite dish is a concave (parabolic) surface and hence it bounces the signal to the focal point.

(7) The electrons at the focal point in the recieving antenna oscillate back and forth about the antenna in a wave pattern, the same pattern as was leaving the the transmission antenna at step (1).

(8) This signal is then amplified in the circuitry of the reciever... end transmission!

----------

*Note: The particular gif in image (2) shows a slight doppler shift (as if moving to the right because there is a larger frequency in that direction and a smaller one to the left), I chose this gif instead of the other one I was looking at merely because you can see the peaks and valleys easier than the other one I had looked at where the wave frequencies were depicted as uniform in all directions and the view was straight on instead of at an angle.

Friday, June 20, 2008

Piano Notes and Exponential Frequencies



1) The C-notes on a piano double in frequency each time you move up one measure (7 white keys). This can be visually seen in the picture below by counting the amount of nodes on the sinousoidal wave ( i.e., ^v -> 2 nodes, ^v^v -> 4 nodes, ^v^v^v^v -> 8 nodes, etc.) for playing each C-note.

2) We could say the frequency has an exponential relationship, or that it doubles every 7 notes you go upscale (as in exponential growth) or that it halves for every 7 notes you go downscale (as in exponential decay)

3) The equation for this would be ...

... frequency = [some constant] x 2^(note / 7)


4) Whereby the notes in this example are defined as follows:

C4=0, D4=1, E4=2, F4=3, G4=4, A5=5, B5=6; then C5=7; C6=14; C7=21; etc... (just the number of notes from the bottom of the pictured keyboard, which happens to start at Middle-C or C4)

Therefore the frequency for C4 is 1, C5 is 2, C6 is 4, C7 is 8, C8 is 16, etc, times the constant 261.63 Hz (hertz). Or going downscale (not shown on picture), the frequency of C3 is 1/2, C2 is 1/4, and C1 is 1/8 times the constant 261.63 Hz.




Note - People who like this post will also like The Best So Far