Saturday, June 23, 2007

V-3: Centripetal Acceleration

Centripetal Acceleration: 
velocity squared divided by radius 
ά =v2/r

Centripetal Force 
A center seeking net force required to keep moving objects in a circular path. If the requirement is not met, then objects move into larger curved paths or go off on a tangent as they follow Newton's First Law.

Centrifugal Force
An outward seeking force in reaction to the centripetal force. Of course, centrifugal force can be used in spinning space habitats to simulate Earth's gravity.

Driving Forces cause motion.
For a freely falling object, Earth's gravity pulls the object toward the center of our planet.

Linear Acceleration: 
a = v/t

Newton's First Law (The Law of Inertia) 
In the absence of acting forces, an object continues with its present velocity (direction and speed).
If at rest, it stays at rest. 
If moving, it maintains velocity.
(JimOnote: Objects traveling at constant velocity in a circular path are not in accordance with this law; thus, there must be forces causing this circularity.)

Human Factors Axiom: 
After countless millenia, humans have grown so accustomed to Earth's gravity that it's become essential to our long term health. Extended periods in space under zero g conditions result in discomfort and other more severe reactions which range from discomfort to bone loss to swelling and of course reduce our ability readapt back to 1-g conditions (many books have been written by experts on this). Thus, extended time in space mandates a 1-g environment.

Two known ways to simulate gravity in space:
1. Straight line acceleration at 10 m/sec2. I've written many blog entries about this.
2. Circular (or angular) acceleration of 10 m/sec2 for larger habitats which must orbit a celestial body or must travel at extended periods at a constant velocity.

Many experiments have demonstrated that most humans are comfortable being on a surface that rotates 3 degrees per second or less. Extended time on quicker angular velocities will cause discomfort to most people, especially when the rotation stops and one's bodily equalibrium must readjust. Thus, the inner ear will have gotten used to the spin and will have to spend some time adjusting to no spin. Results range from dizziness to nausea, etc.

Thus, let's determine some dimensions of a human habitat. We have following requirements.
1. Cylindrical for the ample living space on the the inside of the outer surface.
Rotate this cylinder about its longitudinal axis
2. ---for angular velocity of 3 degrees.
3.---centripetal force equals Earth gravity.

First let's work on angular velocity.

Broad Range of Angular Velocities
Radius Angular
m/s m Rad/sec Deg/sec
1 0.1 10 573.25
IV r=v2/g ω = v/r ω * 180/π

Not having any idea how big a cylinder must be to fit above requirements, let's create a table to get us in the ball part.

Linear Velocity: Independently Vary the surface velocity of the cylinder as it rotates about its long axis. We'll arbitrarity start at one meter per second and increase by an order of magnitude as we search for the value which brings near an angular velocity near 3 deg /sec.

Radius depends on linear velocity because the radius must be the value which will produce centripetal acceleration of 10 m/s/s (same acceleration that freely falling objects undergo near Earth's surface; i.e., subject to Earth's gravity. Recall that we want to spin the cylinder in such a way to simulate gravity for those people living on inside of cylindrical surface.)

Angular Velocity (radians per second) is easily computed by multiplying radius times linear velocity.

Angular Velocity (degrees per second) is easily computed by converting rad/sec to degress per second by using conversion constant of 180/3.14 = 57.3.

Linear Angular Rotation
Velocity Radius Circumference Velocity Period
m/s km km Deg/sec Min
IV r = v2/g C = 2 π r (v/r) * (180/π) 6/(ω180/π)

Upcoming: features.
THE HIGH FRONTIER: Human Colonies in Space was written in 1976 by a brilliant scientist, Gerald O'Neill.
Sadly, Dr. O'Neill has since passed, but the 3rd edition was accomplished by his son, Roger O'Neill. Among the many concepts advocated in this book are the ideas of Islands One, Two and Three. Three classes of human habitats which are essentially orbiting cylinders of huge dimensions which rotate around their longitudinal axis and are not limited to orbiting Earth, but Dr. O'Neill has them orbit throughout the Solar System.
After some searching throughout the text of the 3rd edition, I discovered following specificiations:
HabitatLinear VelocityDiameterRotation PeriodCircumference
Island One1,500 ft. (pg 97)31 sec (pg 59)
Island Two1,800 meters (pg 93)4 miles (pg 93)
Island Three400 mph (pg 38)4 miles (pg 37)120 sec (pg 38)

Above table extracts values from O'Neill's book. Following table performs following conversions.

1. Radius vs. diameter. (Recall r = d /2).

2. 4 miles = 6.44 kilometers

3. 6.44 kph * 1 hr/3600 secs * 1000m/km = 178.88 meters per sec.

Acceleration from Centripetal Force
Island 1
225 m
47.1 m/sec
1,414 m
30 sec
Island 2
900 m
94.2 m/sec
5,655 m
60 sec
Island 3
120 sec
Given (r×g) 2 π r



g = v2/r = 9.80665 m/sec2Thus, v = (r×9.80665m/sec2)

NOTE: While we concentrate on such parameters as radius and angular velocity (which translates centripetal acceleration and thus to simulating Earth gravity), we have not yet metioned length of habitat. We have not done so because length of cylinder does not directly bear on the basic design dimensions needed to simulate Earth-g.
On the other hand, it does directly bear in that habitat length directly affects mass; thus, more length leads to more habitat mass which leads to more force needed to attain required spin for desired centrifugal force to simulate gravity.

Later text should note following items:
Habitat length is variable and even adjustable.

Possible way(s) to impact spin to the cylindrical habitat.

(in work) Impart Spin with Mass Driver

For space habitat to impact Earth like gravity to its human occupants, it should be shaped like a cylinder and should spin about its longitudinal axis. The resultant centrifugal force would force occupants against the inside of the outer cylindrical wall (or hull to use nautical terms). With centain combinations of angular velocity and radius (of circular end planes, known as "top and bottom" of cylinder), g-force accleration is achieved. While the principle of conservation of angular momemtum may maintain required spin once "g" is achieved, moving a huge habitat from zero spin to g-force spin will take a significant amount of power.

Significant power is available from high speed mass drivers (see links at tail of this blentry). While using mass drivers to impart spin to a habitat is a new concept (at least, I couldn't find it documented anywhere), it does have certain tradeoffs.

Mass limitations

Eventually upgrade to accelerated particles, because mass availability will have certain limitations which conversion to plasma then acceleration will considerably ameliorate.
To impart spin to the cylindrical habitats, I envisage high speed mass ejecting from outer surface of cylindrical hull such that mass particles travel tangentially to the habitat's circular outer surface. This would be even more effective if there were two such forces being applied at 180 degrees from each other on same plane (two forces applying forces opposite to each other on opposite sides of a circular object, known as "couple". Of course, this concept could be expanded with more tangential forces being applied at more equiangular locations.)

BACKGROUND MATERIAL from the web. Following sources discuss straight line mass drivers.

Space Studies Institute 
Advantages of mass drivers: Rugged design that uses off-the-shelf parts
Variable specific impulse
Performance degrades gracefully if individual units are faulty
Can use any mass as fuel (that is, as reaction mass)
Very high efficiency in conversion of electrical energy to kinetic energy
Mass drivers have benefited from the advances in solid-state switching and in high-performance ultracapacitors
Mass drivers do not require a nuclear power source, a solar power source similar to that used on the international space station will be sufficient.

++A launch capsule accelerates to high speed in a vacuum tube via electromagnetic levitation.
++Goes into orbit by an array of lasers (vs. legacy rocket which needs to carry its fuel).
++Escape V of Earth = 11.4kps
++Electric locomotive. supertrain. maglev. superconducting electro-magnets. Super-conductors levitate because they repel lines of magnetic force.
++Payload capsule rides magnetic force wave in superconducting track ahead of the capsule attract it, magnets behind repel it.
++Electromagnetic accelerator. "Mass drivers" in an evacuated tube.
++High-speed airlocks. allow a permanent vacuum. launch tube 125km. 10gs.
++Cargo capsules - colonizing space requires huge tones of mundane materials , immune to acceleration 45km. Linear induction motors. More voltage to handle higher magnetic thrust. 50000 m/s
++A kilogram of payload to orbit for less than a $1 of electricity.
++Ring capacitor w/ a volume of 182,000 m**3 required to store the 300,000 kwh needed to launch a 14 ton capsule. 50,000 kwh capacitor needs superconducting cable just over 6' in diameter.
++From the launch tube, elevation of 6000 meters, 5 km/s. Launch capsule will ride an array of laser beams.
++Beam of light and a block of ice. Ice block, and water superheated to 10,000 C flashes to steam. Superheated steam expands at 10.000 m/s.
++Rocket thrust, divided by the mass of fuel- specific impulse Isp.
++Laser induced specific impulses at high as 2,000 seconds.
++Readily available , environmentally safe, and low molecular weight. Water, in the form of ice, meets all these criteria.
++4ton slab of ice, 40cm (16") thick.
++at 40% efficiency, laser will produce 100N of thrust / megawatt of laser power. -- Insurance Plans for Humanity's Survival
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PERMANENT - Transportation - Mass Drivers
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mass driver
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Mass Drivers and Electromagnetic Launch Systems
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The Space Settlement Art Gallery
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