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 that is 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 is 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 A fancy word meaning line like or along a line.

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 it is at rest, it will stay at rest. If it is moving, it will maintain its 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
Linear
Velocity
Radius Angular
Velocity
Angular
Velocity
m/s m Rad/sec Deg/sec
1 0.1 10 573.25
10
10
1
57.32
100
1000
0.1
5.73
1000
100000
0.01
0.57
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
100
1
6.3
5.73
1.0
200
4
25.1
2.87
2.1
300
9
56.5
1.91
3.1
400
16
100.5
1.43
4.2
500
25
157.0
1.15
5.2
600
36
226.1
0.96
6.3
700
49
307.7
0.82
7.3
800
64
401.9
0.72
8.4
900
81
508.7
0.64
9.4
1000
100
628.0
0.57
10.5
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
HabitatRadiusLinear
Velocity
Circum-
ference
Period
Angular
Velocity
rvCTω
Island 1
225 m
47.1 m/sec
1,414 m
30 sec
12°/sec
Island 2
900 m
94.2 m/sec
5,655 m
60 sec
6°/sec
Island 3
3,600m
188.5m/sec
22,619m
120 sec
3°/sec
Given (r×g) 2 π r
C

v
v

r
×180°

π
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.


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