After the introduction (by more recent posters) of the “artificial gravity” and “buoyancy” of the centrifuge and the discussion of the direction of the flow of liquids (such as a pilot’s blood) in this environment, I’d like to try a more complete discussion of the centripetal/centrifugal system. I hope this is not too long winded.
The thing that struck me (I hadn’t thought of it earlier) was the display of “gravitational” properties within the centripetal/centrifugal system. What struck me was why denser material (such as in the lab centrifuge) settles towards the “outside” of the arc away from the center of rotation, and why lighter material (such as the helium balloon in the car) settles toward the “center” of the arc. That sounds right to us instinctively, but in a “real” gravity environment the denser material settles toward the “center” of the gravitational field, and not towards the outside of it. I really had to think about this for a day or more to understand why centripetal force created “gravity” seems to work in a manner that is opposite from “real” gravity. The answer turned out to be very interesting.
First I discovered I had to agree with heedm, that the rotation of an object around an arc is in fact being subjected to an “acceleration” force, which is the centripetal force. But I accepted this fact with certain qualifications, which I’ll explain. The following quote is from the online version of Microsoft’s Encarta Encyclopedia, from an article entitled “Acceleration (velocity)”.
“Acceleration (velocity), also known as linear acceleration, rate at which the velocity of an object changes per unit of time. Acceleration is a vector—that is, it has both magnitude and direction. Acceleration is uniform if the rate of change of an object's velocity is the same over successive and equal time intervals. For example, an object that is released and allowed to fall freely towards the ground is accelerated uniformly. An object tied to a string and swung at a constant speed in a circle above a person's head is also accelerated uniformly; in this case, the acceleration vector points along the string toward the person's hand.”
The quote is confusing in that it basically states that changes in velocity (distance covered per unit of time) is a fundamental characteristic of “acceleration”. Then the article states that tying an object to a string and swinging it over your head at a constant speed is also imparting uniform acceleration to that object. My question regarding the over-the-head description was, “what happened to the “change in velocity” characteristic of acceleration?
Then it hit me, there must be in fact 2 different types of acceleration, “velocity” acceleration, and “directional” acceleration. Pure “velocity” acceleration is applied at 0 degrees (or 180 degrees) relative to an object’s current line of travel, so that only its velocity is changed but its direction remains unchanged. Pure “directional” acceleration is applied at 90 degrees (or 270 degrees) relative to an object’s current line of travel, so that only its direction is changed, but its velocity remains unchanged. Any application of “acceleration” between 0 and 90 degrees applies a combination of both types.
Pure “directional” acceleration, applied at right angles to the line of travel, does NOT alter the velocity of a moving object, only its direction. Pure “directional” acceleration also has to maintain this 90-degree angle of applied force, even while the object’s direction of travel is in the process of being changed. A mechanical system of rotation, with a mechanical means of applying a centripetal force around a constant radius, satisfies this requirement perfectly, and produces pure “directional” acceleration. This acceleration however, has to overcome the inertia of the object being accelerated “directionally”, just as it does when an object is accelerated with pure “velocity” acceleration.
Why physicists don’t describe “acceleration” as existing in both types I don’t know. In a field like ballistics for example, both types of acceleration ARE dealt with, and both are often dealt with separately. In ballistics, longitudinal acceleration and vertical acceleration (from the force of gravity) are at right angles to each other relative to the projectile, just as “velocity” and “directional” acceleration are at right angles to a moving object’s line of travel.
Now that “centripetal acceleration” has been dealt with, let’s move on to the relationship of mass, inertia, gravity and how gravitational properties are displayed in a centripetal/centrifugal system.
The reason that centripetal/centrifugal systems demonstrate gravitational properties is hinted at in the following quote from another Encarta article entitled “Mechanics”.
“A massive object will require a greater force for a given acceleration than a small, light object. What is remarkable is that mass, which is a measure of the inertia of an object (inertia is its reluctance to change velocity), is also a measure of the gravitational attraction that the object exerts on other objects. It is surprising and profound that the inertial property and the gravitational property are determined by the same thing. The implication of this phenomenon is that it is impossible to distinguish at a point whether the point is in a gravitational field or in an accelerated frame of reference. Einstein made this one of the cornerstones of his general theory of relativity, which is the currently accepted theory of gravitation.”
The quote from the “Mechanics” article points out the remarkable direct relationship between an object’s mass (or inertia) and its gravitational attraction. A lighter object has low inertia, but also have low gravitational attraction to other objects. A heavy object has higher inertia, and a proportionately higher gravitational attraction to other objects. I’d like to use 2 simple examples to illustrate the direct relationship between an object’s inertia and its gravitational attraction. The first example will illustrate the inertia of 2 objects, and the second example will illustrate the gravitational attraction of the same 2 objects.
The first example is 2 steel balls on a flat low friction surface. One ball weighs 1kg and the other weighs 10kg. Both balls are at rest (relative to the flat surface). Now lets apply a 1 Newton force (as pure “velocity” acceleration) to each ball for 1 second. Recall the basic formula for acceleration which is: a=F/m (F=Newtons, m=mass, a=acceleration). Also lets use the following formula to help calculate the resulting acceleration: v2=v1+at (v2=final vel, v1=init vel, a= accel, t=time). Recall that 1 Newton is equal to 1kg/m/s.
For the 1kg ball, the final velocity is v2 = v1 (0) + a (F (1kg/m/s)/ m(1kg)) * t (1 sec), which is 1 meter/sec.
For the 10kg ball, the final velocity is v2 = v1 (0) + a (F (1kg/m/s)/ m(10kg) * 1 (1 sec), which is .1 meters/sec.
The inertia of the 10kg steel ball offered 10 times more resistance to the “acceleration” force than the 1kg steel ball, resulting in 1/10th the final velocity of the 1kg ball. You can call the resistance to acceleration a “reaction force” if you’d like, but that would just be semantics, as inertia is a very real force. Again, this inertia is just as real in “directional” acceleration as it is in “velocity” acceleration.
Now let’s use the same steel balls for the second example. This time lets drop them from a height of 10 meters. Discounting air resistance, we know that if you release the steel balls at the same time, the steel balls will hit the ground at the same time. We also know that if you remove the air from a glass cylinder and do the same experiment with a feather and a steel ball, they will also both hit the ground at the same time.
Now here’s the remarkable thing about this second example. We know that gravity is an acceleration force, and has an acceleration value of 9.8 m/s. We also know from the first example that the 10kg ball has 10 times the inertia (resistance to acceleration) that the 1kg ball has. The difference in inertia would be even greater between the feather and the steel ball. So why do all the objects hit the ground at the same time, given that the same acceleration is being applied to all the objects, all of which have different masses and inertia? The answer is that the amount of gravitational attraction exerted by each object towards the earth varies directly with its mass. The greater the mass, the greater the gravitational attraction. This means each object will experience the same units of acceleration (velocity change), because the amount of gravitational attraction between each object and the earth will vary based on each object’s mass. So the more inertia an object has, the greater the attraction to overcome that inertia. The differences in attraction MUST be directly proportional to the differences in mass (or inertia) or else the objects could NOT hit the ground at the same time.
In a normal gravity environment where buoyancy is concerned, the denser objects (density expressed as kg/cm^2) will go to the bottom, due to the greater gravitational attraction of the denser objects. In an artificial gravity environment created by a centripetal force, there is no gravitational attraction pulling any of the objects towards the center, since the acceleration around the arc is mechanical and not gravitational. Thus the mass of the object(s) cannot use their innate gravitational attraction to help draw them towards the center of the arc. Instead, the inertia of the objects resists being drawn into travel around an arc, because the “directional” acceleration is turning them away from straight-line travel. The greater the mass, the greater the inertia and the greater the resistance to being drawn into the arc by the “directional” acceleration of the centripetal force. So in this environment, the greater inertia of the denser objects (or material) causes them to settle towards the outside of the arc. That’s how the lab centrifuge works.
BTW, the “centrifugal” force is real, and is nothing more than the inertia of an object resisting the purely “directional” acceleration of the “centripetal” force. Again remember the formula for the centripetal force, which is:
F = m * v^2 / R (F = centripetal force, m = mass, v = velocity, R = radius)
Note the things in the formula that increase the value of the centripetal force. Increasing the mass will increase the force, because of the greater inertia of the greater mass. Increasing the velocity will increase the force squared, since increasing the velocity increases the rate of the “directional” acceleration of the same mass around the arc (more degrees of arc are covered per second). The greater the change rate in direction, the greater the resistance to that change, and this term is squared. To me, nothing more clearly illustrates the effect of inertia on a centripetal/centrifugal system than a velocity change. Decreasing the radius also increases the force, as this also increases the rate of the “directional” acceleration (again, more degrees of arc are covered per second).
P.S. I wish it was easier to write formulas in this forum.
(edited for typos and greater clarity)
[ 31 December 2001: Message edited by: Flight Safety ]</p>