Despite its daunting size, the huge structure was in fact a very simple machine, essentially a massive slingshot exploiting the rotation of the KBO to hurl objects into space. Slugs of refined, processed matter were loaded into open-topped buckets at the KBO’s surface. For the first hundred kilometres, they were hoisted up the length of the flinger by electric induction motors, until they passed through a point at which gravitational and centripetal effects were exactly balanced. After that, the flinger’s own rotation did the rest of the work.
Baxter and Reynolds are describing the mining of a Kuiper Belt Object in the outer solar system. This asteroid-like body is spinning on its axis, and a tall tower (the “flinger”) has been erected on its surface at its equator. The tower is so high that the KBO’s rotation swings its upper end around at faster than orbital velocity. So the tower is being twirled around like a stone on the end of a string. If you can move buckets of refined ore up the tower far enough against the KBO’s slight gravity, they’ll soon get to a point at which they are impelled up to the top of the tower and launched into space without any more input of energy. That transition point corresponds to the middle bucket in my little diagram:It’s the point at which a bucket, released from the tower, would just hang around in orbit, right where it was. It’s also the point Baxter and Reynolds describe as the “point at which gravitational and centripetal effects [are] exactly balanced”.
But wait. We all know that gravity is a central force—it pulls inwards, and in my little diagram it’s pulling the buckets to the left, down the tower, towards the centre of the KBO. But centripetal means “centre-seeking”—a centripetal force is one that pulls inwards, towards the centre. So gravity is a centripetal force, and there simply can’t be a point at which “gravitational and centripetal effects” balance, because they’re the same thing.
What Baxter and Reynolds meant to say was that gravity and centrifugal effects are exactly balanced, because centrifugal means “centre-fleeing”. Where gravity and centrifugal effects balance, the middle bucket in my diagram experiences no net inward or outward force, and stays in orbit. Lower down the tower, gravity wins, so the buckets need to be pushed up that section. Higher on the tower, centrifugal force wins, and the buckets slide higher, against gravity, and eventually fly off into space.
Now, between them, Baxter and Reynolds have multiple degrees in maths, engineering and physics. They know this stuff. Why then did they choose the word centripetal instead of centrifugal? I suggest that it was because, in some quarters, the use of the word centrifugal is thought to mark you out as someone ignorant of physics. It’s toxic. Famous physicists and astronomers who make public statements using the word “centrifugal” find themselves being loftily denounced on social media. So writers will try to work around it, even when it means writing something nonsensical instead.
For generations, guilt about the word “centrifugal” was one of the few things people took away from their physics classes at school. I can still picture Mr Anderson (a very fine physics teacher) tapping his desk with the corner of the blackboard eraser (yes, it was that long ago) and intoning: “There’s! No! Such! Thing! As! Centri! Fugal! Force!”
He was making an important point, which is this:
If we spin something around in a circle (a stone on a string, a satellite in orbit), the only force it experiences is centripetal.
An object will move in a straight line unless acted on by a force. To make it move in a circle, it has to be pulled out of its straight-line path and made to accelerate constantly towards the centre of the circle. The centripetal force to generate that acceleration is provided by tension in a string (for the swinging stone), or by gravity (for the orbiting satellite). The centrifugal component is revealed to be just the tendency of the circling object to head off in a straight line, tangent to the original circle, as soon as the centripetal force is released (by letting go of the string, for instance). There’s no force pulling the object outwards. Hence my physics teacher’s emphatic attempt to drive the maxim, “There’s no such thing as centrifugal force,” into our reluctant little heads.
But what happens if we climb inside a rotating reference frame and rotate along with it? What does physics look like in that situation? Here’s Stanley Kubrick‘s gorgeous (and, for our purposes, accurate*) evocation of life inside a spaceship centrifuge, for 2001: A Space Odyssey:
It’s worth just clicking on the movie to see how eerie it looks. Of course if we step outside the centrifuge (and listen again to my physics teacher hammering away at his desk), then we can see that the jogging astronaut is whirling around in circles, and would be heading off in a straight line if the floor of the centrifuge wasn’t applying a centripetal force to his feet.
But from inside, with our viewpoint anchored to rotate along with the centrifuge, there certainly seems to be a force sticking the astronaut to the floor, doesn’t there? Indeed, if Isaac Newton had lived his life inside some gigantic space-borne centrifuge, without ever knowing that he was rotating, he’d have been able to formulate his Laws of Motion just fine, but with the addition of a centrifugal force. (He’d also have need to add another force, acting to deflect objects in motion relative to the rotating coordinates of his centrifuge—that one is called Coriolis force, and it’s the topic for another post, I think. [Note: I’ve now written that post.])
So if we choose to do physics in a rotating reference frame, then we find we have these extra forces to contend with—centrifugal and Coriolis.
Of course, they’re rather odd forces—they only crop up because we’ve chosen to use a particular kind of accelerating reference frame rather than an inertial reference frame. And they act to produce a specific acceleration, irrespective of mass—as if the force tuned itself to match the mass of the object it had to accelerate. For this reason they are sometimes called “pseudo-forces”, “fictitious forces” or simply “effects”. But when you figure with them, they work just like real forces. And, interestingly, gravity works just like one of these “pseudo”-force, always producing a specific acceleration—heavy objects fall no faster than light objects. That fact provided Einstein with the insight that led to General Relativity, and a way of treating gravity as being the result of a specific choice of accelerating reference frame. That insight is now a century old, but we’re strangely free of physics teachers hammering on their desks, saying, “There’s! No! Such! Thing! As! Gravity!”
But are there really situations where we, for preference, adopt a rotating reference frame? There sure are, and you’re sitting in one right now. For most purposes we treat the Earth as if it were stationary, despite our knowledge that it rotates. And that means that people whose job it is to calculate trajectories for rockets and missiles do so relative to a “stationary” Earth, while factoring in the effects of centrifugal and Coriolis forces. And meteorologists routinely deal with Coriolis force as it deflects air masses moving across a “stationary” Earth. There’s also sometimes benefit to be had in celestial mechanics, from adopting a rotating reference frame. The effects of gravity and centrifugal force are mathematically combined into a surface of “effective potential”, over which objects move subject to Coriolis force. That’s what’s happening in this contour map of the Lagrange points of the Earth and Sun, for instance:
So it’s all about adopting an appropriate reference frame—centrifugal and Coriolis forces are required in a rotating reference frame, forbidden in a non-rotating frame. Baxter and Reynolds, in my opening quote, were free to invoke centrifugal force as they followed the buckets up the length of the rotating flinger. But they seem to have become so nervous at the prospect of typing “centrifugal” that they just stuffed in the word centripetal instead, hoping no-one would notice.
So it’s fine to say “centrifugal”, as long as you are talking about a rotating reference frame. And actually, most times people use the word, they don’t really nail down the reference frame tightly enough to lay themselves open to justified criticism, anyway.
Finally, just in case you’re still anxious about this, I’m going to haul a few physics textbooks off the shelves, and take a look at their indexes:
Murray & Dermott, Solar System Dynamics. Four pages mentioning centrifugal acceleration, three centrifugal force, and four centrifugal potential. None concerning centripetal.
French, Newtonian Mechanics. One page on centrifugal force, one on centrifugal potential energy, and (by way of balance) two on centripetal acceleration.
Frautschi et al., The Mechanical Universe. Nine pages on centrifugal force, three on centripetal acceleration and four on centripetal force.
Feynman, The Feynman Lectures On Physics. Two pages on centrifugal force (in the second, as part of a discussion of how pseudo-forces arise from coordinate choices).
It really is okay to say “centrifugal”.
Postscript: For (much) more about centrifugal gravity in spacecraft, as featured in 2001: A Space Odyssey, see my post on Coriolis Effect In Rotating Space Habitats, and its Supplement. For a digression on really big, really fast rotating habitats, see my post Relativistic Ringworlds.
* Kubrick’s centrifuge film-set rotated to keep the astronaut at the bottom during filming, so it necessarily depicted a centrifugal force of one Earth gravity. This would be difficult to achieve with a centrifuge just 35 feet in diameter, the value given in the novel and depicted in the film (see here for a painstaking effort to retrieve the diameter of Kubrick’s centrifuge from movie footage and production stills). Such a centrifuge would need to complete 13 rotations per minute. Coriolis forces would be strong at that speed of rotation—deflecting limb movements and (more importantly) inducing abnormal fluid shifts in the semicircular canals of the inner ear during head movements. It might just be possible to adapt to the motion sickness induced in such a rapidly rotating environment—see Clément et al. for a recent review—but slower would be better.
In the novel of 2001: A Space Odyssey, Arthur C. Clarke addressed this problem by having the Discovery centrifuge rotate at 6 rpm, producing an approximation to lunar gravity. Clarke based his rotation speed on the hard data available in the 1960s, which suggested that 6 rpm was the limit of human tolerance. Whether lunar gravity is high enough to maintain long-term health is unknown—but there would seem to be some physiological wiggle room that would allow the centrifuge depicted in 2001 to rotate fast enough to maintain health, but not so fast as to be nauseating for its occupants.