The contact patch on a bike is behind the point at which the steering axis would meet the road (negative trail). When the bike falls left, the contact patch stays where it is (held by the road surface) and because the steering axis is in front of the contact patch, this induces a left turn. This throws the bike to the right and back upright.
The end result is a self correcting mechanism for staying upright.
It’s an easy thing to imagine in your mind’s eye once you see it, but a hard thing to describe.
From my reading of SDSAB Karen’s Staff Report, what she refers to as “trail” is similar to castor angle, except that she states it as a (presumably linear) difference plus the minus the yoke offset rather than an angle. The two would be related per (R[sub]wheel[/sub]-O[sub]yoke[/sub])ּcos(θ[sub]castor[/sub])=L[sub]trail[/sub]. That form is less general (it incorporates both the wheel radius and angle inseperably into one number), which may be simpler or useful for Dr. Klein’s formulation of the dynamic model for a bicycle; however, in general use for vehicle dynamics the castor (or caster, as my Marks’ Standard Handbook for Mechanical Engineers states it) angle is what is specified. Here’s a site that gives an example of the terms stated (though he equates rake angle with castor, which is inconsistant with experience; however, rake doesn’t seem to have a technical definition that I can find so it’s possible that this is the more general definition).
In any case, the castor angle and yoke offset/rake angle/trail is a critical parameter in vehicle control, but again beyond the scope of what I was attempting to describe; I was merely trying to indicate, in general terms, why gyroscopic forces from the spinning wheels (and cranks, and anything else in rotational motion) are not a dominant factor in bicycle balance.
An excellent question, which unfortunately is difficult to answer simply. The tidal forces on the Moon are such that they would cause the Moon to speed up: They’re trying to match the Moon’s orbital period to the Earth’s rotational period, and the Earth rotates faster than the Moon orbits. But you can’t just speed up, in place, in an orbit. If you’re in a stable orbit and you speed up, that’ll put you into a higher orbit. However, in the process of going into a higher orbit, you’ll slow down, due to conservation of energy: You now have more potential energy, being higher up, so you have to have less kinetic energy (and therefore speed) to make up for it. In fact, you actually lose more speed in the process of getting to the higher orbit than the speed you gained in the first place that caused you to change orbits. So on net, it’s actually a decrease of speed.
“The most cock-eyed, contrary to all common sense, difficult aspect of ballistics around a planet is this: To speed up, you slow down; to slow down, you speed up.” – Richard Colin Ames Campbell, The Cat Who Walks Through Walls (Robert Heinlein)
Every stable circular orbit has a characteristic speed; in order to expand the orbit, you have to reduce the speed. I speak of circular orbits 'cause it makes the math easier, but the same applies to elliptical orbits, save that the speed varies with respect to distance between the bodies. Bytegeist is correct that the Earth and Moon will tidally lock to each other as Pluto and its “moon” Charon are. See this site for a simple explaination of how tides work. Using their figure of a loss of 1 second per 50k years, a back-of-the-envelope calculation gives on the order of 4.3 billion years before the Earth and Moon are tidally locked together, by which time we should all be converted to giant energy clouds roaming around the universe or pan-dimensional beings or whatever the next stage in evolution is according to Star Trek.
However, Bytegeist is incorrect, or at least imprecise, in referring to friction as the agent for this action. Although some of the work done by tidal forces is lost to structural hysteresis (Earth/Moonquakes and other geological phenomena), the vast majority is converted from rotational momentum to orbital momentum; every time Earth’s tidal bulges pass 'round they get pulled by/pull on the Moon, which ‘slings’ the Earth/Moon a little further out. (It is traditional, for the sake of simplicity, to assume that the Moon is revolving strictly about the Earth, but in actually both orbit about their common center of mass, called the barycenter, so this ‘slinging’ has an effect on both bodies.) Although the term friction is often used in common discourse to refer to any nonconservative loss of energy from a system, in technical terms it has a very specific–if often contended–meaning which does not pertain to tidal effects.
It’s painful but I have to disagree with Stranger slightly on this one. The earth’s tidal bulge doesn’t rotate with the earth. It stands still relative to the moon and the earth rotates through it.
The moon is getting further away because the earth’s rotation is gradually pulling it along faster in its orbit, which in a seeming paradox results in its getting further away and slowing down.
If anything isn’t clear, and probably much isn’t, further questions will be answered, if possible.
On going over the cite I notice that the final paragraph is a little garbled. It should read “The rotation of the earth is slowed by the friction of the earth rotating through the tidal bulge.”
Never fear; I stand corrected. I was trying to simplify in my own mind the dynamics of the problem and ended up making a blatant misstatement. You are correct that the tidal bulges remain (mostly) stationary with respect to the Moon. This results in ocean tides; in addition, it also creates a tidal bulge in solid crust. This bulge, however, leads the moon, slinging it outward in from its orbit every so slightly. The energy for this comes from the Earth’s rotational momentum, which is consequentially reduced. [url=]Here is perhaps a better description and diagrams of what occurs.
The reason it leads the moon is because of what you term structural hysteresis–the bulge doesn’t subside before it can rotate out from “under” the moon. You differentiate that from “friction”–you mention that that does not encompass every “nonconservative loss of energy from a system” but I don’t really see why we can’t invoke friction here unless you want to call it viscosity or something.
That’s my take on it too. The earth moves the tidal bulge in the direction of its rotation because of the resistance of the mass of the earth (including the oceans) to deformation. The bulge is thus pushed forward until the gravitational pull of the moon balances the push of the earth. This simultaneously speeds up the moon and slows the rotation of the earth.
Maybe whether it’s “friction” or “structural hysteresis” is just a matter of semantics. I favor friction because there is frictional loss when the material of the earth flexes as it rotates through the bulge like the body of a tire rotates through the flat spot in contact with the road surface.
Because I’m a mechanical engineer and am pedantically required to differntiate between friction (the tangetial shearing contact force between two solid objects or a solid and a viscous fluid) and other forms of nonconservative mechanical loss which are often lumped under the term “internal friction” or somesuch. I also squirm at the use of the terms “air friction” and “frictional heating” as used to describe aerodynamic drag and ram pressure heating, respectively, even though ‘everybody knows’ what is meant by these terms when they read it in a CNN.com article. I just gotta be me, I guess. Carry on.
I guess that depends on how you want to model the system. If you want to think of the Earth as being composed of discrete little pieces of stuff, then you can consider the action and loss between them to be friction. In general, though, it’s going to be modelled as a bulk continuum, and nothing like what a tribologist could quantify as friction occurs. Like I said, though, I’m being pedantic owing to my training and vocation. If you want to refer to it as friction, the majority of the world will get the general sense of what you mean. If you see the guy over in the corner nashing his teeth, never mind him: he’s just a frustrated sturctural mechanics or materials science major.
We need a smilie for a light bulb going on, like in the comics. I see why you thought hysteresis. Take an automobile axle. As the wheel rotates the axle is subjected to stress reversals. The sections that are under tension elongate and those under compresseion shorten so there is shear in the material but no slipping of parts of the material past one another, hence no friction. Yet the axle gets hot, why? Hysteresis in the material, of course.
I think that in the earth there would be both hysteresis and friction. As I look at the face of the Sierra Nevada range I see big slabs of granite. Those slabs would act like beams, bending in response to the tidal bulge but there would be no slipping of layers past one another in the slab so no friction but there could be hysteresis.
However, looking closer at the mountain’s face I see several granite slabs layered, one atop another. Ther could be some slipping of those layers past each other resulting in friction.
