A reader writes:
Is it possible to spin an object with two axes of rotation from just one initial impulse?
Specifically the question is about spinning a cricket ball.
If we take the X axis to be running between the wickets, Y to be the other horizontal axis, and Z to be vertical; spin bowlers generally spin the ball on a single axis somewhere between X and Y, depending on whether it's a break or a top-spinner. But, would it not also be possible to spin the ball simultaneously on the X and Z axes, so that it would be virtually impossible to tell which way the ball will turn until it lands? It would start off spinning on Z and X, but as it rotates about the Z axis the other axis of rotation would change.
I've been attempting to do this but it does not seem to work. The ball only seems to take one axis. I'm wondering if it's not actually possible to do it with just one spin, and it would require an initial spin followed by a separate impact?
James
Not only is it not possible to spin an object with two axes of rotation from just one initial impulse, but it's not possible for a rigid body to revolve simultaneously about more than one axis, at all, per Euler's rotation theorem. If a rigid body is spinning around an axis you call X, and you apply a force to it that, were it motionless, would cause it to start spinning around an axis you call Y that's at right angles to X, you'll end up with it spinning about some single axis between the two, the axis and speed of rotation being determined by the forces that've acted on the object.
An ongoing force acting on a rigid object, though, can cause an ongoing change in its axis of rotation. The axis will always pass through the object's centre of gravity, but it can be moved around in numerous ways.
In the case of a cricket ball, the axis of rotation can change slowly as a result of aerodynamic forces - which can also push the ball off the perfect ballistic trajectory it'd have if there were no air - and suddenly when the ball bounces off the ground, as it usually does in cricket.
(In an interesting piece of US/Commonwealth parallel evolution, cricket balls and baseballs are actually extremely similar in size, mass and construction. Apart from colour, the principal difference between the two is that a cricket ball has "equator" stitching while a baseball is made from two saddle-shaped pieces of leather, and the cricket ball has a harder surface. Cricket balls are also generally bowled somewhat slower than baseballs are pitched, and lose some momentum when they bounce off the ground. This lamentable reduction in lethal potential is, thankfully, largely compensated for by the cricket ball's harder surface and the bowler's ability to bounce it right up into the batsman's chin at ninety miles an hour.)
There are many other situations in which a spinning object can appear to have more than one axis of rotation, but they're all the result of forces acting on the object. A flying ball experiences aerodynamic forces, a billiard ball skids across the table with a spin different from its direction of travel, a comet is pushed around by gas from its own melting ice, a gyroscope precesses because of the pull of gravity, the wobble ("nutation") of the Earth's axis is the result of tidal forces from the moon and sun, and so on.
Regarding your own bowling experiments: Sorry, but you cannae change the laws o' physics. Even if you cheat, the ball's still only going to have one axis of rotation at a time, and that axis is only going to change in response to aerodynamics and hitting the ground.
Psycho Science, as I have brilliantly decided to call it, is a new regular feature here. Ask me your science questions, and I'll answer them. Probably.
And then commenters will, I hope, correct at least the most obvious flaws in my answer.
13 January 2012 at 6:52 am
As I recall Hyperion http://en.wikipedia.org/wiki/Hyperion_(moon) tumbles chaotically - can't recall why (space course was a long time ago) but it seems to be not accounted for in the post.
I remember a demonstration where the lecturer threw a wooden block in the air, spinning. Each of its three dimensions were different (like a pack of cards); it would spin stably on its short or long axes but not the middle one where, even in the short time it spent in the air, you could see it starting to tumble.
17 January 2012 at 9:41 pm
The situation is actually much messier, though spheres are a special case for which what you said is true.
A rigid object spinning in space with no external forces will have constant angular momentum. But the direction of rotation is detemined by the angular velocity - at any instant, the object appears to be rotating around the angular velocity vector. Angular momentum is simply the angular velocity multiplied by the moment of inertia - but the moment of inertia is a tensor. For a symmetrical object like a sphere or a cube, the tensor is diagonal so that it behaves like a scalar and the angular momentum is parallel to the angular velocity and the object just spins on one axis. For a non-symmetrical object, it's messier.
What do I mean by non-symmetrical? Here we get into rather deeper waters. Since the moment of inertia tensor is symmetric, there is a coordinate system - three orthogonal axes - along which it is diagonal. These are called the principal axes, and for an object like a book, they are aligned in the three obvious directions: height width and thickness of the book. The inertia tensor can then be described with three numbers, a moment of inertia about each axis. If the three numbers are equal, then any axes will do, the moment of inertia is a scalar, and the object rotates like a ball. If they're not all equal, though, things are messier.
If the three numbers are not equal, the angular velocity can point in a different direction from the angular momentum. The angular momentum remains fixed, but the object rotates around the angular velocity vector. Once it has rotated a little bit around the angular velocity vector, the three principal axes will have moved. Since the angular momentum is constant, this forces the angular velocity vector to point in a different direction. The precise motion is messy, and requires a good parametrization of orientation (which is messy because rotations are not commutative). But there are some special cases that can be easily described.
If you spin a triaxial object (that is, one whose moments of inertia about the three principal axes are all different) around any of the three principal axes, the angular momentum and angular velocity will line up, and the object will spin like a ball. Any other orientation and it will follow a rather messy set of differential equations, but you can look at what happens if you're close to but not exactly on a principal axis. Interestingly, if you're on the principal axis with the largest or the smallest moment of inertia, it spins almost regularly around the axis, and little perturbations tend to move it towards the axis. But if you spin it around the middle axis, any off-axis tendency grows, encouraged by most perturbations, so the object begins to tumble messily. You can try this by putting an elastic band around a (non-square) book and throwing it in the air; it'll spin nicely in the plane of the book or around the vertical axis, but around the horizontal axis it'll tumble.
To fully answer the question, as video game programmers are increasingly forced to, you need to start working with quaternions, a lovely generalization of the complex numbers in which a*b does not equal b*a. They were supposedly thought of by Alexander Hamilton on a bus ride, whereupon he hopped out and scribbled their defining relation i^2 = j^2 = k^2 = ijk = -1 on the stones of a bridge so he wouldn't forget.