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Sobeita
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I've seen dozens of examples of well-made games and demonstrations that seem to handle this concept perfectly well, and yet I can't find a documented method to approaching it. Basically, a tire on an incline will begin to roll rather than slide down the slope, but a spinning tire dropped on a level surface will be inclined to move in the direction it's spinning. Slippery surfaces and bald tires make both transitions slower, hence slipping, spinning out, skidding, etc. I've established these general rules with a thought experiment:
1. The two basic properties of the circle are X and theta (position and rotation). They can both be derived over time to form velocity, acceleration, and so on. Both are completely independent properties, but there is interaction due to other conditions in the system (see below.)
2. The circle will seek a state of equilibrium in which V of theta and V of x are equal. Equilibrium is achieved through (some function of) friction between the circle and the plane.
(Corollary: all forms of friction can help the circle achieve equilibrium, including air friction. A baseball, for instance, curves as its rotational velocity affects its trajectory.)
Predictions of interaction:
a) If V of theta exceeds V of x, the circle will slip as it spins faster than it moves until velocities level.
b) If V of x exceeds V of theta, the circle will slide as it moves faster than it spins until velocities level.
c) If V of x and V of theta are different across the origin, the circle will move in the direction of V of x but spin in the direction of V of theta - thus achieving Slip 'n' Slide.
(If V of theta equals V of x, the circle is in "pure roll".)
So, what model does the interaction follow? I am almost certain there's a simple method. Just look at Fantastic Contraption for example.
I was able to find this page: http://cnx.org/content/m14312/latest/
...however, it only handles pure roll conditions. I even found some eighth-year college videos - http://ocw.mit.edu/OcwWeb/Physics/8-01Physics-IFall1999/VideoLectures/index.htm - which were really, really high quality, but still never left pure roll. There were some mind-boggling gyroscope demos, but nothing to help my little problem.
1. The two basic properties of the circle are X and theta (position and rotation). They can both be derived over time to form velocity, acceleration, and so on. Both are completely independent properties, but there is interaction due to other conditions in the system (see below.)
2. The circle will seek a state of equilibrium in which V of theta and V of x are equal. Equilibrium is achieved through (some function of) friction between the circle and the plane.
(Corollary: all forms of friction can help the circle achieve equilibrium, including air friction. A baseball, for instance, curves as its rotational velocity affects its trajectory.)
Predictions of interaction:
a) If V of theta exceeds V of x, the circle will slip as it spins faster than it moves until velocities level.
b) If V of x exceeds V of theta, the circle will slide as it moves faster than it spins until velocities level.
c) If V of x and V of theta are different across the origin, the circle will move in the direction of V of x but spin in the direction of V of theta - thus achieving Slip 'n' Slide.
(If V of theta equals V of x, the circle is in "pure roll".)
So, what model does the interaction follow? I am almost certain there's a simple method. Just look at Fantastic Contraption for example.
I was able to find this page: http://cnx.org/content/m14312/latest/
...however, it only handles pure roll conditions. I even found some eighth-year college videos - http://ocw.mit.edu/OcwWeb/Physics/8-01Physics-IFall1999/VideoLectures/index.htm - which were really, really high quality, but still never left pure roll. There were some mind-boggling gyroscope demos, but nothing to help my little problem.
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