I'm terrible at math, but I had this thought while playing around in Blender. The car near the pole needs to turn the wheel to maintain distance, but the car on the equator drives in a "straight" line. At what distance from the pole is the transition? I hope this makes the question clear, I ran into a character limit in the title.
Not homework. Planet size mass. Two cars driving, one following equator, the other a circle near the North pole. How far away does the driver need to turn the wheel to stay the same distance away?
11 Comments
Cort@lemmy.world · 9 pts · 4d
Anything not on the equator needs to turn the steering wheel.
You would have to specify which car is being driven to determine how far the wheel would need to be turned due to differences in steering control mechanisms. Some cars use rack & pinion gears for steering while others are street by wire. The latter have no fixed relationship between steering wheel being rotation angle and the amount the tires are being turned (that ratio changes based on speed)
shoomemer@lemmy.world · 5 pts · 4d
I think we could probably rephrase the question to specify what angle the front car tires have to be turned to make the circle. The answer would then be simply whatever angle is needed to have the car tires be tangent to the curve.
Cort@lemmy.world · 3 pts · 3d
Gotta narrow it further. Which tire, both? When turning the inner tire turns more than the outer (see Ackerman geometry).
shoomemer@lemmy.world · 1 pts · 1d
Nah, you right I forgot that inner turns slightly more aggressively. Can we instead argue that the midline of the car has to be tangent on the line of longitude that we'd be traveling on to maintain radial position?
Side note: it's not radial position on the surface skin of a sphere but I couldn't think of a better term. Any ideas for a better wording?
DarrinBrunner@lemmy.world · 2 pts · 4d
This seems obvious now that I read it.
Image, because that's how I understand things:
DjangoFett@lemmy.zip · 4 pts · 4d
I would also note that turning ratios aren't the only consideration, but distance traveled and therefore speeds.
Also not at all a mathologist.
SmoothOperator@lemmy.world · 1 pts · 3d
Note that the lines you drew are straight, no turning needed, but they are all equivalent to driving on the equator if you're on a sphere.
You don't get the north pole route by turning the equator route, but by translating it northwards.
Fermion@feddit.nl · 5 pts · 4d
The transition is continuous and doesn't occur at any singular distance from the pole. The angle of the steering wheel would likely go as something like cosine(theta)/r where theta is the angle from the pole (90-latitude), and r is the radius of the circle from the pole which would go as R*sine(theta) where R is the radius of the earth. The reasoning for picking cosine is that the effect you are describing comes from how well aligned the gravity vector is with the acceleration vector to go on a circle. When near the pole, the tires do all the acceleration. When near the equator, gravity does the acceleration. Putting that together the steering angle would be proportional to cotangent(90-latitude)/R. Of course this approximation assumes an absurdly smooth sphere. Any local topology completely overwhelms this effect.
Lysergid@lemmy.ml · 3 pts · 4d
From someone who is bad at math too my, probably wrong, guess is something like angle between longitude and latitude minus 90. Why - 90, because at equator angle is 90. 90-90=0 which is the angle at which equator car wheel is turned (not turned at all)
Edit: I realized I’m being dumb angle between longitude and latitude is always 90. Which means my math kinda checks out. Both cars will have wheels straight. Curvature of sphere will make sure car on pole is on the same distance because inner wheels are taking shorter distances than outer wheels
phailhaus@piefed.social · 1 pts · 4d
Your edit is incorrect. A straight path on a sphere creates a great circle. If you are on the equator and aligned with it, you'd stay on the equator. Elsewhere, you'd follow an identically-sized circle around the sphere. Starting near the pole with it off to your left, you'd drive all the way down to the other pole, passing it on your right, before driving back up.
Earth isn't really a sphere, so the circles wouldn't be quite identical in reality but close enough.
Zephyr@sh.itjust.works · 1 pts · 1d
Depends on the car. You could build a car just to do that circle so the turn would be 0°. Of course as others have pointed out, and ideal normal car aka straight is straight and it drives perfectly straight wouldn't need to turn at all at the equator.