{"id":2962,"date":"2020-10-11T15:25:47","date_gmt":"2020-10-11T20:25:47","guid":{"rendered":"https:\/\/laurentlessard.com\/bookproofs\/?p=2962"},"modified":"2020-10-11T16:05:21","modified_gmt":"2020-10-11T21:05:21","slug":"turning-radius","status":"publish","type":"post","link":"https:\/\/laurentlessard.com\/bookproofs\/turning-radius\/","title":{"rendered":"Turning radius"},"content":{"rendered":"

This week’s Riddler Classic<\/a> is a simple-looking question about the turning radius of a truck.<\/p>\n

\nSuppose I’m driving a very long truck (with length L) with two front wheels and two rear wheels. (The truck is so long compared to its width that I can consider the two front wheels as being a single wheel, and the two rear wheels as being a single wheel.)<\/p>\n

Suppose I can also rotate the front wheels by $\\alpha$ and the back wheels \u2014 independently from the front wheels \u2014 by $\\beta$. What is the truck’s turning radius?\n<\/p><\/blockquote>\n

Here is my solution:
\n[Show Solution]<\/a><\/p>\n

\n

When operating normally (no slippage), wheels can never slide sideways. They can only lead to motion in the direction they are pointing. In a steady turn (front and rear wheel angles constant), the wheels each trace out a circle and each wheel is tangent to its circle. Thinking about this a bit more carefully, we come to the conclusion that the front and rear wheels cannot trace out the same circle, since that would cause either the front or rear to slip. Therefore, there are actually two<\/em> turning radii (inner and outer). Here is a diagram illustrating the situation:<\/p>\n

\"\"<\/a><\/p>\n

By the law of sines, we have:
\n\\[
\n\\frac{r}{\\sin(\\tfrac{\\pi}{2}-\\alpha)} = \\frac{R}{\\sin(\\tfrac{\\pi}{2}-\\beta)}
\n\\]and by the law of cosines, we have:
\n\\[
\nL^2 = R^2 + r^2-2rR \\cos(\\alpha+\\beta).
\n\\]Putting these two together and solving for $r$ and $R$, we obtain:<\/p>\n

$\\displaystyle
\n\\frac{r}{L} = \\frac{\\cos(\\alpha)}{\\sin(\\alpha+\\beta)}
\n\\quad\\text{and}\\quad
\n\\frac{R}{L} = \\frac{\\cos(\\beta)}{\\sin(\\alpha+\\beta)}
\n$<\/span><\/p>\n

We have a few cases depending on the relative size of $\\alpha$ and $\\beta$:<\/p>\n