What is the truck’s turning radius?

Question

Suppose 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.)

Question 1: Suppose I can rotate the front wheels up to 30 degrees in either direction (right or left), but the rear wheels do not turn. What is the truck’s turning radius?

Question 2: Suppose I can also rotate the rear wheels — independently from the front wheels — 30 degrees in either direction. Now what is the truck’s turning radius?

Answer 1

We'll have to make a few more assumptions here:

1. Turning radius is measured from the inner front wheel; during the turn, all four wheels follow different arcs, so – knowing nothing about cars – I'm just picking the most intuitive wheel to measure
2. Length L of the very long truck is the distance between the front wheel and back wheel; imagine the difference in turning radius between two trucks of equal length, one with 100 ft between axles and the other with 1 ft between axles

With those additional assumptions, we can use triangles and trig calculations to find the turning radius. Picture both axles as infinite lines in space...

1. Turning the front wheel(s) 30° creates a right triangle with the front axle as the hypotenuse (aka turning radius), length L as the height, and the rear axle as the base. Use a trig function to find the hypotenuse...
2. Same idea here, except instead of a right triangle, turning the back wheel(s) 30° (obviously along the same curve as the front) we have an isosceles triangle. Use some basic division and a trig function to find the hypotenuse...

Hope this helps!