Re: Which wheels on a tractor turn faster when the front are smaller.

Let's say the front wheels have a radius of R_f, while the back wheels have a radius of R_b. If the tractor is going in a straight line at a linear (ground) speed v then the angular speed of the front wheels is
w_f = v / R_f
and the angular speed of the back wheels is
w_b = v / R_b
so if R_f < R_b then the angular speed of the front wheels is greater than that of the back wheels.

Below is a picture of a top view of a tractor that is in a turn.

The smaller rectangles represent the smaller front wheels, while the larger rectangles represent the larger back wheels. It is obvious that the linear (ground) speed of each of the four wheels is different because while in the turn the turn radius is different for each wheel, those radii being represented by the arcs R1, R2, R3, and R4.

We can say that the angular speed of the center of the tractor in the turn is roughly
w_c = v_c / R_avg
where v_c is the linear (ground) speed of the center of the tractor through the turn and R_avg is the average radius of the arcs R1 through R4. Note that this angular speed is qualitatively different than the angular speed of each of the wheels, because this is the turn angular speed of the tractor in the top view.

Now that we have the tractor's turn angular speed w_c we can calculate the angular speed of each of the four wheels. Since the linear (ground) speed of each wheel is
v_n = w_c * R_n
where R_n is one of the turn radii R1 through R4, then the angular speed of the "nth" wheel is
w_n = v_n / R_f
for the front wheels and
w_n = v_n / R_b
for the back wheels.

Therefore
w₁ = w _c * R₁ / R_b (the righthand back wheel)
w₂ = w _c * R₂ / R_f (the righthand front wheel)
w₃ = w _c * R₃ / R_b (the lefthand back wheel)
w₄ = w _c * R₄ / R_f (the lefthand front wheel)

The only way one of the back wheels can turn with larger angular speed than one of the front wheels is if the ratio R₃ / R_b is larger than the ratio R₂ / R_f. I am here talking about only these particular wheels because R2 is smaller than R4 and R3 is larger than R1, so I am here considering only the slowest-turning front wheel and the fastest-turning back wheel.

This depends not only on the relative sizes of the front and back wheels, but also on how tight the turn is. Other factors affecting the details for a particular tractor are the distance between the front and back axles, the separation of the front wheels, the separation of the back wheels, and things like whether or not the drive axle has a differential.

John Link, MadSci Physicist