Maximum angle that a wheeled body can climb at I've only recently been introduced to the concept of static vs dynamic friction and I was looking to check my understanding by determining the max angle a wheeled body can climb. In the diagram below the car is moving up the hill.

So by resolving parallel to the hill, the max angle a car could climb at a constant velocity is when
$$
N \mu_{s} = N\mu_{k} + mg\sin\theta
$$
Because $ N = mg\cos\theta $ , the equation can be re-arranged to give
$$
\tan\theta = \mu_{s} - \mu_{k}
$$
So is the tan of the max angle simply the difference between the two coefficients? This confused me because a car is capable of climbing a much steeper hill than a train however the difference between rubber and asphalt coefficients is not much greater than the difference between steel on steel and would only give a angle difference of 1 or 2 degrees, so I am wondering if I have made a mistake or a wrong assumption?
 A: At just under 38 degrees the static equation shows that the force down the slope (mgsin) is equal to the resistance due to friction (mu mgcos), ie the vehicle starts to lose grip.  Even if does have an engine this makes it difficult.
A: If the tires are well inflated and the car is moving uphill at a low constant speed, then the only significant friction is the static friction force from the road which is pushing the car up the hill (as a result of the engine turning the wheels). Then the maximum angle occurs when: $μ_S$N – (mg)sin(θ) = 0  and: N - (mg)cos(θ) = 0 (Assuming weight is evenly distributed on four driven wheels.)
A: I assume your question is about the grip required to move up a hill, and that the car's engine has sufficient power and torque to do the job.
The car will not be able to progress up a hill if it is so steep that the force of gravity puling the car down the direction of the slope is greater than the frictional force that prevents the car from sliding.
The gravitational force is simply Wsin, where W is the weight of the car. The simplest model is to take the frictional force to be Wcos, in which case the two forces become just equal to each other in magnitude when tan = .
That is a simplified explanation, since it assumes, amongst other things, that the weight of the car is evenly distributed over the driving wheels. That might hold roughly for a four wheel drive vehicle, but would not for a two wheel drive.
A: The steepest angle a vehicle can climb is arctan(mu)
Or, tan^-1(coefficient of friction between tire and surface)
COF = 1,   Angle = 45°
COF = .7.  Angle = 35°
COF = .5.  Angle = 26°
COF = .25. Angle = 14°
COF = 0.   Angle = 0°
For adhesion:
COA = 2.   Angle = 63°
COA = 4.   Angle = 76°.  Oops, flips over backwards if GC is 18” above wheel patch line.
