If banked turns with friction are covered in your course, why not incorporate a "real" turn into the discussion? A scale drawing of the Indianapolis Speedway is shown below (1 lap = 2.5 miles), with each of the quarter mile turns being banked at 9.2°. A standard (and good) question is the maximum safe speed if the radius, angle, and coefficient of friction are known. Since the turns total one mile it is easy to show that 2pR = 1609 m, so R = 256 m. In the analysis it is assumed that the car rides the turns at a constant elevation, which is not consistent with how those cars take the turns!

Fnsinq + msFncosq = mv2/R, or

Fn(sinq + mscosq) = mv2/R, and (1)

Fncosq = msFnsinq + mg, or

Fn(cosq - mssinq) = mg (2)

v = (3)

Substituting R = 256 m, g = 9.8 m/s2, q = 9.2°, and ms = 1, v = 59 m/s = 133 mi/h, which is clearly less than the speed Indy cars travel through the turns. Aside from the fact that R is adjusted by entering a turn high, dropping down, and then exiting high, as the tires get hotter they become somewhat tacky and ms increases.