Acar rounds a curve that is banked inward. the radius of curvature of the road is r = 124 m, the banking angle is θ = 28°, and the coefficient of static friction is μs = 0.34. find the minimum speed that the car can have without slipping.

The minimum speed that the car can have without slipping is

Explanation:

First it’s important to remark two things 1) If the car doesn’t round the curve with enough speed it will slip down the curve, so to avoid that the static friction between the car and the street should be upwards the incline (see Figure 1) 2) Because of the car is moving on a curve it’s describing a circular motion and in this kind of motion there is always a radial acceleration pointing towards the center of the circumference (see figure 2). Knowing that, we sketch our free-body diagram in Figure 1 and use Newton’s second law on the y-axis:

(1)

(the acceleration on the y-axis is zero because if the car doesn't slip it is not moving in the y direction)

(2)

With Ny = N*cos(28°) the y component of the normal force, W=mg the weight of the car and fsy=fs*sin(28°) the y component of static friction, the magnitude of the static friction (fs) is μs*N = 0.34*N, so fsy=0.34*sin(28°). Using that on equation (2)

(3)

Solving for N:

(4)

Similar using Newton’s second law on the x-axis:

(5)

(6)

(7)

(8)

Using the fact that and using (4) in (8):

(9)

Solving for v with :

(10)

That is the minimum speed that the car can have without slipping.

probably not if net force is zero