Acar is driving around a banked curve, with the road surface at an angle of 10.0º. if the radius of curvature of the road is 30.0 m and the coefficient of static friction between the tires of the car and the road is 0.65, what is the maximum speed (in km/hr) the car can go without skidding?

maximum speed 56 km/h

Explanation:

To apply Newton's second law to this system we create a reference system with the horizontal x-axis and the Vertical y-axis. In this system, normal is the only force that we must decompose

       sin 10 = Nx / N

      cos 10 = Ny / N

      Ny = N cos 10

     Nx = N sin 10

Let's develop Newton's equations on each axis

X axis

We include the force of friction towards the center of the curve because the high-speed car has to get out of the curve

     Nx + fr = m a

     a = v2 / r

     fr = mu N

     N sin10 + mu N = m v² / r

     N (sin10 + mu) = m v² / r

Y Axis  

     Ny -W = 0

     N cos 10 = mg

Let's solve these two equations,

    (mg / cos 10) (sin 10 + mu) = m v² / r

    g (tan 10 + μ / cos 10) = v² / r

    v² = r g (tan 10 + μ / cos 10)

They ask us for the maximum speed

   v² = 30.0 9.8 (tan 10+ 0.65 / cos 10)

   v² = 294 (0.8364)  

   v = √(245.9)

   v = 15.68 m / s

Let's reduce this to km / h

   v = 15.68 m / s (1 km / 1000m) (3600s / 1h)

   v = 56.45 km / h

This is the maximum speed so you don't skid