A curve of radius 151 m is banked at an angle of 11°. An 838-kg car negotiates the curve at 86 km/h without skidding. Neglect the effects of air drag and rolling friction. Find the following.
(a) the normal force exerted by the pavement on the tires
kN ?

(b) the frictional force exerted by the pavement on the tires
kN ?

(c) the minimum coefficient of static friction between the pavement and the tires?

a) 8.52 kN

b) 0.74 kN

c) 0.087

Explanation:

a)

There are 3 forces acting on the car on the banked curve:

- The weight of the car, , vertically downward

- The normal force of the pavement on the tires, N, upward perpendicular to the road

- The force of friction, , down along the road

Resolving the 3 forces along two perpendicular directions (horizontal and vertical), we obtain the equations of motions:

x- direction:

(1)

y- direction:

(2)

where

is the angle of the ramp

is the force of friction

m = 838 kg is the mass of the car

r = 151 m is the radius of the curve

is the speed of the car

Solving eq.(2) for Ff and substituting into eq.(1), we can find the normal force:

From (2):

(3)

Substituting into (1) and re-arranging,

b)

The frictional force between the pavement and the tires, , can be found by using eq.(3) derived in part a):

where we have:

is the normal force

is the angle of the ramp

m = 838 kg is the mass of the car

is the acceleration due to gravity

Substituting the values, we find:

c)

The force of friction between the road and the tires can be rewritten as

where

is the coefficient of static friction

N is the normal force exerted by the road on the car

In this problem, we know that

N = 8.52 kN is the normal force

is the frictional force

Therefore, the minimum coefficient of static friction between the pavement and the tires is: