Question

A civil engineer wishes to redesign the curved roadway in the figure in such a way that a car will not have to rely on friction to round the curve without skidding. In other words, a car moving at the designated speed can negotiate the curve even when the road is covered with ice. Such a ramp is usually banked, which means that the roadway is tilted toward the inside of the curve. Suppose the designated speed for the road is to be 11.8 m/s (26.4 mi/h) and the radius of the curve is 34.0 m. At what angle should the curve be banked?

SOLVE IT Conceptualize The figure shows the banked roadway, with the center of the circular path of the car far to the left of the figure. Notice that the horizontal component of the normal force participates in causing the car's centripetal acceleration.

Categorize The car is modeled as a particle in equilibrium in the vertical direction and a particle in uniform circular motion in the horizontal direction.

Analyze On a level (unbanked) road, the force that causes the centripetal acceleration is the force of static friction between tires and the road. If the road is banked at an angle θ as in the figure, however, the normal force n has a horizontal component toward the center of the curve. Because the road is to be designed so that the force of static friction is zero, only the component nx = n sin θ causes the centripetal acceleration. Write Newton's second law for the car in the radial direction, which is the x direction: (1) Fr = n sin θ = mv2 r Apply the particle in equilibrium model to the car in the vertical direction: Fy = n cos θ − mg = 0 (2) n cos θ = mg Divide Equation (1) by Equation (2): (3) tan θ = v2 rg Solve for the angle θ: θ = tan−1 (11.8 m/s)2 (34.0 m)(9.80 m/s2) = 22.68 Correct: Your answer is correct. °

Finalize Equation (3) shows that the banking angle is independent of the mass of the vehicle negotiating the curve. If a car rounds the curve at a speed less than 11.8 m/s, the centripetal acceleration decreases. Therefore, the normal force, which is unchanged, is sufficient to cause two accelerations: the lower centripetal acceleration and an acceleration of the car down the inclined roadway. Consequently, an additional friction force parallel to the roadway and upward is needed to keep the car from sliding down the bank (to the left in the figure). Similarly, a driver attempting to negotiate the curve at a speed greater than 11.8 m/s has to depend on friction to keep from sliding up the bank (to the right in the figure). For the above example, suppose that the coefficient of static friction between the tires and the road is 0.41.

What maximum speed could a car negotiate the exit ramp without skidding?

Answer 1