Question 12.SP.7: Determine the rated speed of a highway curve with a radius o......
Determine the rated speed of a highway curve with a radius of ρ = 400 ft banked through an angle θ = 18°. The rated speed of a banked highway curve is the speed at which a car should travel to have no lateral friction force exerted on its wheels.
STRATEGY: You are given information about the lateral friction force— that is, it is equal to zero—so use Newton’s second law. Use normal and tangential components, because the car is traveling in a curved path and the problem involves speed and a radius of curvature.
MODELING: Choose the car to be the system. Assuming you can neglect the rotation of the car about its center of mass, treat it as a particle. The car travels in a horizontal circular path with a radius of ρ. The normal component a_{n} of the acceleration is directed toward the center of the path, as shown in the kinetic diagram (Fig. 1); its magnitude is a_{n} = v²/ρ, where v is the speed of the car in ft/s. The mass m of the car is W/g, where W is the weight of the car. Because no lateral friction force is exerted on the car, the reaction R of the road is perpendicular to the roadway, as shown in the freebody diagram (Fig. 1).
ANALYSIS: You can obtain scalar equations by applying Newton’s second law in the vertical and normal directions. Thus,
+↑\Sigma F_{y}=0\colon \qquad\qquad R\cos\theta-W=0\qquad R=\frac{W}{\cos\theta} (1)
\underleftarrow{+}\Sigma F_{n}=m a_{n}\colon \qquad\qquad R\sin\theta=\frac{W}{g}a_{n} (2)
Substituting R from Eq. (1) into Eq. (2), and recalling that a_{n}= v²/ρ, you obtain
\frac{W}{\cos\theta}\sin\theta=\frac{W}{g}\frac{v^{2}}{\rho}\qquad v^{2}=g\rho\tan\theta
Finally, substituting ρ = 400 ft and θ = 18° into this equation, you get v² = (32.2 ft/s²)(400 ft) tan 18°. Hence,
v = 64.7 ft/s v = 44.1 mi/h ◂
REFLECT and THINK: For a highway curve, this seems like a reasonable speed for avoiding a spin-out. For this problem, the tangential direction is into the page; because you were not asked about forces or accelerations in this direction, you did not need to analyze motion in the tangential direction. If the roadway were banked at a larger angle, would the rated speed be larger or smaller than this calculated value?
