EXAMPLE 7.7 Daytona International Speedway GOAL Solve a centripetal force problem involving two dimensions. 1= a mg n cos 0 2 n sin e mg As the car rounds a curve banked at an angle 8, the centripetal force that keeps it on a circular path is supplied by the radial component of the normal force. Friction also contributes, although that is neglected in this example. The car is moving forward, into the page. (a) Force diagram for the car. (b) Components of the forces. PROBLEM The Daytona International Speedway in Daytona Beach, Florida, is famous for its races, especially the Daytona 500, held every February. Both of its courses feature four-story, 31.0 banked curves, with maximum radius of 316 m. If a car negotiates the curve too slowly, it tends to slip down the incline of the turn, whereas if it's going too fast, it may begin to slide up the incline. (a) Find the necessary centripetal acceleration on this banked curve so the car won't slip down or slide up the incline. (Neglect friction.) (b) Calculate the speed of the race car. STRATEGY Two forces act on the race car: the force of gravity and the normal force n. (See figure.) Use Newton's second law in the upward and radial directions to find the centripetal acceleration a Solving a = v/ for then gives the race car's speed. SOLUTION (A) Find the centripetal acceleration. Write Newton's second law for the car. Use the y-component of Newton's second law to solve for the normal force n. Obtain an expression for the horizontal component of n, which is the centripetal force F in this example. Substitute this expression for F into the radial component of Newton's second law and divide by m to get the centripetal acceleration. (B) Find the speed of the race car. Apply the centripetal acceleration equation. ma = = + mg n cos 0 mg 0 mg n = cos e = n sin e mg sin e = = mg tan 6 cos 0 Fen ma = ac ac = = mg tan e - g tan e m m = 5.89 m/s (9.80 m/s)(tan 31.0) LEARN MORE v = rac = (316 m)(5.89 m/s) = 43.1 m/s REMARKS In fact, both banking and friction assist in keeping the race car on the track. QUESTION What three physical quantities determine a minimum and maximum safe speed on a banked racetrack? (Select all that apply.) the angle the road surface makes from the horizontal the radii of the curves the coefficient of kinetic friction between the tires and the road surface the weight of the vehicle the coefficient of maximum static friction between the tires and the road surface the mass of the vehicle EXERCISE HINTS: GETTING STARTED | I'M STUCK! A racetrack is to have a banked curve with radius 240 m. What should be the angle of the bank if the normal force alone is to allow safe travel around the curve at 57.5 m/s? EXAMPLE 7.7 Daytona International Speedway GOAL Solve a centripetal force problem involving two dimensions. 1= a mg n cos 0 2 n sin e mg As the car rounds a curve banked at an angle 8, the centripetal force that keeps it on a circular path is supplied by the radial component of the normal force. Friction also contributes, although that is neglected in this example. The car is moving forward, into the page. (a) Force diagram for the car. (b) Components of the forces. PROBLEM The Daytona International Speedway in Daytona Beach, Florida, is famous for its races, especially the Daytona 500, held every February. Both of its courses feature four-story, 31.0 banked curves, with maximum radius of 316 m. If a car negotiates the curve too slowly, it tends to slip down the incline of the turn, whereas if it's going too fast, it may begin to slide up the incline. (a) Find the necessary centripetal acceleration on this banked curve so the car won't slip down or slide up the incline. (Neglect friction.) (b) Calculate the speed of the race car. STRATEGY Two forces act on the race car: the force of gravity and the normal force n. (See figure.) Use Newton's second law in the upward and radial directions to find the centripetal acceleration a Solving a = v/ for then gives the race car's speed. SOLUTION (A) Find the centripetal acceleration. Write Newton's second law for the car. Use the y-component of Newton's second law to solve for the normal force n. Obtain an expression for the horizontal component of n, which is the centripetal force F in this example. Substitute this expression for F into the radial component of Newton's second law and divide by m to get the centripetal acceleration. (B) Find the speed of the race car. Apply the centripetal acceleration equation. ma = = + mg n cos 0 mg 0 mg n = cos e = n sin e mg sin e = = mg tan 6 cos 0 Fen ma = ac ac = = mg tan e - g tan e m m = 5.89 m/s (9.80 m/s)(tan 31.0) LEARN MORE v = rac = (316 m)(5.89 m/s) = 43.1 m/s REMARKS In fact, both banking and friction assist in keeping the race car on the track. QUESTION What three physical quantities determine a minimum and maximum safe speed on a banked racetrack? (Select all that apply.) the angle the road surface makes from the horizontal the radii of the curves the coefficient of kinetic friction between the tires and the road surface the weight of the vehicle the coefficient of maximum static friction between the tires and the road surface the mass of the vehicle EXERCISE HINTS: GETTING STARTED | I'M STUCK! A racetrack is to have a banked curve with radius 240 m. What should be the angle of the bank if the normal force alone is to allow safe travel around the curve at 57.5 m/s?