You have a car traveling on a circular track of radius r and a velocity v. The t

Question

You have a car traveling on a circular track of radius r and a velocity v. The track is banked at an inclination of angle . The coefficient of static friction is s. Gravity is 10 m/s2. In each of the following situations, does the car:

A) Fly off the top of the track

B) Slide down and off the bottom of the track

C) None of the above

Show all work that shows this. Guesses do not receive any credit.

1. = 45 degrees, r = 1000 m, v = 90 m/s, s = 0.5

2. = 20 degrees, r = 1000 m, v = 90 m/s, s = 0.2

3. = 70 degrees, r = 800 m, v = 200 m/s, s = 0.7

4. = 30 degrees, r = 800 m, v = 5 m/s, s = 0.3

5. = 30 degrees, r = 800 m, v = 78 m/s, s = 0.3

Explanation / Answer

Let us consider car is moving ina circular path of radius R at an angle with the bottom at a velocity v. If mass of the car is m, then the forces acting on it are: Fc = mv2/R is the centripetal force towards the center N = smg is the normal force with coefficient of kinetic friction k If we resolve the forces into components, the horizontal components are: Fc = mv2/R Nsin = smgsin For the car to be in equilibrium these two forces must be balanced. case 1: = 45o, R = 1000 m, v = 90 m/s, s = 0.5 Fc = 8.1m N Nsin = 3.53m N As Fc > Nsin , car moves towards the center OPtion(B) case 2: = 20o, R = 1000 m, v = 90 m/s, s = 0.2 Fc = 8.1m N Nsin = 0.68m N As Fc > Nsin , car moves towards the center OPtion(B) case 3: = 70o, R = 800 m, v = 200 m/s, s = 0.7 Fc = 50m N Nsin = 6.58m N As Fc > Nsin , car moves towards the center OPtion(B) case 4: = 30o, R = 800 m, v = 5 m/s, s = 0.3 Fc = 0.031m N Nsin = 1.5m N As Fc < Nsin , car moves away the center OPtion(A) case 5: = 30o, R = 800 m, v = 78 m/s, s = 0.3 Fc = 7.6m N Nsin = 1.5m N As Fc > Nsin , car moves towards the center OPtion(B) Let us consider car is moving ina circular path of radius R at an angle with the bottom at a velocity v. If mass of the car is m, then the forces acting on it are: Fc = mv2/R is the centripetal force towards the center N = smg is the normal force with coefficient of kinetic friction k If we resolve the forces into components, the horizontal components are: Fc = mv2/R Nsin = smgsin For the car to be in equilibrium these two forces must be balanced. case 1: = 45o, R = 1000 m, v = 90 m/s, s = 0.5 Fc = 8.1m N Nsin = 3.53m N As Fc > Nsin , car moves towards the center OPtion(B) case 2: = 20o, R = 1000 m, v = 90 m/s, s = 0.2 Fc = 8.1m N Nsin = 0.68m N As Fc > Nsin , car moves towards the center OPtion(B) case 3: = 70o, R = 800 m, v = 200 m/s, s = 0.7 Fc = 50m N Nsin = 6.58m N As Fc > Nsin , car moves towards the center OPtion(B) case 4: = 30o, R = 800 m, v = 5 m/s, s = 0.3 Fc = 0.031m N Nsin = 1.5m N As Fc < Nsin , car moves away the center OPtion(A) case 5: = 30o, R = 800 m, v = 78 m/s, s = 0.3 Fc = 7.6m N Nsin = 1.5m N As Fc > Nsin , car moves towards the center OPtion(B) If we resolve the forces into components, the horizontal components are: Fc = mv2/R Nsin = smgsin For the car to be in equilibrium these two forces must be balanced. case 1: = 45o, R = 1000 m, v = 90 m/s, s = 0.5 Fc = 8.1m N Nsin = 3.53m N As Fc > Nsin , car moves towards the center OPtion(B) case 2: = 20o, R = 1000 m, v = 90 m/s, s = 0.2 Fc = 8.1m N Nsin = 0.68m N As Fc > Nsin , car moves towards the center OPtion(B) case 3: = 70o, R = 800 m, v = 200 m/s, s = 0.7 Fc = 50m N Nsin = 6.58m N As Fc > Nsin , car moves towards the center OPtion(B) case 4: = 30o, R = 800 m, v = 5 m/s, s = 0.3 Fc = 0.031m N Nsin = 1.5m N As Fc < Nsin , car moves away the center OPtion(A) case 5: = 30o, R = 800 m, v = 78 m/s, s = 0.3 Fc = 7.6m N Nsin = 1.5m N As Fc > Nsin , car moves towards the center OPtion(B) case 2: = 20o, R = 1000 m, v = 90 m/s, s = 0.2 Fc = 8.1m N Nsin = 0.68m N As Fc > Nsin , car moves towards the center OPtion(B) case 3: = 70o, R = 800 m, v = 200 m/s, s = 0.7 Fc = 50m N Nsin = 6.58m N As Fc > Nsin , car moves towards the center OPtion(B) case 4: = 30o, R = 800 m, v = 5 m/s, s = 0.3 Fc = 0.031m N Nsin = 1.5m N As Fc < Nsin , car moves away the center OPtion(A) case 5: = 30o, R = 800 m, v = 78 m/s, s = 0.3 Fc = 7.6m N Nsin = 1.5m N As Fc > Nsin , car moves towards the center OPtion(B) case 3: = 70o, R = 800 m, v = 200 m/s, s = 0.7 Fc = 50m N Nsin = 6.58m N As Fc > Nsin , car moves towards the center OPtion(B) case 4: = 30o, R = 800 m, v = 5 m/s, s = 0.3 Fc = 0.031m N Nsin = 1.5m N As Fc < Nsin , car moves away the center OPtion(A) case 5: = 30o, R = 800 m, v = 78 m/s, s = 0.3 Fc = 7.6m N Nsin = 1.5m N As Fc > Nsin , car moves towards the center OPtion(B) case 4: = 30o, R = 800 m, v = 5 m/s, s = 0.3 Fc = 0.031m N Nsin = 1.5m N As Fc < Nsin , car moves away the center OPtion(A) case 5: = 30o, R = 800 m, v = 78 m/s, s = 0.3 Fc = 7.6m N Nsin = 1.5m N As Fc > Nsin , car moves towards the center OPtion(B) = 30o, R = 800 m, v = 78 m/s, s = 0.3 Fc = 7.6m N Nsin = 1.5m N As Fc > Nsin , car moves towards the center OPtion(B)

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