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 theta. Static friction is Mk. Gravity is 10m/s^2. In each of the following 5 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. 1. Theta= 45 deg, r= 1000m, v=90, Mk=0.5 2. Theta= 20 deg, r= 1000m, v=90, Mk=0.2 3. Theta= 70 deg, r= 800m, v=200, Mk=0.7 4. Theta= 30 deg, r= 800m, v=5, Mk=0.3 5. Theta= 30 deg, r= 800m, v=78, Mk=0.3. (If this is too much work for 1 question then only do maybe the first 2 or 3 situations)

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 = kmg 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 = kmgsin For the car to be in equilibrium these two forces must be balanced. case 1: = 45o, R = 1000 m, v = 90 m/s, k = 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, k = 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, k = 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, k = 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, k = 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 = kmgsin For the car to be in equilibrium these two forces must be balanced. case 1: = 45o, R = 1000 m, v = 90 m/s, k = 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, k = 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, k = 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, k = 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, k = 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, k = 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, k = 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, k = 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, k = 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, k = 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, k = 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, k = 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, k = 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, k = 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, k = 0.3 Fc = 7.6m N Nsin = 1.5m N As Fc > Nsin , car moves towards the center OPtion(B)

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