Physics 220
You are a highway engineer who wants to design a freeway segment that has a radius of curvature of 500 m. The road is to be banked so that cars travelling at 25 m/s have no friction on the tires. What should the bank angle be?
Side view
Draw a free body diagram showing all the forces acting on the car. Then decide on the direction of the acceleration and draw it in your diagram too. Hint: where is the center of the circle?
Now choose your coordinates.
Fill out the table showing the components of the forces in terms of their magnitudes.
Force |
x-component |
y-component |
n |
nsinq |
ncosq |
W |
0 |
-mg |
Sum of force components |
å F_{x} = nsinq |
å F_{y} = ncosq - mg |
Acceleration |
a = mv^{2}/r |
0 |
Now apply Newton's second law. You should have two equations in terms of two unknowns. (Hint: the mass of the car should be considered a known quantity.) Eliminate the one unknown you don't want to find the angle of the bank.
nsinq = mv^{2}/r
ncosq - mg = 0
Eliminate n by dividing:
tanq = v^{2}/rg = (25 m/s)^{2}/(500 m)(10 m/s^{2}) = 0.125
So
q = 7°