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# Banked Curves

### Uniform Circular Motion

Uniform circular motion means that a particle is traveling in a circular path at constant speed. For an object of mass m to execute uniform circular motion with speed v and radius r, it must be subjected to a net force that
1. is always directed centripetally, i.e. in towards the center of the circular path;
2. has a constant magnitude equal to mv2/r.

Unless both these conditions are true, the particle is not traveling with uniform circular motion. Any kind of force can satisfy these conditions, which together are called the centripetal condition. Although most paths are not circular, most paths have parts that are approximately circular. And, while most circular motion is not uniform, the motion on a sufficiently small portion of the path is likely to be approximately uniform. Thus it is useful to study uniform circular motion even if a full circle is never executed by an object on the path.

### Banked Curves

To negotiate turns at high speed, where frictional effects are not sufficient to maintain circular motion, we often resort to a banked curve. Let's consider some examples.

Larry Gladney is Associate Professor of Physics and Dennis DeTurck is Professor of Mathematics, both at the University of Pennsylvania.

Example 1. A highway that curves around the base of a large hill is designed so that cars can execute the curve without the help of friction (along the radial direction). How is this possible?

Solution. Consider a banked roadway, as compared to an unbanked curve. The following animation shows the difference between the two.

The flat curve at the beginning of the video needs a static frictional force to satisfy the centripetal condition, as that is the only force acting in the horizontal direction of the curve radius. Compare the force diagrams for a car on an unbanked and on a banked roadway surface in the following figures. In both cases the curve bends to the left so the car needs a net acceleration to the left. The velocity of the car is directed into the page and is constant in magnitude.

In the first case static friction acts, since the car would travel to the outside of the curve and eventually leave the roadway if it were traveling in a straight line. The driver turns the steering wheel to negotiate the curve. On the banked roadway, if the bank angle (q) is appropriate, then the driver need do nothing to stay on the road. In this case the normal force of the roadway surface maintains a vertical component against gravity and a horizontal component that satisfies the centripetal condition. To find the value of the bank angle, we resort to the freebody diagram and proceed as follows.

The force equation for the y direction is

 FN, y - mg = 0 FN cos q = mg ==> FN = mg/cos q

We can now find the bank angle by looking at the x force equation:

 FN, x = mv2/r FN sin q = mv2/r mg tan q = mv2/r ==> q = tan-1[v2/(gr)]

Example 2. Airplanes must also execute banked turns, since the air does not provide nearly enough friction to turn a massive plane moving a high speed. An examination of the forces involved in this case are explained in this digital video.

Problems

1. An airplane is circling an airport by traveling a horizontal circular path at a speed of 400 km/h. The bank angle of the wings is 20° . What is the radius of the circular path the plane is flying?
2. A jet fighter is also flying around the airport with a speed of 800 km/h along a circular path with a radius of 2 km. What g force is the pilot experiencing? At what angles must the wings of the plane be banked?
3. A curved roadway has a radius of curvature of 200 meters and a bank angle of 10° If the coefficient of static friction for car tires on the road surface is 0.2, what is the highest speed at which a car can round the curve safely?
4. An old streetcar goes around a corner on unbanked tracks. If the radius of the curve is 10 meters and the streetcar speed is 5 km/h, what angle with respect to the vertical will be made by hand straps hanging from the ceiling of the streetcar?
5. You are asked to design a curved section of a highway such that, when the road is icy and the coefficient of static friction is 0.08, a car at rest will not slide down the curve slope and, if the car is traveling at 60 km/h or less it will not slide to the outside of the curve. What minimum radius of curvature and what bank angle does the curve need to have?