# University of Wisconsin Green Bay

A 540 kg car is merging onto the interstate on a banked curve. The curve is banked 7.1o from the horizontal and is rated at 35 mph. The car takes the turn at 52 mph (23 m/s). What sideways frictional force is required between the car and the road in order for the car to stay in its lane? The radius of the curve is 210 m.

• In this problem, you are asked to relate motion (the car moves in a circle) to force (friction). Force and motion of a single object are always related through Newton’s Second Law, so this is a force or 2nd Law problem.

Even if no forces were mentioned, and you were asked, for example, for the degree to which the curve is banked, you know that it takes a net inward force to make an object move in a circle and so forces are the appropriate interactions to consider.

• Step 1

Your FBD is not yet finished, because tension has both x- and y- components. Continue down to step 2 when you are ready to continue.

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Step 2

In the final FBD drawn here, all forces are divided into components. The contribution each force makes in the x-direction (in the plane of the circle) is shown explicitly, as is the contribution each force makes in the y-direction. The FBD is now a visual representation of ∑F=ma in each direction.

• The key equation for any problem that relates forces and motion is Newton’s Second Law. Regardless of what quantity you are asked to find, begin with the Second Law. If additional information is needed, it will become apparent as you proceed.

• Step 1

At this point, it seems that you have two equations and two unknowns (fr and n). Scroll down to continue the mathematical solution.

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Step 2

At this stage in the problem, we have two unknowns, n and fr, and two unsolved equations:

fr cos(7.1o) + n sin(7.1o) = 1360 N
n cos(7.1o) – 5290 N – fr sin(7.1o) = 0

One approach that always works is to solve one equation for one of the variables and substitute it into the other.

Friction is the only unknown quantity that was requested in this problem. No further mathematical solution is necessary.

• In this problem, a car is traveling in a circle on a banked incline. The normal force not only balances against gravity (as seen in the y-equation) but also pushes the car inward around the circle (as seen in the x-equation.)

Because the car is traveling faster than the rated speed, normal force is not enough to keep the car moving in a circle. If there was no friction present, the car would move outward in the curve (up along the incline.) If friction is present, therefore, it will act to prevent the tires from sliding out. In other words, friction will act in along the incline. The x-component of friction, then, provides additional force to keep the car in its circular path.