Cars and a banked ramp

Ok, here is another problem. First, I assume that you already did the “find the speed limit” for the off ramp problem.

What if I wanted to rebuild that ramp? Suppose I want to build a similar ramp with a similar radius (around 57 meters) for cars that will be moving at 15 m/s. I want to make this ramp so that it will work no matter what the road conditions are like. That is to say that the car will not need any friction to go around the curve.

One way to accomplish this is to put a bank to the ramp. It think the actual ramp I used in the previous problem is banked a little. However, you need to find out how to bank it so that you don’t need any friction to turn the car. Think of the tracks for NASCAR. Like that. Here is a similar picture (but not NASCAR).

Mercedes Benz amg sls Roadster.jpg (660×402)

So, what would be the best banked angle for this on-off ramp?

Questions

Let me start by assuming that you finished the flat-ramp version.

1. How would you change your free body diagram from the flat version of this problem?

2. What force is exerted on the car that accelerates it towards the center of the circle? Before it was friction, but it isn’t in this case.

3. Is the normal force equal to the weight of the car? Why or why not?

4. After drawing your free body diagram, what would the force-acceleration equations look like for the x- and y-directions?

Hints

  • Put your x-axis in the direction of the center of the circle (just like the flat ramp problem). This way the acceleration will be along one of the axis.
  • After writing down your force equations, you should have two equations with two unknowns. The unknowns should be the magnitude of the normal force and the angle the normal force is from the vertical. With two equations and two unknowns, you can algebraically solve for the angle.
  • If you get to a point where you have sin(θ)/cos(θ) you can re-write this as tan(θ).

The video

Ok, here is the video solution.

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