Banked turn
Swinging buckets, racecars on a burm, and airplanes banking are all common intoductions to centripetal force and circular motion problems. I have not written the rotation page yet, so the proof for the centripetal accelration equation is below.
To maintain a circular path with a constant magnitude velocity (not directon) and constant distance from the centerpoint, an acceration pointing inward is needed.
Applying Newtons laws, we know that the net force on the object must point inward to the center of the circular path, this allows us to setup free body diagrams in the in-plane, and out-of-plane directions. The velocity is into the page.
The setup for a swinging bucket only differs from the airplane in that you might care about the legnth of the rope. The car is also the same on a frictionless track, but the FBD for a non frictionless track is a bit diffrent, so its setup further down.
The banking turn problem is more interesting when the racecar has static friction holding it on the circular path; if its going to slow on a steep bank, it could slip down, or if it goes to fast, it gets launced off the track. We can define the min and max velocity based on the max staic friction force ponting up or down the slope.
