Common examples include:
We use DID TASC to analyze all these
problems.
Step 1: Draw a diagram and coordinate system. Step 2: Isolate system.
Step 3: Draw all forces. Step 4: Take components.
At rest:
Crucial feature of uniform circular motion in the vertical plane:
Step 5: Apply Newton's second law.
At the bottom (y=up): ΣF_{r} = N  mg = ma_{r} = m v^{2}/r N = m v^{2}/r + mg
At the top (y=down) ΣF_{r} = N + mg = ma_{r} = m v^{2}/r N = m v^{2}/r  mg
Step 6: Solve. How slow can you go at the top and not fall off?
The limit is N =0! The corresponding speed is: v^{2} = gr
Halfway up or down: ΣF_{r} = N = ma_{r} = m v^{2}/r N = m v^{2}/r
At these points, mg provides the tangential acceleration that slows the
object down (on the way up) or speeds it up (on the way down).
What is the normal force on a car when it moves with speed v at the bottom of a circular valley of radius r?
At the bottom:  N  =  mg + 

What is the normal force on a car when it moves with speed v at the top of a circular hill of radius r?
At the top:  N  =  mg  

The normal force equals your apparent weight. This is why you feel heavier at the bottom and lighter at the top.
A key difference between the car and water bucket examples: The
normal is always up for the car. For the bucket, the normal force points
from the bottom of the bucket toward the top of the bucket.