#### Vertical circular motion

The situation of vertical circular motion is fairly common. Examples include:

• roller coasters
• water buckets
• cars traveling on hilly roads

The analysis of all these problems is similar, although there's a difference between cars on hilly roads and the other two examples. Let's start with the roller coaster / water bucket example.

As usual, begin with a free-body diagram.
Follow this up with an appropriate choice of coordinate system.

At rest, the free-body diagram is simple, with an upward normal force and a downward force of gravity. These are the only two forces in the system even when circular motion is going on. The force of gravity has a constant magnitude and direction. The normal force, however, changes both magnitude and direction.

In the radial direction (toward the center) there is a net force as long as the object is moving along the circle. This object also slows down on the way up, and speeds up on the way down, so there is an acceleration in the tangential direction, too.