AP® Physics 1: Algebra-Based · Unit 2 · Topic 2.9
Vertical Circles
In a vertical circle, gravity and the track's normal force both point toward the center but add differently at the top and bottom, so the net centripetal force—and how fast you must go—changes around the loop.
Centripetal force just means whatever net force points toward the center, keeping something on a circular path. On a vertical loop, two forces can contribute: gravity, which always points straight down, and the normal force from the track, which always points toward the center (that's what tracks and loops do). At the top of the loop, both gravity and normal force point downward, toward the center, so they add together to supply the centripetal force. At the bottom, gravity points down and away from the center, so the track's normal force has to work against gravity and still provide enough leftover force to curve the path. That's why the normal force is weakest at the top and strongest at the bottom.
The minimum speed condition happens at the top. As speed drops, the cart needs less centripetal force, so the track pushes less. At some speed, the track's push hits zero and gravity alone provides exactly the centripetal force needed. Go any slower and gravity would overshoot what's needed to keep the cart curving, pulling it off the track and into free fall. That critical speed is v = √(g r), found by setting gravity equal to the required centripetal force m v2 / r and solving.
This builds directly on Newton's second law applied to circular motion: sum of forces toward the center equals m v2 / r. It also connects to the idea that normal force is not fixed—it adjusts to whatever the situation demands, from zero at the top of a loop to several times an object's weight at the bottom, which is the same reasoning used for apparent weight and g-forces on riders.
Key terms
- Centripetal force
- The net force directed toward the center of a circular path, required to keep an object curving rather than moving straight.
- Normal force
- The push a surface exerts perpendicular to itself, which adjusts in size depending on what else is happening in the motion.
- Minimum speed condition
- The slowest speed at the top of a loop where gravity alone, with zero normal force, still supplies the needed centripetal force.
- Apparent weight
- The force a surface exerts on an object, which a rider feels as heavier or lighter than their actual weight.
Watch it explained
- How it worksHow do gravity and the track force team up differently at the top and the bottom of a loop?
- Worked exampleFind the minimum speed at the top of a 2 m radius loop for a cart to stay on the track.
- Common mistakeWhy isn't the normal force zero everywhere on a loop just because it vanishes at the top?
- Going furtherWhat does a rider feel at the bottom of a vertical loop compared with the top?
Check yourself
End of topic.