I have had a cactus for a little over a year now. It has parts that stick out with fuzz and a fish hook on the end. It is barrel shaped, not very tall, and is in its own pot. A little while ago, it started to curve, and then the newly grown section fell off.
Cactus plants need a lot of sun and not a lot of water. If your plant is starting to drop off it is easy to plant them again. All you basically do is put them in a pot and allow them to root again. Cactus plants can easily root again as long as they have the right conditions to grow and root in. I cut off pieces of my cactus plants all the time and just put them in a pot. I give them a lot of sun and not so much water. If it rains a lot here, I don't water them again for about a week or so.
This is a page about rooting a cactus pad. Many types of cactus propagate in nature by their ability to have a pad or segment grow into a new plant. You can do this too, at home, and increase your cactus garden plantings.