I found an answer to your question. This answer is from a scientist named Kevin Thorpe; he is an entomologist (a person that studies insects), and he works at USDA-ARS Insect Bio-control Laboratory in Beltsville, Md.

Caterpillars make their cocoons from strands of silk that they produce from glands near their mouths.

They use their mouth parts to weave the silk, which is sticky at first, into a cocoon. Some caterpillars make very loose, open cocoons, while others build very strong, tightly-woven cocoons.

Caterpillars secrete silk which they can then wrap around their bodies to make a cocoon. The silk hardens when it comes into contact with air. First they stick themselves to the underside of a stalk, and then they spin silk around their bodies all the way around to their head (since they are hanging upside down that's actually the bottom!). Now they are sealed inside the cocoon and ready to metamorphose into their adult form.

Note from ScienceLine Coordinator

The following is a contribution from Barbara, a reader of ScienceLine:

"The caterpillar does use some sticky silk to mount their tail onto the location they want to transform - a branch, a butterfly house, etc. But they do not 'spin' a cocoon. The chrysalis is already in place under their skin, which they then split and wiggle from until it falls off.

My 9 year old children and I were curious how our very large striped (future black swallowtail butterfly) caterpillars suddenly less than hour or so later managed to so quickly become encased in a soft green chrysalis much smaller than their body was an hour earlier, when the were very inactive and not moving for hours after they silk mounted their tails to the wall of the butterfly house. We tried each time to catch them spinning it. We couldn't, because there was no spinning - I noticed the rumpled shed striped skins beneath on the bottom of the butterfly house, like a discarded t-shirt. Mystery solved.

The soft green chrysalis then hardens up and turns brown. You may see the chrysalis pulse and wiggle at times. The transformation takes about 3 weeks or so before the butterflies begin to emerge. We still have several more not ready to go into a chrysalis yet, so perhaps I will video them doing this."