Why is the spider in its own web swinging it back and forth?

John R makes a good point, but in your case the spider is an orb weaver in the genus araneus.

(likely a barn spider)

When a flying insect gets caught in the web (or if the spider thinks this) often the insect will lay quiet.

The spider will shake her web which often makes the insect move. the spider can feel this, and find the insect in the web.

This is a behavior I have seen consistently with this type of spider. I feed this one from time to time, and she will do the same thing.

http://www.flickr.com/photos/32083154@N02/60124442...

Spider Web Swing

Some spiders (most notably Cellar Spiders) shake their webs, often at astonishingly high frequencies, in order to discourage predators. It doesn't sound like this is what you're seeing, though, and I must say that I've never noticed that in the orb-weavers I've ever seen. One other possibility, especially at this time of year, is that you have a male spider tugging on the web in a certain pattern in order to get the female's attention, but as a potential partner rather than as a meal. Check for a smaller spider at the rim when you notice the web moving like that. I doubt that this is what you see either, since you say it happens a large part of the day, but it's worth eliminating possibilities. Apart from those two things, I'm afraid I don't come up with anything. See if you can get a photo of the spider, and maybe a quick video clip of the web moving. You may have come across something interesting, and perhaps some spider person in your area would want to see it.

Bolas spiders produce a web consisting of a blob of "glue" on a long thread. The spider swings this back and forth until it sticks to an insect. The gluey blob is often scented in a way which will attract moths, etc. Mastophora alachua is a species found in the USA; perhaps that is what you are seeing.

don't see why not.