Is it normal for my eyes to get watery when i look at a computer screen for a long time?

You bet!: When we stare at something for a long time (computer screen, book, wall) we decrease the amount of blinking per minute. Blinking pushes water on the eye (reverse windshield wiper) to coat it. When you don't blink, the eye dries and sends a signal to the tear gland to produce more tears in order to protect the eye. It is a sign that you are staring too long. Try a break or artificial tears.

Yes : Studies show that when intensely concentrating on a visual task (such as reading or working on a computer), the normal blink reflex is blunted. This means you don't blink as frequently and your eyes tend to dry out. As dryness turns to irritation, the tear reflex goes overboard and can cause overflow tearing. Try placing systane in the eye prior to using the computer and tape "blink" to screen.