how can i convince my mom?

No need to convince her, if you live in the states. Just suck up to a friend that can drive to get them to take you to planned parenthood or a local free clinic. If you google planned parenthood you should be able to find listings for your area. There you can aquire free (if you're go to a free clinic) or very low cost based on a sliding scale, birth control. But if it really bothers you and you want your mom to approve, then tell her all the stuff you just told us.