Hiylo

If you do well in school and do exercise and still have enough free time to play or go on the internet, I don't see why you shouldn't. My time on the computer varies day to day because of school but, once I have done exercise and my hw/study I usually go and play with my friends for a few hours.

It really depends on your situation, and also what you do with the computers. If you spend time just dicking around and playing video games 16 hours a day, then it's a problem and you should probably try and move to doing something more productive, get a job, etc. If you do a load of good stuff on the computer, learning how to program, spending your time creating something that will benefit you and/or other people, then that's a much more acceptable situation, especially if you show your parents what you're working on, so that they understand that you don't spend all the time playing CS:GO or whatever game. A lot of times gaming is good for you, it trains your brain to react to uncertainty, things like that, but as you do or should know, too much of anything is detrimental to your mind. I'm not trying to say limit yourself to a certain number of hours behind a computer screen - that would be hypocritical of me to say - but you should at the very least try to do things that are productive. Learning Linux, learning how to write scripts, programs, games, etc. Things that will help you in the future. And the one thing that I would say is the most important - go outside every now and then. Don't spend all day behind the screen. Go out into the woods from time to time and enjoy nature. And keep your family in the loop about what you're doing. Then they'll be a lot less uneasy about the time you spend on your machine.

Source: I'm a computer addict. My job is sitting in a dark old house and typing things into spreadsheets and pdfs. Then I mess around with computers more or less constantly at home. Spending time outside and/or with family is important to my lifestyle, if I didn't, I would be a nervous wreck NEET with a social anxiety disorder.

Here's an interesting link, talks about how you are affected by media consumption based on what you're doing: http://www.pcadvisor.co.uk/feature/digital-home/how-much-screen-time-is-healthy-for-children-benefits-3520917/

haha, I'm currently growing my hair long. Sometimes idk whether i want to cut it or not

Agreed, I was not particularly social myself before my freshmen year, If the issue is the 2 hour limit you could try to see if your grades and attitude stay fine if you could have additional time on the computer? 
Side note: I did this to keep my long hair all throughout High School.

I spend pretty much 10 hours a day in front of my computer. Back when I was XX, if I wasn't playing piano (spent 8 hours a day on that), I was on the computer. I ate at my computer. 
 
Except I do mostly productive things, like learn coding. And write. And help people.
 
I'm fine.

That you're are actually concerned about what your mum is thinking is a good sign.
That shows me that you're probably quite a bit more mature than other kids in your age.
But back to your questions:
To use a computer for more than two hours is usually not a problem. But you should always try to do something else inbetween and you should also take your time for school and other important stuff. Try to show your mum that you're able to control yourself and help her out in the house, get good notes in school and if she sees that you're doing well then she won't have any problems to let you sit in front of the computer for more than 2 hours.
 
Also keep in mind that sitting a long time in front of a PC isn't good for your muscles, neck and back.