Is it normal to worry about people?

Well, my problem is that.
You may think it is stupid, but i am worried about the people close to me.
And when i say worried, i talk about the fact that if my mom goes to the market and does not answer her phone, i am getting all hyped up and stressed, and starting to bomb-phone her (a concept i made for a non-stop-calling in to a specific phone number) till she answers.
And it is like that all the time.
If i call my sister and she does not answer - i call my mom immediately and tell her to call my sister too, just to make sure everything is okay.
And it was just an example.
Is it normal?
Thanks in advance :)