To start with I need explain that I'm not the brightest when it comes to anything too technical as far as PC's go!

OK... my PC will boot up normally and everything seems fine. Then after a while (not more than 10 minutes) the machine hangs briefly and then ungraciously dies and the machine begins to reboot. The whole family has come to understand that once this has happened once it won't happen again until the machine has been shut down normally.

What makes it the more confusing is that I have recently reformatted the disk and reloaded Windows XP and this happened soon there after.

What would be really useful is to know what I could do just to determine what is causing the problem but that, at this point, is beyond my capabilities!

Possible Reasons may cause this problems:

1. Power Failure due to increased resource processing.
2. Harddisk Failure also cause this if its head unable locate itself on proper address.

Not convinced regarding your option 1. Seems unlikely that a power failure would strike 5 to 20 minutes after initial boot up every time. Especially since the all initial resource demands are over by the time the problem hits.

Is there any way that the hard disk can be checked to see if it is having sporadic failures?

Means weak SMPS (power supply) causes if its unable to flow more power when processing increased.
Harddisk failure pointed by hearing the sound like tak-tak in running mode.

Are all your fans operating correctly? Could the PC be overheating?

I happy to believe anything! However if the PC was overheating then on the reboot why does the PC stay up for days! In other words it only fails once and there after it is fine until the machine is powered off via the regular shut down.