1) Computer unplugged from power, computer on a table, computer case open.
Use a can of compressed air for computers, and clean out the inside of the computer, and it's Power Supply.
[Viewing the front of the computer, the Left Side Panel removes. Two thumbscrews in the rear, slide the panel towards the rear of the computer]
2) Observe Anti-Static Precautions.
Your body carries Static electricity.
Static WILL fry out (Short Circuit) hardware components in a computer. (Processor, Ram Memory, etc.)
TOUCH the metal frame of the open computer case to relieve your body of Static.
Should you leave your computer in the middle of working on it, upon your return be SURE to Touch the metal frame again.
3) Remove the Ram Memory modules. (Ram memory 'sticks')
Clean the gold plated contact pins on the bottom of the ram memory module, with a pencil eraser.http://en.wikipedia.org/wiki/File:Generic_DDR_Memory_%28Xytram%29.jpg
You can click on the photo to enlarge. You can enlarge twice.
Rub the eraser on each contact up, and down. You don't have to scrub it with the eraser, just use medium pressure.
(Both sides. All contacts. Just takes a few minutes.
The Acer Aspire T310 uses DDR Sdram ram memory. Just like shown in the link example above.
There are 184 contact pins. 92 on each side)
Once you have finished cleaning use air to remove the eraser dust.
If you have a can of compressed air for computers available, use it.
If not air pressure from your mouth will be sufficient.
Then I suggest you have a bad Power Supply.
Weak voltage power rail.
(There are three main power rails for the power supply used in that Acer.
1) The 3.3 Volt power rail
2) The 5 Volt power rail
3) The 12 Volt power rail.
All are DC voltage.
Two D cell flashlight batteries produce 3 Volts DC)
With a Power Supply having a weak voltage power rail, there is enough power to light LED lights, maybe spin fans, but not enough to turn the Processor on.
A) ALL of the LED lights combined use less than 1 Watt of power.
B) EACH fan uses 2 to 3 Watts
C) A typical Processor uses 51 to 125 Watts. Depends on what Processor it is.
Copy the above link, paste it in your address bar, press the Enter key.
It's the Service Manual in PDF file form, from Acer Support for the Acer Aspire T310.