Hi there,
I have a 2002 Corolla and I noticed last winter that it gave a whining noise when I started the car in morning and evening (WI weather). This also happens in cold damp weather;I have no problems in summer. The noise only lasts 10-15 seconds and so I was only able to locate that it came somewhere from the serpentine belt area. My car has around 67K miles and so I felt it was time to replace the belt. The noise vanished and then came back 
It was suggested to me that the tensioner was going bad. So my theory is that something is unable to take the tension in the belt and running the car run for 10 seconds heats the belt and it expands thus reducing the tension?
Thanks,
Indu
You may have just answered your question.
Remove the belt and with a bit of side/angled hand pressure, rotate the tensioner pulley back and forth. If there is wear in the bearing, you should hear a slight squeaking/scratching noise.
If no noise is heard there, apply the same test to all the other pulleys. Don’t forget to check the alternator pulley.
While the belt is off, closely inspect it for cracks, not only ones that may show running across the belt but also cracks running the same direction as the grooves. Bend the rubber backwards to check.
With at least some belts, cracks that run across the ribs are considered normal.