So my RCA (Thomson) CD alarm clock stopped reading discs. I was playing a CD in it a few days ago. It worked fine. But when I put the same CD in to play today, it makes a loud rumbling noise as the disc turns and then it says "No Disc". I tried a different disc, and tried cleaning the lens, but it still won't work. I don't understand why it worked a few days ago but now it won't play same disc. Tried unplugging and plugging back in too. 

Is it dead now? It's about 10 years old, but everything else is working on it.

Check if the laser module is able to slide properly. Its usually mounted to a worm gear (shaft with threads) which when rotated will slide the laser along so it can lock onto the tracks on the cd.

When you mention rumbling sound, that suggests that the laser unit is trying to move along its rail to read the disk and something is blocking it. Might be that it needs grease on thr shaft or there is physical damage from wear and tear /age.

This is just my best assumption and hopefully others can provide their ideas.

One thing I notice is I can hear the laser moving to find the track. But I can see through the window that the rumbling happens when the CD is spinning. It almost sounds like it's rubbing against something. But I can't see why because nothing seems to be out of place.

The laser part does seem to slide, and I noticed black grease somewhere near it.

Oh I see, sorry.

Perhaps the motor that spins the disk has something stuck in it. Some of these use a step motor and sometimes a crumb or speck of dirt will get in between the copper coils and the magnets causing friction or jamming of the motor. Run your finger on the cd hub and see if you feel any grinding when its moved.

Or the optics had just accumulated too much dust and it rumbles as the electronics is moving with everything around trying to find some signal.
Or check whether the focusing is smooth or whether some lens hinges are not broken (some makers used plastic flex hinges on the lens mechanism, which tend to have limited life due to fatigue; the thing becomes worse when in too dry environment - plastics are less flexible and less fatigue resistant when the humidity, normally diffused in the material, dries out).
Or it could be dried out electrolytics (if it is very old unit), causing ripple on supply and improper time constant in the focusing and tracking analog servo loops, making them unstable; in that way you would see the lens vibrating wild instead of smooth slowl motion.

Well guys I figured it out!

I tried taking the unit apart, but couldn't get the case apart. But then I noticed that the spinner (not sure the name of it) that the CD snaps onto to rotate, seemed tight, and sounded raspy, like it was scraping something. It also looked like it was pushed too far back on the shaft. So I tried pulling it forward, but it popped off the shaft.

So I put it back on, but not pushing all the way back, put the CD back in and it worked! So it was an easy fix. But I don't understand why they would make it so easy to push the spinner back too far.

To be honest, I found Thomson's electronics to not be all the durable. Or at least the newer ones. The ones in the 80s and 90s might have been better made. But the newer stuff seems to be finicky and have issues.

Hmmm, I have one of those open CD players from Amazon in a drawer at work, and upon pressing play, it spins maybe 1/4 turn at the most before giving up. I wonder if I could try this... I might have to take a look on Monday.