So suppose you have an OM-1N with one of the following problems:
These problems have a number of origins, but if the shutter ring assembly is involved then this page should tell you how to fix it.

Anyway, let's get started. First, and most importantly, SET THE SHUTTER SPEED TO 1/1000. In theory you don't have to do this, but it'll make things much easier later if you fuck up. Next, remove your lens and unscrew the three screws in the metal plate on top of the shutter speed ring.

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Remove the plate, and you should see this:

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Three things you should take note of: see the notches in that strip of rubbery stuff on the top half of the inside of the ring? That's what causes the shutter speed ring to stop exactly on the numbers listed on the outside. A small bearing is what gets pushed into each notch. We're going to be lifting the shutter speed ring out in a second, and nothing's going to be holding that bearing in place. For the love of god don't lose it.

There's also a free-floating gear near the bearing. I'm not sure what this attaches to, but take note that there's nothing really holding it in place, so it gets bumped out of its socket easily.

Lastly, note the notch in the bottom half of the shutter speed ring. There's a tab in the shutter speed sensing ring that fits in this notch.

Okay, enough blabbing, take the shutter speed ring and gently but firmly pull it up off of the camera body. If all goes well, the bearing won't pop out and you'll now have direct access to the shutter speed sensing ring.

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Pause for a moment here, and take a look at the underside of the shutter speed sensing ring. See those teeth? Those grip a gear at the bottom of the camera, which is what actually sets the shutter speed. Now consider this: Nothing holds the shutter speed sensing ring in place. Suppose you accidentally lift it up and don't manage to put it back exactly as it was before. Well, guess what? Your shutter speeds are now all off by some factor. Now, suppose this happened at some point. The way to check is to delicately lift off the shutter speed sensing ring, and shine a flashlight at the metal gear at the bottom of the assembly. If your shutter speed is currently set to 1/1000, what you should see is a small circular hole in the dead center of the gear. If you don't see this hole, your shutter speeds are off. Take a small screwdriver or something and turn the gear gently until you see the hole and it's in the center. You may want to put everything back at this point and time your shutter speeds to confirm that they last for the correct durations. (I do this for 1s to 1/8s using an audio recording application).

Anyway, if shutter speeds were your only problem, then you're done at this point. If they're still wonky and the gear was properly oriented then your problems lie elsewhere in the camera. Now on to part B: the aperture sensing ring and how it might be screwing up your light meter:

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So beneath the shutter speed sensing ring is the aperture sensing ring. The way it works is that there's a little metal catch, and above it a metal tab that gets pushed around by another tab on your lens. It's pretty easy to bump this ring and position it improperly when messing around with the shutter speed apparatus. Here's how the tab should be positioned:

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If you push the tab around with your finger, you can see that its movement is somewhat limited, it only moves from about 7 o'clock to 1 o'clock. If you had it positioned at 1 o'clock, your light meter would work backwards or not at all. Anyhow, if your light meter's still messed up and the tab's in the right position then your problems lie elsewhere. That concludes our tour of the OM-1N's shutter speed assembly, I hope this was helpful in some way.