Bicycling Question, why are there are two types of Deore XT M772 rear derailleur the SGS and GS?

The reason for long, mid and short cages is to make sure you have enough cage length for chain take up. If you use a short cage with a large rear cassette along with a three ring gear crank you will not be able to use all the gear ratios available due to chain tension problems. A long cage in this case would allow you to have proper chain tension in all the combinations. If you run a single front chain ring you could use a shorter cage derailleur as there would be less chain take up needed.

Rear Derailleur Types

As others have noted, the shorter cage models don't cover as large a gear range, which can be limiting

The advantages of the shorter cage are twofold:

1) It's a tiny bit lighter

2) Both the chain and the derailleur are further off the ground, which can be especially useful if you're riding over really aggressive terrain.

Short cage = road type usage with small rear gear cluster.

Long cage = touring / mountain bike type used with larger rear gear clusters. You can tell these because they are physically larger than the road type. The long cage can handle shifts that the short cage simply can't.

So why the short cage at all? Because it's smaller, less likely to get hit, lighter, and shifts slightly faster.

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long is easier when changing to low gear, its for aggresive riders.