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With that method, your bow is still lifting the weight of the chain and the stuck anchor. And the forces and directions are all still the same. I don't see any benefit in it.
If I wasn't set up with a trip line, I might use a bouy and line to mark where the anchor is, so I (or a diver like erps) could go back and get it.
Maybe we should add this to the how-to-make-money-while-cruising thread. Retrieving stuck anchors and selling old anchors that you find.
My guess, looking at the buoy/ring thing in erps' link, is that the idea is to get the ring as close to the anchor itself as possible.
Say you have a 25 lb anchor and, I dunno, 100 feet of chain at (I'm making numbers up) 1 lb/ft. You take the bitter end of the chain, stick it through the ring, and -- gulp -- drop it overboard. Not mentioned, but I expect it would have to go into a bag or something.
Suppose the anchor is fouled in 20 ft of water. You keep pulling the chain through the ring and into the bag and eventually you have 80 feet of "bitter end" on one side of the ring in a bag, which weighs 80 lbs, and 20 ft plus a 25 lb anchor on the other end. The 80 lb bag is now a counterweight that, along with the buoyancy of the ring buoy, is sufficient to lift the 45 lbs of chain and anchor out of whatever it's stuck in. The closer you can get the ring to the anchor, the easier it gets.
What your anchorless boat is supposed to be doing at this time is not clear to me.
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