How much longer could Titanic have stayed afloat if it gotten rid of its anchor and chain right after hitting the iceberg? I am wondering how much longer the RMS Titanic could have stayed afloat if the crew had allowed the ship's anchor and anchor chain to fall to the bottom of the ocean immediately after the ship had hit the iceberg. (I am not even sure if a ship's anchor chain can be unfastened from a ship, but let's just say for the sake of this question that it can be unfastened.)
The combined weight of Titanic's anchor and anchor chain was approximately 116 tons according to this Wikipedia article:
"...In 1911, the company manufactured the anchors and chain for the ocean liner RMS Titanic. The largest of the anchors weighed 15.5 tons and on completion was drawn through the streets of Netherton on a wagon drawn by 20 shire horses.[15] The chain and fittings for the anchors weighed around 100 tons..."
https://en.wikipedia.org/wiki/N._Hingley_%26_Sons_Ltd
Since the Titanic went down bow first, and the anchor and chain was located in the bow section, immediately getting rid of 116 tons in the bow section would have increased the time it had stayed afloat before it sank.
 A: A detailed question to this answer will, as so often, require a detailed numerical simulation or, better, a scaled-down experiment. But being the gung-ho physicists we are on this site here, we can do a simple guesstimate: According to a quick internet search, the total mass of the titanic is somewhere in the ballpark of $5\times 10^7$kg, so the mass you are talking about is only a small $10^{-3}$ fraction. So to first order one might guess that this would delay the sinking by such an amount: With the total time between hitting the iceberg and sinking being about 2.5 hours (says wikipedia), $10^{-3}$ of that is of order 10 seconds. Rather not a lot.
More detailed studies have undoubtedly been done and lead to estimates like the one quoted on, again, wikipedia, of 7 tons of water flooding the titanic per second. That means the mass of anchor and chain is made up entirely within just a few seconds also, again giving an unfortunately also negligible delay estimate of some 16 seconds.
A: Titanic weighed 46,000 tons, so removing say 120 tons would have made almost no difference.
She had several long tears well below the water line.  Flooding was such that there was no means that could have prevented her sinking.  It is quoted as about 7 tons per second.  It would have taken just 17 seconds to replace all the weight of anchors and chains.
It's worth noting that her chief (?) architect and captain would both have considered counter-flooding in order to keep her afloat.  This would be an automatic thought to them.  They did not do these things because they knew it would make no practical difference to her.  She was mortally wounded.  Such measures would also have made launching boat difficult and greatly increased panic.
