A boat containing some piece of material is floating in a pond. What will happen to the level of water in the pond if on unloading the pieces in the pond, the piece (a) floats, (b) sinks?

Explanation

If `M` is the mass of boat and `m` of pieces in it, then initially as the system is floating.
`(M+m)g=V_(D)sigma_(W)g`
i.e., the system displaces water
`V_(D)=M/(sigma_(W))+m/(sigma_(W))`
When the pieces are dropped in the pond, the boat will still float, so it displaces water `M=V_(1)sigma_(W),` i.e., `V_(1)=(M//sigma_(W))`
a. Now if the unloaded pieces float in the pond, the water displaced by them
`m=V_(2)sigma_(W)`
`implies implies V_(2)=(m//sigma_(W))`...........i
So the total water displaced by the boat and the floating pieces.
`V_(1)+V_(2)=M/(sigma_(W))+m/(sigma_(W))`.........ii
which is same as the water displaced by the floating system initially (Eq. i) so the level of water in the pond will remain unchanged.
b. Now if the unloaded pieces sink, the water displaced by them will be equal to their own volume, i.e.,
`V_(2)^(')=m/(rho)` [as `rho=m/V`]
and so in this situation the total volume of water displaced by the boat and sinking pieces will be
`V_(1)+V_(2)^(')=(M/(sigma_(W))+m/(rho))`............iii
Now as the pieces are sinking `rhogtsigma_(W)`, so this volume will be lesser than initial water displaced by the floating system (Eq. i), so the level of water in the pond will go down (or fall).