A piece of ice is floating in water. What will happen to the level of water when all ice melts? What will happen if the beaker is filled not with water but with liquid
a. denser than water,
b. lighter than water?

Explanation

If M gram of ice is floating in a liquid of density `rho_(L)` then for its equilibrium
Weight of ice = thrust, i.e., `Mg=V_(D)rho_(L)g`
So the volume of liquid displaced by the floating ice, `V_(D)=(M//rho_(L))`
Now if M gras ice melts completely, water formed will have mass M grams (as mass is conserved). Now if `rho_(w)` is the density of water, the volume of water formed will be `V_(F)=(M//rho_(w))`
Here the liquid is water, i.e., `rho_(L)=rho_(w)` , so water displaced by floating ice is equal to water formed by melting of whole ice and hence the level of water will remain unchanged. Furthermore:
(a) If `rho_(L)gtrho_(w),(M//rho_(L))lt(M//rho_(w)),i.e.,V_(D)ltV_(F)`
i.e., water displaced by floating ice will be lesser than water formed and so the level of liquid in the beaker will rise.
(b) If `rho_(L)ltrho_(w),(M//rho_(L))gt(M//rho_(w)),i.e.,V_(D)gtV_(F)`
i.e., water displaced by floating ice will be more than more formed and so the level of liquid in the breaker will fall.