Question

A piece of ice is floating in a jar containing water. When the ice melts, then the level of water :

• A. Rises
• B. Falls
• C. Remains unchanged
• D. Rises of Falls depending upon the mass of ice

Answer 1

Answer: (C)

The correct an option is C.

According to the Archimedes' principle, it states that the weight of water displaced will equal the upward buoyancy force provided by that water. In this case,

The weight of water displaced $=m_{waterdisplaced}g=\rho Vg=\rho Ahg$

where V is the volume of water displaced, ρ is the density of water, A is the surface area of the glass and g is acceleration due to gravity.

Therefore, the upward buoyancy force acting on the ice is $\rho Ahg$.

Now, the downward weight of ice is $m_{ice}g$.

Now, because the ice is neither sinking nor floating, these must balance. That is:

$\rho Ahg=m_{ice}g$

Therefore,

$h=\frac{m_{i}ce}{\rho A}$

Now when the ice melts, this height difference due to buoyancy goes to 0. But now an additional mass mice of water has been added to the cup in the form of water. Since mass is conserved, the mass of ice that has melted has been turned into an equivalent mass of water.

The volume of such water added to the cup is thus:

$V=m_{ice}\rho$

and therefore,

$Ah=\frac{m_{ice}}{\rho }$

So,

$h=\frac{m_{ice}}{\rho A}$

That is, the height the water has increased due to the melted ice is exactly the same as the height increase due to buoyancy before the ice had melted.