Question

A piece of ice is floating in a jar containing water. When the ice melts than the level of water:
A. Rises
B. Falls
C. Remains unchanged
D. Rises or falls depending upon the mass of ice

Answer 1

Archimedes’ principle states that any object submerged in a fluid, whether completely or partially, at rest is acted upon by an upward force. The magnitude of this upward force, also known as buoyant force, is equal to the weight of the fluid displaced by the body.
Use Archimedes’ principle to obtain the level of water.

Formula used: mass, $m=\rho V$, buoyant force = weight displaced

Complete step by step answer:
According to Archimedes’ principle, an upward buoyant force acts on an object submerged in a liquid.
According to the question, water applies an upward buoyant force on ice. Since, ice is neither sinking nor rising. The buoyant force is equal to the weight of water displaced.
Weight of water = ${{m}_{dw}}g=\rho Vg=\rho Ahg$
Where ${{m}_{dw}}$ is the mass of displaced water, V is the volume of water displaced, $\rho$ is the density of water, A is the surface area of the glass and g is acceleration due to gravity.
We can say that, the upward buoyancy force acting on the ice is $\rho Ahg$
Now, the downward weight of ice ${{m}_{ice}}g$ is equal to upward buoyancy force acting on the ice $\rho Ahg$. Therefore,
$\rho Ahg={{m}_{ice}}g$
$\Rightarrow h=\dfrac{{{m}_{ice}}g}{\rho A\text{g}}=\dfrac{{{m}_{ice}}}{\rho A}$
When the ice melts, this height difference due to buoyancy tends to 0 but an additional mass ${{m}_{ice}}$ of water has been added to the jar in the form of water. Since, the mass of ice that has melted has been turned into an equivalent mass of water, the volume of water added to the cup is
$V=\dfrac{{{m}_{ice}}}{\rho }=Ah'$
$h'=\dfrac{{{m}_{ice}}}{\rho A}$
We can observe that, $h=h'$ i.e. 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.

So, the correct answer is “Option C”.

Note: The downward force on the object is its weight. The upward force on the object is the buoyant force stated by Archimedes' principle. Therefore, the net force on the object is the difference between the magnitudes of the buoyant force and its weight. If this net force is greater than zero, the object rises; if less than zero, the object sinks; and if zero, the object remains in place without either rising or sinking.