Question

Place a glass beaker, partially filled with water, in a sink. The beaker has a mass of 390gm and an interior volume of $500c{{m}^{3}}$. You now start to fill the sink with water and you find, by experiment, that if the beaker is less than half full, it will float; but if it is more than half full, it remains on the bottom of the sink as the water rises to its rim. What is the density of the material of which the beaker is made?

Answer 1

Hint:The buoyant force is greater than the gravitational force acting on the beaker, until the beaker is less than half full. And when it is half full, the gravitational force overcomes the buoyant force. This means that gravitational force is equal to the buoyant force. Calculate the total volume of the beaker and the water in it by equating the two forces.

Formula used:
${{F}_{g}}=mg$
where ${{F}_{g}}$ is the gravitational force on the body of mass m and g is acceleration due to gravity.
${{F}_{B}}={{\rho }_{l}}g{{V}_{S}}$ where ${{F}_{B}}$ is the buoyant force on a body submerged in a liquid of density ${{\rho }_{l}}$ and ${{V}_{S}}$ is the volume of the body submerged in the liquid.
$m=\rho V$
where m is mass, $\rho $ is density and V is volume of the body.

Complete step by step answer:
When a body is put in a liquid, the liquid exerts a force called force of buoyancy on the body. the direction of the buoyant force is always in the opposite direction, opposing the gravitational force acting on the body. It is given that if the glass beaker is more than half full with water, then it will sink in the water of the container. Otherwise, if the beaker is less than half full, it will float on the surface of the water.

This means that the buoyant force is greater than the gravitational force acting on the beaker, until the beaker is less than half full. And when it is half full, the gravitational force overcomes the buoyant force. This means that gravitational force is equal to the buoyant force.
I.e. ${{F}_{g}}={{F}_{B}}$
$\Rightarrow mg={{\rho }_{W}}g{{V}_{S}}$ .…. (i)
When the beaker is half full, the total mass of the beaker is equal to the sum of its mass and the mass of water in it.
i.e. $m={{m}_{B}}+{{m}_{W}}$.
Here, ${{m}_{B}}=390gm$ and ${{m}_{W}}={{\rho }_{W}}{{V}_{W}}$.
Here, ${{\rho }_{W}}=1gc{{m}^{-3}},{{V}_{W}}=250c{{m}^{3}}$.
$\Rightarrow {{m}_{W}}=1\times 250=250gm$.
$\Rightarrow m={{m}_{B}}+{{m}_{W}}=390+250=640gm$.
Substitute the known values in (i)
$\Rightarrow 640g=(1)g{{V}_{S}}$
$\Rightarrow {{V}_{S}}=640c{{m}^{3}}$.

This is the total volume submerged (glass and water in it) in the water of the container.
$\Rightarrow {{V}_{S}}={{V}_{B}}+{{V}_{W}}=640c{{m}^{3}}$
But ${{V}_{W}}=500c{{m}^{3}}$
$\Rightarrow {{V}_{B}}+500c{{m}^{3}}=640c{{m}^{3}}$.
$\Rightarrow {{V}_{B}}=140c{{m}^{3}}$.
$\Rightarrow{{\rho }_{B}}=\dfrac{{{m}_{B}}}{{{V}_{B}}}\\
\Rightarrow{{\rho }_{B}}=\dfrac{390}{140}\\
\therefore{{\rho }_{B}}=2.78gc{{m}^{-3}}$

Therefore, the density of the material by which the beaker is made is $2.78gc{{m}^{-3}}$.

Note: In the given case, first placing the half filled beaker in empty sink and then filling up the sink is similar to first fill up the sink and then place the half filled beaker in itYou can analyse anyone of the cases and both will give the same result.Note that the beaker and water in it act as a single body and the only the water outside the beaker exert the buoyant force.