Place a glass beaker, partially filled with water, in a sink. The beaker has a mass $390 g m$ and an interior volume of $500 {c m}^{3}$ . 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?

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Solution

It is given that when the beaker is half full, it will float, that means weight of water and beaker is supported by upthrust. Lets say $V$ is total volume of water displaced in the sink.
Mass of water in half filled beaker $= 250 {c m}^{3} \times 1 g m / {c m}^{3} = 250 g m$
So applying the law of floatation, we have
$weight of water and beaker = weight of water displaced in the sink \Rightarrow (\frac{390 + 250}{1000}) \times 9.8 = V \times 1000 \times 9.8 \Rightarrow V = 640 \times 10^{- 6} m^{3} = 640 {c m}^{3}$
So volume of solid mass of beaker $= 640 {c m}^{3} - 500 {c m}^{3} = 140 {c m}^{3}$
And we know the mass of the beaker $= 390 g m$
$\Rightarrow Density of beaker = \frac{390 g m}{140 c m^{3}} = 2.7857 g m / c m^{3} \approx 2.79 g m / c m^{3}$

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