Question 8: Properties of Matter

Question

A cubical block of density $\rho_b=600 \mathrm{~kg} / \mathrm{m}^3$ floats in a liquid of density $\rho_{\mathrm{e}}=900 \mathrm{kg} / \mathrm{m}^3$. If the height of block is $H=8.0 \mathrm{~cm}$ then height of the submerged part is
$\_\_\_\_$ cm .

Accepted Answer

When an object floats in a liquid, the weight of the object is exactly balanced by the buoyant force (up thrust) exerted by the liquid.
Weight of block is $=\mathrm{W}=\mathrm{m} \cdot \mathrm{g}=\left(ho_{\mathrm{b}} 	imes \mathrm{V}_{	ext {total }}ight) 	imes \mathrm{g}$
Buoyant force ( $\mathrm{F}_{\mathrm{B}}$ ) acting on a submerged body is equal to the weight of displaced liquid,
$$ F_B=\left(ho_1 V_{	ext {submerged }}ight) 	imes g $$
For flotation :
$$ \mathrm{W}=\mathrm{F}_{\mathrm{B}} $$
$$ ho_{\mathrm{b}} \mathrm{~V}_{	ext {total }} \mathrm{g}=ho_{\mathrm{l}} \mathrm{~V}_{	ext {submerged }} \mathrm{g} $$
For a cubical block with a uniform cross-sectional area (A):
Total Volume $=\mathrm{V}_{	ext {total }}=\mathrm{AH}$
Submerged Volume $\mathrm{V}_{	ext {submerged }}=\mathrm{Ah}$ (where h is the submerged height)
$$\Rightarrow $$ $$ ho_{\mathrm{b}}(\mathrm{AH})=ho_{\mathrm{l}}(\mathrm{Ah}) \Rightarrow \mathrm{h}=\left(\frac{ho_{\mathrm{b}}}{ho_{\mathrm{l}}}ight) \mathrm{H} $$
Putting the given values,
$$ ho_{\mathrm{b}}=600 \mathrm{~kg} / \mathrm{m}^3, ho_{\mathrm{l}}=900 \mathrm{~kg} / \mathrm{m}^3 	ext { and } \mathrm{H}=8.0 \mathrm{~cm} $$
$$\Rightarrow $$ $$ \mathrm{~h}=\left(\frac{600}{900}ight) 	imes 8.0 \mathrm{~cm} \approx 5.33 \mathrm{~cm}  $$
Therefore, the height of the submerged part of the block is approximately 5.3 cm.
Hence, the correct option is (D).

Answer

6.3

Answer

4.3

Answer

7.3

Answer

5.3

Properties of Matter

A cubical block of density \rho_b=600 \mathrm{~kg} / \mathrm{m}^3 floats in a liquid of density \rho_{\mathrm{e}}=900 \mathrm{kg} / \mathrm{m}^3. If the height of block is H=8.0 \mathrm{~cm} then height of the submerged part is \_\_\_\_ cm .

Choose the correct answer: