Question

Two unequal blocks placed over each other of different densities ${\sigma }_1$ and ${\sigma }_2$ are immersed in a fluid of density of $\sigma$. The block of density ${\sigma }_1$ is fully submerged and the block of density ${\sigma }_2$ is partly submerged so that ratio of their masses is 1/2 and $\sigma /{\sigma }_1=2$ and $\sigma /{\sigma }_2=0.5$. Find the degree of submergence of the upper block of density ${\sigma }_2$

• A. 50% submerged
• B. 25% submerged
• C. 75% submerged
• D. fully submerged

Answer 1

Answer: (D)

fully submerged

Let the volumes of two blocks be $V_1$ and $V_2$

Let the fraction $f$ of the second block is submerged.

Balancing weight and buoyancy,

$V_1{\sigma }_{1}g+f{V}_2{\sigma }_{2}g=(V_1+f{V}_2)\sigma g$

also, $V_1{\sigma }_1/V_2{\sigma }_2=m_1/m_2=1/2\Rightarrow V_2{\sigma }_2=2V_1{\sigma }_1$

$V_2{\sigma }_2+f\times 2V_1{\sigma }_1=(V_1+f\times 2V_1{\sigma }_1/{\sigma }_2)\sigma g$

$\Rightarrow {\sigma }_1+2f{\sigma }_2=(1+2f{\sigma }_1/{\sigma }_2)\sigma$

${\sigma }_1/\sigma +2f{\sigma }_1/\sigma =(1+2f{\sigma }_1/{\sigma }_2)$

$1/2+2f/2=(1+2f/4)$

$\Rightarrow f=1$

So second block is submerged completely.