Search any question & find its solution
Question:
Answered & Verified by Expert
A concrete sphere of radius $R$ has a cavity of radius $r$ which is packed with sawdust. The specific gravities of concrete and sawdust are respectively 2.4 and 0.3 for this sphere to float with its entire volume submerged under water. Ratio of mass of concrete to mass of sawdust will be
Options:
Solution:
1325 Upvotes
Verified Answer
The correct answer is:
4
Let specific gravities of concrete and saw dust are $\rho_1$ and $\rho_2$ respectively.
According to principle of floatation weight of whole sphere = upthrust on the sphere
$\frac{4}{3} \pi\left(R^3-r^3\right) \rho_1 g+\frac{4}{3} \pi r^3 \rho_2 g=\frac{4}{3} \pi R^3 \times 1 \times g$
$\Rightarrow R^3 \rho_1-r^3 \rho_1+r^3 \rho_2=R^3$
$\Rightarrow R^3\left(\rho_1-1\right)=r^3\left(\rho_1-\rho_2\right) \Rightarrow \frac{R^3}{r^3}=\frac{\rho_1-\rho_2}{\rho_1-1}$
$\Rightarrow \quad \frac{R^3-r^3}{r^3}=\frac{\rho_1-\rho_2-\rho_1+1}{\rho_1-1}$
$\Rightarrow \quad \frac{\left(R^3-r^3\right) \rho_1}{r^3 \rho_2}=\left(\frac{1-\rho_2}{\rho_1-1}\right) \frac{\rho_1}{\rho_2}$
$\Rightarrow \frac{\text { Massof concrete }}{\text { Massof saw dust }}=\left(\frac{1-0.3}{2.4-1}\right) \times \frac{2.4}{0.3}=4$
According to principle of floatation weight of whole sphere = upthrust on the sphere
$\frac{4}{3} \pi\left(R^3-r^3\right) \rho_1 g+\frac{4}{3} \pi r^3 \rho_2 g=\frac{4}{3} \pi R^3 \times 1 \times g$
$\Rightarrow R^3 \rho_1-r^3 \rho_1+r^3 \rho_2=R^3$
$\Rightarrow R^3\left(\rho_1-1\right)=r^3\left(\rho_1-\rho_2\right) \Rightarrow \frac{R^3}{r^3}=\frac{\rho_1-\rho_2}{\rho_1-1}$
$\Rightarrow \quad \frac{R^3-r^3}{r^3}=\frac{\rho_1-\rho_2-\rho_1+1}{\rho_1-1}$
$\Rightarrow \quad \frac{\left(R^3-r^3\right) \rho_1}{r^3 \rho_2}=\left(\frac{1-\rho_2}{\rho_1-1}\right) \frac{\rho_1}{\rho_2}$
$\Rightarrow \frac{\text { Massof concrete }}{\text { Massof saw dust }}=\left(\frac{1-0.3}{2.4-1}\right) \times \frac{2.4}{0.3}=4$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.