A thin uniform cylindrical shell, closed at both ends, is partial… - Vidyadip

MCQ

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If $\rho_0$ is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

✓
more than half filled if $\rho_c$ is less than $0.5$
B
more than half filled if $\rho_c$ is less than $1.0$
C
half filled if $\rho_c$ is less than $0.5$
D
less than half filled if $\rho_c$ is less than $0.5$

✓

Answer

Correct option: A.

more than half filled if $\rho_c$ is less than $0.5$

a
Let outer volume of shell is $V_0$

Let inner volume of shell is $V_1$

Let volume of water inside the shall is $v$.

$\Rightarrow 1 Vg +\rho_{ c }\left( V _0- V _1\right) g =\frac{1 V _0}{2} g $

$V +\rho_{ c }\left( V _0- V _1\right)=\frac{ V _0}{2}$

[Equlibrium]

$\rho_c\left(V_0-V_1\right)=\frac{V_0}{2}-V $

$\rho_c=\frac{\frac{V_0}{2}-V}{V_0-V_i} $

if $\rho_c<\frac{1}{2} \quad \Rightarrow \quad \frac{\frac{V_0}{2}-V}{V_0-V_i}<\frac{1}{2} $

$\frac{V_0}{2}-V<\frac{V_0}{2}-\frac{V_i}{2} $

$-V<-\frac{V_i}{2} $

$V >\frac{ V _{ i }}{2} \quad \text { so (A) } $

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