Question

The volume of the hollow portion of a sphere is $3/4$ of the external volume of the sphere. If it floats in a liquid of relative density $3/2$, half of its external volume immersed, the relative density of the material of the solid is:

• A. $1.8$
• B. $2.4$
• C. $3$
• D. $2$

Answer 1

The correct option is C $3$

We know that, $B=Vpg$
Let $R$ be external radius of sphere and $r$ be radius of $c$.
Given,

$\frac{4}{3}\pi r^3=\frac{3}{4}\left(\frac{4}{3}\pi R^3\right)$

$r^3=\frac{3}{4}{R}^3\dots \left(i\right)$

$\frac{p_l}{p_w}=\frac{3}{2}\dots \left(ii\right)$

According to given condition,
Upthrust (Weight of liquid displaced) = Weight of body.

$V{p}_{l}g=\frac{4}{3}\pi (R^3-r^3)pg$

Using $\left(i\right)$ and $\left(ii\right)$,

$=\frac{1}{2}\times \frac{4}{3}\pi R^3\times \frac{3}{2}{p}_w\times g$

$=\frac{4}{3}\pi (R^3-r^3)pg$

$\frac{p}{pw}=3$