Question

$A$ solid sphere of radius $r$ made of $a$ material of bulk modulus $K$ is surrounded by $a$ liquid in $a$ cylindrical container. $A$ massless piston of area $a$ floats on the surface of the liquid. When $a$ mass $m$ is placed on the piston to compress the liquid, the fractional change in the radius of the sphere $\left(\right.Δr/r\left.\right)$ is

• A. $ka/mg$
• B. $ka/3mg$
• C. $mg/3ka$
• D. $mg/ka$

Answer 1

Answer: (C)

The volume of sphere in liquid,
$V=\frac{4}{3}\pi r^{3}$
When mass $m$ is placed on the position, the increased pressure
$p=\frac{mg}{a}$

since, this increased pressure is equally applicable to all directions on the sphere, so there will be decrease in volume of sphere, due to decrease in its radius
$ΔV=4/3\pi \times 3r^{2}Δr=4\pi r^{2}Δr$
$\Rightarrow \frac{ΔV}{V}=\frac{4 \pi r^{2} Δr}{4 / 3 \pi ^{3}}=\frac{3 \Delta }{r}$
$\therefore $ Bulk modulus,
$K=\rho /σV/V=\frac{mg}{a}\times \frac{r}{3 Δr}$
$\therefore Δr/r=mg/3ka$