Question 173: Properties of Matter

Question

A solid sphere of radius r made of a soft 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, covering entire cross
section of cylindrical container. When a mass m is placed on the surface of the piston to compress the
liquid, the fractional decrement in the radius of the sphere, $$\left( {{dr \over r}} \right)$$ is:

Accepted Answer

Because of m mass the extra pressure created is,
$$\Delta $$P = $${{mg} \over a}$$
And Bulk modulus, $$\beta $$ = $${{\Delta P} \over {{{\Delta V} \over V}}}$$
Given $$\beta $$ = K
$$	herefore\,\,\,$$ K = $${{{{mg} \over a}} \over {{{\Delta V} \over V}}}$$
We know volume of sphere,
V = $${4 \over 3}\pi {r^3}$$
$$	herefore\,\,\,$$ $${{dV} \over V}$$ = 3 $${{dr} \over r}$$
$$	herefore\,\,\,$$ K = $${{{{mg} \over a}} \over {3{{dr} \over r}}}$$
$$ \Rightarrow $$$$\,\,\,$$ $${{dr} \over r}$$  =  $${{mg} \over {3Ka}}$$

Answer

$${{mg} \over {Ka}}$$

Answer

$${{Ka} \over {mg}}$$

Answer

$${{Ka} \over {3mg}}$$

Answer

$${{mg} \over {3Ka}}$$

Properties of Matter

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