Question

For a constant hydraulic stress on an object, the
fractional change in the object’s volume
(ΔV /V ) and its bulk modulus (B) are related as :
(a) ΔV ∝ B
V
(b) ΔV ∝ 1
V B
(c) ΔV ∝ B²
V
(d) ΔV ∝ B⁻²
V

Answer 1

$\Large\boxed{\sf{(b)\ \dfrac{\Delta V}{V}\propto\dfrac{1}{B}}}$

According to Hooke's Law, we have,

$\longrightarrow\sf{Modulus\ of\ Elasticity=\dfrac{Stress}{Strain}}$

which is constant. Thus we have, bulk modulus,

$\longrightarrow\sf{B=\dfrac{Volume\ Stress}{Volume\ Strain}}$

$\longrightarrow\sf{B=\dfrac{Volume\ Stress}{\left(\dfrac{\Delta V}{V}\right)}}$

In case of constant stress,

$\longrightarrow\sf{B\propto\dfrac{1}{\left(\dfrac{\Delta V}{V}\right)}}$

$\longrightarrow\sf{\underline{\underline{\dfrac{\Delta V}{V}\propto\dfrac{1}{B}}}}$