In a hydraulic system, the pressure P of a working liquid varies with its volume V according to the relation P = P₀ + V^2 , where P₀ and are positive constants. If the liquid is subjected to a small additional pressure P , the elastic potential energy stored per unit volume of the liquid is

Step-by-step solution

First, find the bulk modulus B of the liquid. The given relation is P = P₀ + V⁻² . Differentiating with respect to V : dP dV = -2 V⁻³ = - 2 V^3 The bulk modulus is given by: B = -V dP dV = -V (- 2 V^3 ) = 2 V^2 From the given pressure relation, we have V^2 = P - P₀ . Substituting this into the expression for B : B = 2(P - P₀) The elastic potential energy stored per unit volume (energy density) u is given by: u = 1 2 stress strain Since bulk modulus B = stress strain , we can write strain as stress B . Here, the str