A solid sphere is taken to a depth of 1500 m in an ocean. If the bulk modulus of the material of the sphere is 5 10^9 N m ⁻² , the percentage increase in its density is x 10⁻¹ % . The value of x is _____. (Take the density of ocean water as 10^3 kg m ⁻³ and g = 10 m s ⁻² )

Step-by-step solution

The increase in hydrostatic pressure at a depth h is given by: P = _w g h P = 10^3 10 1500 = 1.5 10^7 N m ⁻² The fractional change in volume is related to the bulk modulus B by: V V = P B V V = 1.5 10^7 5 10^9 = 3 10⁻³ Since the mass of the sphere remains constant, the density = M V . For small changes, the fractional increase in density is approximately equal to the fractional decrease in volume: = V V = 3 10⁻³ The percentage increase in density is: 100 % = 3 10⁻³ 100 % = 0.3 % Comparing this with x 10⁻¹ % , we ge