A solid metal sphere of volume 0.2 m ^3 and bulk modulus 2 10¹⁰ N m ⁻² is lowered to a depth of 1000 m in an ocean. The density of ocean water is 1000 kg m ⁻³ and acceleration due to gravity is 10 m s ⁻² . The elastic potential energy stored in the sphere due to hydrostatic compression is _______ J.

Step-by-step solution

The hydrostatic pressure P at a depth h is: P = g h = 1000 10 1000 = 10^7 N m ⁻² The elastic potential energy U stored in a compressed solid is given by: U = 1 2 stress strain volume Since Bulk modulus B = stress strain = P ( V V ) , we have strain = P B . Thus, the energy stored is: U = 1 2 P ( P B ) V = ( P)^2 V 2B Substituting the given values: U = (10^7)^2 0.2 2 (2 10¹⁰) U = 10¹⁴ 0.2 4 10¹⁰ U = 0.2 4 10^4 U = 0.05 10000 = 500 J Answer: 500