A rubber ball of bulk modulus 5 10^8 N m ⁻² is taken to the bottom of a lake. The volume of the ball decreases by 0.05 % . If the density of lake water is 1000 kg m ⁻³ and acceleration due to gravity is 10 m s ⁻² , the depth of the lake is _______ m.

Step-by-step solution

The percentage decrease in volume is 0.05 % , so the fractional change in volume is: V V = 0.05 100 = 5 10⁻⁴ The excess pressure (hydrostatic pressure) P exerted on the ball is given by the bulk modulus formula: P = B ( V V ) P = (5 10^8) (5 10⁻⁴) = 25 10^4 N m ⁻² The hydrostatic pressure at depth h is also given by: P = g h Substituting the values: 25 10^4 = 1000 10 h 10^4 h = 25 10^4 h = 25 m Answer: 25