Question

A waterproof rubber ball is submerged under water to a depth of \(59.01 \mathrm{~m}\). The fractional change in the volume of the ball is \(2.937 \cdot 10^{-2}\) What is the bulk modulus of the rubber ball?

Short answer

Answer: The bulk modulus of the rubber ball is 19.72 GPa.

Step-by-step solution

Write down the known values

We know the depth (h) is 59.01 meters, the fractional volume change is \(2.937 \cdot 10^{-2}\), density of water (ρ) is \(1000 \mathrm{~kg/m^3}\), and the acceleration due to gravity (g) is \(9.81 \mathrm{~m/s^2}\).

Calculate the pressure at given depth

Using the formula for pressure at a depth: Pressure = ρgh, we can calculate the pressure: Pressure = \((1000 \mathrm{~kg/m^3}) (9.81 \mathrm{~m/s^2})(59.01 \mathrm{~m}) = 579579 \mathrm{~Pa}\)

Calculate the bulk modulus

The formula for bulk modulus (B) is: \(B = \frac{Pressure}{\frac{ΔV}{V_0}}\) Where ΔV/V₀ is the fractional volume change. We are given this value: \(\frac{ΔV}{V_0} = 2.937 \cdot 10^{-2}\) Now we can replace the values into the formula for bulk modulus and solve for B: \(B = \frac{579579 \mathrm{~Pa}}{2.937 \cdot 10^{-2}} = 19722000 \mathrm{~Pa}\) or \(19.72 \mathrm{~GPa}\)

State the final answer

The bulk modulus of the rubber ball is \(19.72 \mathrm{~GPa}\).