Question

If a rubber cube is taken $50m$ deep in a lake its volume reduces by $0.5\%$ the bulk modulus of rubber is nearly _______.

Answer 1

Step 1: Given Data

Depth $h=50m$

Change in volume $△V=-0.5\%=-\frac{0.5}{100}\times V$

Let the initial pressure be $P_0$.

Let $△P$ be the increase in pressure.

Let the density be $\rho ={10}^{3}kg/m^3$.

Let $g=10m/s^2$ be the acceleration due to gravity.

Let $V$ be the initial volume.

Step 2: Formula Used

We know that bulk modulus is given as,

$B=-\frac{△P}{{\displaystyle \raisebox{1ex}{$△V$}\!\left/ \!\raisebox{-1ex}{$V$}\right.}}$

Step 3: Calculate the Bulk Modulus

The final pressure can be given as,

$P=P_0+\rho gh$

$\therefore △P=\rho gh$

We know that bulk modulus is given as,

$$\begin{gathered} B=-\frac{△P}{{\displaystyle \raisebox{1ex}{$△V$}\!\left/ \!\raisebox{-1ex}{$V$}\right.}} \\ =-\frac{\rho ghV}{△V} \\ =-\frac{{10}^3\times 10\times 50\times V\times 100}{-0.5\times V} \\ ={10}^{8}N/m^2 \end{gathered}$$

Hence, the bulk modulus of rubber is nearly ${10}^{8}N/m^2$.