Physics - Elasticity Question with Solution | TestHub

The bulk modulus $K$ of the material is related to Young's modulus $Y$ and Poisson's ratio $\sigma$ by the formula $K=\frac{Y}{3(1-2\sigma )}$.

When the sphere is lowered from the surface to a depth $h$, the pressure acting on it increases. The change in pressure $\Delta P$ is given by the hydrostatic pressure formula: $\Delta P=(P_0+{\rho }_{w}gh)-P_0={\rho }_{w}gh$. The definition of bulk modulus is $K=-\frac{\Delta P}{\Delta V/V}$, where $\Delta V/V$ is the fractional change in volume. Therefore, the fractional change in volume is $\frac{\Delta V}{V}=-\frac{\Delta P}{K}$. Substituting the expressions for $\Delta P$ and $K$:$$\frac{\Delta V}{V}=-\frac{{\rho }_{w}gh}{Y/(3(1-2\sigma ))}=-\frac{3{\rho }_{w}gh(1-2\sigma )}{Y}$$ The negative sign indicates a decrease in volume (compression).