A body completely immersed in a liquid floats at some depth in equilibrium. What physical parameters determine the stability of this equilibrium? Write down the condition of stability of equilibrium, considering that the temperature of the liquid does not depend on the depth.
Decision:
A body immersed in a liquid is subject to the downward force of gravity $mg$ and the upward force $\frac{ \rho_{1}gm}{ \rho_{2}}$ (here $m$ is the mass of the body; $\rho_{1}$ and $\rho_{2}$ are the densities of the liquid and the body at some depth; obviously, the volume of the body is equal to $\frac{m}{ \rho_{2}}$). If the body is in equilibrium, the reciprocal of these forces is zero, hence $\rho_{1}=\rho_{2}$. Let us move the body from the equilibrium position, e.g. downwards, to some depth. The densities of the fluid and the body will change ($\tilde{\rho_{1}}$ and $\tilde{\rho_{2}}$). If the equilibrium is stable, the body will tend to return to its previous position, hence $\frac{ \tilde{\rho_{1}} mg}{ \tilde{\rho_{2}}} > mg$, i.e. $\tilde{\rho_{1}} > \tilde{\rho_{2}}$. This indicates that the compressibility of the fluid is greater than the compressibility of the body (obviously, this is sufficient for equilibrium stability).
Decision:
A body immersed in a liquid is subject to the downward force of gravity $mg$ and the upward force $\frac{ \rho_{1}gm}{ \rho_{2}}$ (here $m$ is the mass of the body; $\rho_{1}$ and $\rho_{2}$ are the densities of the liquid and the body at some depth; obviously, the volume of the body is equal to $\frac{m}{ \rho_{2}}$). If the body is in equilibrium, the reciprocal of these forces is zero, hence $\rho_{1}=\rho_{2}$. Let us move the body from the equilibrium position, e.g. downwards, to some depth. The densities of the fluid and the body will change ($\tilde{\rho_{1}}$ and $\tilde{\rho_{2}}$). If the equilibrium is stable, the body will tend to return to its previous position, hence $\frac{ \tilde{\rho_{1}} mg}{ \tilde{\rho_{2}}} > mg$, i.e. $\tilde{\rho_{1}} > \tilde{\rho_{2}}$. This indicates that the compressibility of the fluid is greater than the compressibility of the body (obviously, this is sufficient for equilibrium stability).