\(\rho = \frac{m}{V}\)\n\(\partial \rho = ?\)
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True or False The mass density $\rho$ of a fluid is defined as mass per unit volume $\left(\mathrm{kg} / \mathrm{m}^{3}\right)$ and is a constant that depends on the type of fluid.
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Find the density $\rho$ of a fluid at a depth $h$ in terms of its density $\rho_{0}$ at the surface. If a mass $m$ of fluid has volume $V_{0}$ at the surface, then it will have volume $V_{0}-\Delta V$ at a depth $h$. The density at depth $h$ is then which gives $$ \begin{array}{c} \rho=\frac{m}{V_{0}-\Delta V} \quad \text { while } \quad \rho_{0}=\frac{m}{V_{0}} \\ \frac{\rho}{\rho_{0}}=\frac{V_{0}}{V_{0}-\Delta V}=\frac{1}{1-\left(\Delta V / V_{0}\right)} \end{array} $$ However, from Chapter 12, the bulk modulus is $B=P /\left(\Delta V / V_{0}\right)$ and so $\Delta V / V_{0}=P / B$. Making this substitution, we obtain $$ \frac{\rho}{\rho_{0}}=\frac{1}{1-P / B} $$ If we assume that $\rho$ is close to $\rho 0$, then the pressure at depth $h$ is approximately $\rho_{0} g h$, and so $$ \frac{\rho}{\rho_{0}}=\frac{1}{1-\left(\rho_{0} g h / B\right)} $$
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