The density of a non-homogeneous liquid varies a function of...

The density of a non-homogeneous liquid varies a function of depth $h$, such that $\rho=(620+50 h) \mathrm{kg} / \mathrm{m}^{3},$ wher $h$ is in meters. Plot the variation of the pressure (vertic: axis) versus depth for $0 \leq h<16 \mathrm{~m}$. Give values for incre ments of $4 \mathrm{~m}$.

Key Concepts

Integration to Determine Pressure Variation

When the density of a liquid is not constant, the pressure at a given depth is obtained by integrating the density function over the relevant depth range. This approach, based on calculus, accumulates the contributions of the variable weight of the liquid layers above the point of interest, resulting in the total pressure at that depth.

Graphical Representation of Physical Quantities

Graphical representation involves plotting the relationship between two variables—in this case, pressure versus depth—to visualize how one quantity changes in response to the other. By evaluating the pressure at specific increments of depth, it becomes easier to understand and communicate the physical behavior of the system, providing useful insights in both analysis and presentation.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity. It is determined by the weight of the fluid above a point and for a variable-density liquid, the pressure is calculated by integrating the density function with respect to depth. This concept is fundamental in understanding how pressure increases as one moves deeper into a fluid.

Non-homogeneous Liquids

Non-homogeneous liquids are fluids whose properties, such as density, vary with position. In many practical applications, such as atmospheric studies or oceanography, the density can change with depth, and this variation must be taken into account when calculating pressure. Recognizing non-homogeneity is crucial for accurately modeling the behavior of the fluid.

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