91Ó°ÊÓ

Gauge pressure measures the pressure in a fluid relative to atmospheric pressure. Unlike absolute pressure, which includes atmospheric pressure in its measurement, gauge pressure only measures the pressure above or below atmospheric pressure levels. This is important in fluid mechanics as it provides a practical measure of the pressure exerted by fluids in real-world applications, such as in the cylinder in this problem.
In this exercise, the initial gauge pressure was due to the mercury. We calculated it with the formula:

• \( P_{mercury} = \rho_{mercury} \cdot g \cdot h_{mercury} \)

where \( \rho_{mercury} = 13,600 \, \mathrm{kg/m^3} \), \( g = 9.81 \, \mathrm{m/s^2} \), and \( h_{mercury} = 0.05 \, \mathrm{m} \). The outcome was a pressure of \( 6,673.2 \, \mathrm{Pa} \). By doubling the gauge pressure, we could determine the extra contribution needed from the water that is added. So, the total gauge pressure was set to be twice the initial mercury pressure.

The density of mercury plays a crucial role in determining the pressure it exerts. With its density being quite high at \( 13,600 \, \mathrm{kg/m^3} \), mercury can exert significant pressure over a small height. This physical property makes mercury ideal for barometers and some pressure measurement devices. When using the formula for pressure, \( P = \rho \cdot g \cdot h \), where \( \rho \) is density, \( g \) is the acceleration due to gravity, and \( h \) is height, the large value of \( \rho_{mercury} \) shows why the pressure at a mere 5 cm height was so substantial. This density factor was evidently useful in doubling the gauge pressure when adding water in the exercise.

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the gravitational force. It increases with the depth of the fluid column. The key formula used to calculate hydrostatic pressure is:

• \( P = \rho \cdot g \cdot h \)

where \( \rho \) is the density, \( g \) is gravitational acceleration, and \( h \) is the height of the fluid column. In this exercise, hydrostatic pressure was calculated initially from mercury, and later from water when added to double the very gauge pressure. Notably, despite water having a lower density (\( 1,000 \, \mathrm{kg/m^3} \)), it contributed significantly to the hydrostatic pressure given the larger height needed.

The calculation of volume is necessary whenever dealing with varying fluid levels in enclosed spaces. It integrates the concepts of cross-sectional area and fluid height. In this particular exercise, after finding the height of the water column needed (0.68 m) to double the gauge pressure, the volume of water was determined.
The formula used was:

• \( V = A \cdot h_{water} \)

where \( A \) is the cross-sectional area (\( 0.0012 \, \mathrm{m^2} \)), and \( h_{water} \) is the newly derived water height. Calculating it gave \( V = 0.000816 \, \mathrm{m^3} \), which converts to \( 816 \, \mathrm{cm^3} \). Volume calculation is critical in many fields that require precise measurements of fluid quantities.