Final answer:
To calculate the gauge pressure at the pipe entrance, sum the hydrostatic pressure due to the oil column's height and density, and the pressure overcoming frictional losses caused by the oil's viscosity.
Step-by-step explanation:
The student's question relates to the calculation of gauge pressure at the entrance of a 50.0-meter-long vertical pipe of an oil gusher. To determine the gauge pressure, one must consider the fluid’s properties, such as its density and viscosity, as well as the geometry of the system, including the length and diameter of the pipe.
Firstly, the hydrostatic pressure due to the column of oil needs to be taken into account. This is calculated using the height of the column and the density of the oil. Secondly, the pressure needed to overcome frictional losses due to the viscosity of the oil must be calculated, which requires the use of principles from fluid mechanics, such as Poiseuille's law for laminar flow in pipes. The gauge pressure can then be found by summing these two pressures.