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Specific gravity (SG) is a measure that helps us understand how dense a fluid is, compared to water. If a fluid has a specific gravity of 1, it means it has the same density as water. Oil in our problem has a specific gravity of 0.9. This indicates that oil is less dense than water by 10%.

To find the actual density of any fluid with a given specific gravity, we use water's density as a benchmark. Water is usually used as the reference fluid with a standard density of 1000 kg/m³. Therefore, for oil with an SG of 0.9, the density calculation is:

\[\text{Density of Oil} = 0.9 \times 1000\, \text{kg/m}^3 = 900\, \text{kg/m}^3\]
This provides us a straightforward way to relate specific gravity to actual fluid density, making it easier to work with other calculations.

Angular velocity is a concept used to describe how fast something is rotating. It is expressed in radians per second (rad/s), which denotes how many radians a rotating object covers in one second. In our exercise, the cylindrical tank is spinning at an angular velocity of 40 rad/s.

Understanding angular velocity is essential as it directly affects the pressure distribution within a rotating fluid. As angular velocity increases, the centrifugal effects become more significant, impacting the pressure difference across the fluid. This rotational speed is crucial for calculating the pressure change in the fluid due to rotation, using methods from fluid dynamics.

Pressure difference in a rotating fluid arises due to centrifugal forces. These forces cause the fluid to move outward from the axis of rotation, lowering pressure at the center and increasing it at the periphery.

The pressure difference in the exercise is calculated using the formula:

\[\Delta P = 0.5 \times \rho \times \omega^2 \times r^2\]

Where \(\Delta P\) is the pressure difference, \(\rho\) is fluid density, \(\omega\) is angular velocity, and \(r\) is radius. Plug in the given values to obtain:

\[\Delta P = 0.5 \times 900 \text{ kg/m}^3 \times (40 \text{ rad/s})^2 \times (0.2 \text{ m})^2 = 57600 \text{ Pa}\]

This formula is vital because it allows us to predict how pressure changes within rotating systems, which is important in many engineering applications.

A cylindrical tank in fluid dynamics is often used to contain and study the behavior of fluids. In our exercise, the cylindrical tank has a diameter of 0.4 meters, which we use to calculate the radius (0.2 meters).

In rotating fluid systems, the cylinder's shape significantly influences the fluid motion because the geometry determines how the angular velocity affects radial acceleration and thus the pressure distribution.

Understanding the role of the cylindrical tank helps us comprehend the fluid behavior and consequent pressure differences when the tank rotates. This application is common in various engineering fields like aviation fuel systems and industrial mixers, where cylindrical tanks often ensure uniformity and efficiency in fluid management.