The bearing characteristic number is a dimensionless quantity that is often used in the analysis and design of journal bearings. It is defined as the product of the absolute viscosity of the lubricant, the rotational speed of the journal, and the average bearing pressure.

To understand why the correct answer is option 'A', let's examine the equation for the bearing characteristic number and its individual components:

Bearing Characteristic Number (Nc) = Z * N / P

- **Absolute Viscosity (Z)**: The absolute viscosity of the lubricant represents its resistance to flow. It is a measure of how thick or thin the lubricant is. A higher viscosity means a thicker lubricant, while a lower viscosity means a thinner lubricant. The absolute viscosity is denoted by the symbol Z.

- **Rotational Speed (N)**: The rotational speed of the journal refers to how fast the journal rotates within the bearing. It is usually measured in rotations per minute (RPM) or radians per second (rad/s). The rotational speed is denoted by the symbol N.

- **Average Bearing Pressure (P)**: The average bearing pressure represents the force per unit area exerted on the bearing surfaces due to the load being supported. It is determined by dividing the load by the projected area of the bearing. The average bearing pressure is denoted by the symbol P.

Now, let's analyze the given options:

a) Z * N / P: This is the correct answer. It follows the equation for the bearing characteristic number, where the absolute viscosity (Z) is multiplied by the rotational speed (N) and divided by the average bearing pressure (P).

b) P / (Z * N): This option has the inverse of the correct answer. Dividing the average bearing pressure (P) by the product of the absolute viscosity (Z) and the rotational speed (N) does not yield the correct bearing characteristic number.

c) Z / P: This option only considers the ratio of the absolute viscosity (Z) to the average bearing pressure (P), without including the rotational speed (N). It does not represent the bearing characteristic number correctly.

d) N / (Z * P): This option has the inverse of the correct answer. Dividing the rotational speed (N) by the product of the absolute viscosity (Z) and the average bearing pressure (P) does not yield the correct bearing characteristic number.

In conclusion, the correct answer is option 'A' because it follows the equation for the bearing characteristic number, taking into account the absolute viscosity, rotational speed, and average bearing pressure.