In the context of balancing of rotating and reciprocating masses, which one of the following options is true?

The correct option is: An unbalanced rigid rotor can be completely balanced using two balancing masses attached in two distinct planes.

To understand why this is the correct statement, let's analyze the concepts of static and dynamic balancing for a rigid rotor, as well as the balancing of reciprocating engines:
1. Balancing of a Rigid Rotor:
An unbalanced rigid rotor generally experiences two types of unbalance:
- Static Unbalance: This occurs when the center of gravity of the rotor does not lie on the axis of rotation. It causes a net centrifugal force. Static unbalance can be corrected by adding a single balancing mass in the plane of the unbalance.
- Dynamic Unbalance: This occurs when the axis of rotation does not coincide with one of the principal axes of inertia. Even if the center of mass lies on the axis of rotation (i.e., it is statically balanced), rotation will generate unbalanced couples (moments).
To completely balance a rigid rotor (achieve both static and dynamic balance, meaning both the net centrifugal force and the net centrifugal couple are zero), we require at least two balancing masses placed in two distinct planes. A single mass cannot balance a dynamic couple because it would introduce a net force. Therefore, the second option is correct, and the first option is incorrect.

2. Balancing of Reciprocating Engines (Single-Cylinder Engine):
In a single-cylinder reciprocating engine, the forces generated due to the reciprocating parts (piston, wrist pin, etc.) are linear and vary periodically (primary and secondary forces).
- These reciprocating forces cannot be completely balanced by rotating masses because a rotating mass produces a centrifugal force that has components in all radial directions, whereas the reciprocating forces act only along the line of stroke.
- If we attempt to balance the primary reciprocating force along the line of stroke using a rotating mass, we inevitably introduce an unbalanced force of equal magnitude perpendicular to the line of stroke. Therefore, a single-cylinder internal combustion engine cannot be completely balanced using rotating balancing masses (whether one or two). This makes the third and fourth options incorrect.

Thus, the only true statement is that an unbalanced rigid rotor can be completely balanced using two balancing masses attached in two distinct planes.