Short Answer:

Balancing of a single rotating mass is done by adding another mass in such a way that it produces an equal and opposite centrifugal force to the unbalanced force. This additional mass is placed in the same plane as the unbalanced mass but at an angle of 180° to it. The magnitude and position of the balancing mass are chosen so that the resultant force on the shaft becomes zero, ensuring smooth rotation without vibration or stress on bearings.

This process helps maintain the stability of rotating machines, reduces wear and tear, and prevents energy loss caused by vibration or uneven load distribution. Balancing ensures the machine runs quietly and efficiently.

Detailed Explanation:

Balancing of a Single Rotating Mass

When a body rotates about an axis, every particle in the body tries to move away from the center due to centrifugal force. If the mass is not uniformly distributed around the axis, an unbalanced centrifugal force acts on the shaft and bearings. This force causes vibration, noise, and sometimes damage to the machine. Therefore, to maintain smooth and stable operation, the unbalanced mass must be balanced.

In the case of a single rotating mass attached to a shaft, the unbalanced centrifugal force acts radially outward from the axis of rotation. The magnitude of this force is proportional to the mass, the square of angular velocity, and the radius of rotation. It can be expressed as:

where,
= centrifugal force,
= mass of the rotating body,
= angular velocity, and
= radius of rotation.

If this force is not counteracted, it produces vibration and bending stress in the shaft. To eliminate this, another mass known as the balancing mass is added in the same plane of rotation. The purpose of this balancing mass is to produce an equal and opposite centrifugal force so that both forces cancel each other.

Procedure of Balancing

1. Determine the unbalanced force:
   First, calculate the centrifugal force caused by the unbalanced mass using  .
2. Locate the position for balancing mass:
   The balancing mass must be placed diametrically opposite (i.e., at 180°) to the unbalanced mass in the same plane of rotation.
3. Find the magnitude of balancing mass:
   The magnitude of the balancing mass can be found using the equality of centrifugal forces:

where   and   are the unbalanced mass and its radius, and   and   are the balancing mass and its radius.

1. Attach the balancing mass:
   The balancing mass is then securely fixed to the shaft at the calculated position and radius. When the shaft rotates, the two equal and opposite centrifugal forces neutralize each other, resulting in complete balance.

Importance of Balancing a Single Rotating Mass

Balancing a single rotating mass is very important in all rotating machinery like turbines, engines, and fans. If a rotor is not balanced, the machine may suffer from:

• Excessive vibration and noise,
• Loosening of parts and fasteners,
• Bearing wear and failure, and
• Loss of efficiency and increased maintenance cost.

By balancing, all these problems are avoided, ensuring smooth and safe operation of the machine.

Applications

• In automotive engines to balance crankshafts.
• In electric fans and turbines to prevent vibration.
• In grinding wheels and rotors of compressors.
• In laboratory centrifuges to avoid mechanical damage.

Practical Example

Suppose a rotor has a 4 kg unbalanced mass rotating at a radius of 0.2 m. To balance this mass, a balancing mass must be placed opposite to it. If the balancing mass can be placed at a radius of 0.4 m, then the magnitude of the balancing mass can be calculated as:

 

Hence, a 2 kg mass placed at 0.4 m radius in the opposite direction will balance the rotating system.

Conclusion

Balancing of a single rotating mass is a vital process to ensure that the rotating shaft or body runs smoothly without vibration. It is done by adding a balancing mass of appropriate magnitude and position opposite to the unbalanced mass. This prevents mechanical vibration, reduces bearing loads, and increases the lifespan and efficiency of the machine. Proper balancing ensures reliability and safety in mechanical systems involving rotation.