Effects of unbalances are proportional to the square of the speed because of several reasons. Let's discuss them in detail:

1. Introduction to Static and Dynamic Balancing:
- Balancing is the process of eliminating or minimizing the unbalance forces and moments in a rotating system.
- Static balancing involves the correction of unbalance forces only, while dynamic balancing corrects both unbalance forces and unbalance moments.
- Static balancing is usually sufficient for low-speed rotors, but as the speed increases, the effects of unbalance become more significant, requiring dynamic balancing.

2. Unbalanced Couples at Higher Speeds:
- At higher speeds, the unbalance of a rotor generates unbalanced couples or moments, causing the rotor to vibrate.
- These unbalanced couples result in additional forces acting on the rotor, which can lead to severe vibrations and potential failure.
- Therefore, dynamic balancing becomes necessary to eliminate or reduce these unbalanced couples and minimize the harmful effects.

3. Dangerous Nature of Unbalanced Forces at Higher Speeds:
- Unbalanced forces are always present in a rotating system, even at low speeds.
- However, at higher speeds, the effects of unbalanced forces become more pronounced and potentially dangerous.
- Unbalanced forces can cause excessive vibrations, leading to increased stress levels, fatigue, and failure of the rotor.
- Therefore, it is crucial to address these unbalanced forces through dynamic balancing to ensure the safe and efficient operation of the rotor.

4. Effects of Unbalances Proportional to the Square of the Speed:
- The effects of unbalances in a rotating system are directly proportional to the speed of rotation.
- When the speed increases, the centrifugal forces acting on the unbalanced masses also increase.
- These centrifugal forces cause larger displacements and higher levels of vibration.
- Mathematically, the centrifugal force is proportional to the square of the speed (F = mω²r), where F is the centrifugal force, m is the mass, ω is the angular velocity, and r is the distance from the center of rotation.
- Therefore, the effects of unbalances, including vibrations and stresses, increase exponentially with the square of the speed.
- Dynamic balancing helps minimize these effects by redistributing the unbalance masses to achieve a more balanced and stable rotor.

In conclusion, dynamic balancing becomes necessary as the speed of the rotor increases because the effects of unbalance, including unbalanced couples, become more significant and potentially dangerous. The effects of unbalances are proportional to the square of the speed, emphasizing the importance of dynamic balancing to ensure safe and efficient operation of high-speed rotors.