Introduction:
A spring-controlled governor is a device used to control the speed of an engine or machine by adjusting the flow of fuel or energy. It consists of a rotating shaft with attached weights and springs. The centrifugal force generated by the rotating weights acts against the spring force, creating a controlling force that determines the position of the governor.

Explanation:
The controlling force (F) in a spring-controlled governor is related to the radius (r) by the equation F = ar - b, where 'a' and 'b' are constants.

Understanding the equation:
1. The term 'ar' represents the centrifugal force, which is directly proportional to the radius. As the radius increases, the centrifugal force also increases.
2. The term 'b' represents the spring force, which is a constant value determined by the stiffness of the spring. The spring force acts in the opposite direction to the centrifugal force.
3. The equation F = ar - b represents the net force acting on the governor. If the net force is positive, it means the centrifugal force is greater than the spring force, and vice versa.

Stability of the governor:
For a spring-controlled governor to be stable, it should return to its equilibrium position when subjected to small disturbances. In other words, it should resist any changes in speed and maintain a steady state.

Explanation of the correct option:
The correct option is 'F = ar - b'. This equation satisfies the stability criterion for a spring-controlled governor. Here's why:

1. When the governor is at equilibrium, the net force (F) is zero. This occurs when the centrifugal force (ar) is equal to the spring force (b).
2. If the governor experiences a small increase in speed, the centrifugal force increases, leading to a positive net force (F > 0).
3. The positive net force causes the governor to move towards a lower radius, reducing the centrifugal force and bringing it back to equilibrium.
4. Similarly, if the governor experiences a small decrease in speed, the centrifugal force decreases, leading to a negative net force (F < />
5. The negative net force causes the governor to move towards a higher radius, increasing the centrifugal force and bringing it back to equilibrium.

Conclusion:
In summary, the equation F = ar - b represents the controlling force in a spring-controlled governor. This equation ensures stability by maintaining a balance between the centrifugal force and the spring force. When the governor experiences small disturbances in speed, it adjusts its position to counteract these changes and return to equilibrium.