Spring-controlled governor characteristic: If the controlling force varies as Fc = a·r + b with radius of rotation r,\nthen what is the nature of the governor's stability?

Topic: Governors — controlling force and stability

Given data

• Controlling force: F_c = a·r + b
• Governor is spring-controlled.

Concept / Approach
For stability, the required condition is that controlling force increases with radius (dF_c/dr > 0), i.e., the controlling force line has a positive slope. A spring-controlled governor has a straight-line characteristic of the form a·r + b. When a > 0, the line rises with r; thus the speed increases with radius, ensuring stable equilibrium at each radius.

Step-by-step reasoning
1) Stability criterion: dF_c/dr > 0 ⇒ slope a > 0.2) Given F_c = a·r + b is linear; for practical governors, a is chosen positive.3) With positive slope, an increase in radius demands a higher speed for equilibrium, yielding a unique stable operating point.

Verification
Isochronous behavior would require a straight line through origin (b = 0) with a specific condition relating to speed being constant for all radii; a nonzero intercept b violates that strict condition. Negative slope (a < 0) would indicate instability.

Common pitfalls

• Assuming any linear expression means isochronous; isochronism requires the line to pass through the origin under specific constraints.
• Ignoring the sign of slope a; stability hinges on a > 0.

Final Answer
Stable