Governor characteristics — for a spring-controlled governor with controlling force Fc = a·r + b (where r is radius of rotation), what is the nature of stability?

Introduction / Context: In a spring-controlled centrifugal governor, the controlling force–radius relationship determines whether speed increases cause balls to move inward or outward. Stability requires speed to rise with radius (or equivalently, controlling force to increase with radius appropriately).

Given Data / Assumptions:

• Fc = a·r + b, with a > 0 typically for practical springs.
• r is ball radius of rotation; Fc balances the required centripetal force m·ω²·r.
• At equilibrium, Fc = m·ω²·r.

Concept / Approach: From equilibrium, m·ω²·r = a·r + b ⇒ ω² = (a + b/r)/m. If slope a is positive, Fc increases with r, producing a rising equilibrium speed with radius—the hallmark of a stable governor. Isochronism requires the Fc–r line to pass through the origin (b = 0) with the correct slope so that equilibrium speed is constant for all r.

Step-by-Step Solution:

Verification / Alternative Check: Graphically, a straight Fc–r line with positive slope intersecting the Fc axis at b gives a rising characteristic and thus a stable governor.

Why Other Options Are Wrong:
Unstable — Would require a decreasing Fc with r (negative slope) or inappropriate characteristic.
Isochronous — Needs b = 0 and a specific slope so speed is same for all radii, which is not guaranteed with b ≠ 0.
None of these — Not applicable.

Common Pitfalls: Assuming any affine (linear) law is isochronous; isochronism demands b = 0 and precise tuning.

Final Answer: Stable.