The controlling force curves P, Q and R for a spring controlled governor are shown in the figure, where 𝒓𝟏 and 𝒓𝟐 are any two radii of rotation.

The characteristics shown by the curves are

The correct option is P – Unstable; Q – Isochronous; R – Stable.

Analysis of the Controlling Force Curves:
For a spring-controlled governor, the relation between the controlling force ($F_c$) and the radius of rotation ($r$) is represented by a straight-line equation of the form:

$$F_c=a\cdot r+b$$

Where:
• $a$ is a constant representing the slope of the line.
• $b$ is a constant representing the intercept on the controlling force (vertical) axis.
• $r$ is the radius of rotation.

At any equilibrium state, the controlling force must balance the centrifugal force, which is given by:

$$F_c=m\cdot {\omega }^2\cdot r$$

Where $m$ is the mass of the governor balls and $\omega$ is the angular velocity (speed) of rotation. Equating these two expressions:

$$m\cdot {\omega }^2\cdot r=a\cdot r+b$$

Dividing both sides by $m\cdot r$, we obtain the relationship for the governor's speed:

$${\omega }^2=\frac{a}{m}+\frac{b}{m\cdot r}$$

Let us now analyze each curve based on the value of the intercept $b$ shown in the image:

1. Curve Q (Isochronous Governor):
As seen in the figure, curve Q passes directly through the origin. Therefore, its vertical intercept is zero ($b=0$). Substituting this value into the speed equation:

$${\omega }^2=\frac{a}{m}=\text{constant}$$

Since the equilibrium speed $\omega$ remains constant for all radii of rotation, curve Q represents an isochronous governor.

2. Curve P (Unstable Governor):
Curve P has a positive vertical intercept ($b>0$). Looking at the speed relationship:

$${\omega }^2=\frac{a}{m}+\frac{b}{m\cdot r}$$

As the radius of rotation $r$ increases, the fraction $\frac{b}{m\cdot r}$ decreases. Consequently, the equilibrium speed $\omega$ decreases as the radius $r$ increases. A governor is stable only if its speed increases with an increase in radius. Because the speed decreases here, curve P represents an unstable governor.

3. Curve R (Stable Governor):
Curve R intersects the horizontal axis to the right of the origin, which corresponds to a negative vertical intercept ($b<0$). Letting $b=-c$ (where $c>0$):

$${\omega }^2=\frac{a}{m}-\frac{c}{m\cdot r}$$

As the radius of rotation $r$ increases, the term subtracted ($\frac{c}{m\cdot r}$) decreases. This causes the overall value of ${\omega }^2$ to increase. Since the equilibrium speed $\omega$ increases as the radius $r$ increases, curve R represents a stable governor.