Problems on Hartnell Governor:

Let, $m =$ mass of each ball

$M =$ Mass of sleeve

$r_1$ = minimum radius of rotation

$r_2$ = maximum radius of rotation

$w_1$ = angle speed of governor @ mini radius.

$w_2$ = angle speed of governor @ maximum radius

$s_1$ = Spring force exerted on sleeve @ $w_1$

$s_2$ = Spring force exerted on sleeve @ $w_2$

$fc_1 =$ Centrifugal force @ $w_1$

$fc_2 =$ Centrifugal force $w_2$

$S =$ Stiffness of spring

$X =$ length of vertical/ball arm

$Y =$ length of horizontal /sleeve arm

$r =$ Distance of fulcrum from governor axis/ Radius of rotation when the governor is in mid-position

Diagram

$\frac{h_1}{Y} = \frac{a}{X} = \frac{(r – r_1)}{X}$ - - - -(A)

$\frac{h_2}{Y} = \frac{b}{X} = \frac{(r_2 – r)}{X}$ - - - -(B)

Addition A and B,

$\frac{h_1 + h_2}{Y} = \frac{r-r_1}{X} + \frac{r_2 – r}{X}$

$\frac{h}{Y} = \frac{r – r_1 + r_2 – r}{X} = \frac{r_2 – r_1}{X}$

Obliquity effect: when neglected: $X_1 = X, \ Y_1 = Y, \ mg = 0$

$h = \frac{Y}{X} (r_2 – r_1)$

Stiffness = $S = \frac{S_2 – S_1}{h}$