In a Hartnell governor, the mass of each ball is given as 4 kg. The maximum and minimum centrifugal forces on the basis are 1800 N and 100 N at radii 25 cm and 20 cm respectively. The lengths of the vertical and horizontal arms of the bell-crank levers are the same. We need to determine the spring stiffness in N/cm.

To solve this problem, we can use the principle of equilibrium. The centrifugal force acting on the balls is balanced by the force exerted by the spring. At maximum speed, the centrifugal force is at its maximum value of 1800 N and the radius is 25 cm. At minimum speed, the centrifugal force is at its minimum value of 100 N and the radius is 20 cm.

Let's assume the spring stiffness as 'k' N/cm.

- Determine the maximum and minimum displacements of the spring:
We know that the centrifugal force acting on the balls is given by F = mω²r, where m is the mass of each ball, ω is the angular velocity, and r is the radius. Rearranging this equation, we can solve for the angular velocity ω:
ω = √(F / m * r)
At maximum speed:
ω_max = √(1800 N / 4 kg * 0.25 m) = 30 rad/s
At minimum speed:
ω_min = √(100 N / 4 kg * 0.20 m) = 5 rad/s

- Determine the maximum and minimum displacements of the spring:
At maximum speed:
F_max = k * x_max (equation 1)
At minimum speed:
F_min = k * x_min (equation 2)
Where x_max and x_min are the maximum and minimum displacements of the spring, respectively.

- Determine the maximum and minimum displacements of the spring:
Dividing equation 1 by equation 2, we get:
F_max / F_min = (k * x_max) / (k * x_min)
1800 N / 100 N = x_max / x_min
18 = x_max / x_min

- Determine the maximum and minimum displacements of the spring:
We are given that the lengths of the vertical and horizontal arms of the bell-crank levers are the same. Therefore, the maximum and minimum displacements of the spring are proportional to the radii:
x_max / x_min = r_max / r_min = 25 cm / 20 cm = 1.25

- Determine the maximum and minimum displacements of the spring:
Substituting this value into the previous equation:
18 = 1.25
x_max / x_min = 1.25
x_max = 1.25 * x_min

- Determine the maximum and minimum displacements of the spring:
Substituting this relationship into equation 2:
F_min = k * x_min
100 N = k * x_min

- Determine the maximum and minimum displacements of the spring:
Substituting this relationship into equation 1:
1800 N = k * 1.25 * x_min

- Determine the spring stiffness:
Dividing equation 1 by equation 2, we get:
1800 N / 100 N = k * 1.25 * x_min / (k * x_min)
18 = 1.25