Each arm of a Porter governor is 200 m long and is pivoted on the axis of rotation of governor. The radii of rotation of the balls at the minimum and maximum speed are 120 mm and 160 mm respectively. The mass of the sleeve is 25 kg and that of each ball is 5 kg . What will be the approximate maximum and minimum speed of governor? Assuming negligible friction in the sleeve? [assume, \(g\left(\frac{30}{\pi}\right)^2=900\) ; g = acceleration due to gravity.
 

Concept:

The speed of a Porter governor is determined by balancing the centrifugal force with the gravitational force acting on the balls and the sleeve. The formula for the speed N in rpm is derived from these forces and the geometry of the governor.

The height h of the governor is given by:

\(h = \sqrt{l^2 - r^2}\)

Where l is the length of the arm and r is the radius of rotation.

The speed N is calculated using:

\(N = \frac{30}{\pi} \sqrt{\frac{g (m + M/2)}{m h}} \)

Where g is the acceleration due to gravity, m is the mass of each ball, and M is the mass of the sleeve.

Given the relation \(g \left( \frac{30}{\pi} \right)^2 = 90 \), the formula simplifies to:

\(N = \sqrt{\frac{90 (m + M/2)}{m h}}\)

Calculation:

Given:

Length of each arm, \(l = 200 \, \text{mm} = 0.2 \, \text{m}\)

Radius at minimum speed, \( r_{\text{min}} = 120 \, \text{mm} = 0.12 \, \text{m} \)

Radius at maximum speed, \(r_{\text{max}} = 160 \, \text{mm} = 0.16 \, \text{m}\)

Mass of the sleeve, M = \(25 \, \text{kg}\)

Mass of each ball, m = 5 kg

Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\)

Step 1: Calculate the height h at minimum and maximum speeds

For minimum speed ( \(r = 0.12 \, \text{m}\)):

\(h_{\text{min}} = \sqrt{0.2^2 - 0.12^2} = \sqrt{0.04 - 0.0144} = \sqrt{0.0256} = 0.16 \, \text{m}\)

For maximum speed (\(r = 0.16 \, \text{m}\)):

\(h_{\text{max}} = \sqrt{0.2^2 - 0.16^2} = \sqrt{0.04 - 0.0256} = \sqrt{0.0144} = 0.12 \, \text{m}\)

Step 2: Calculate the minimum and maximum speeds

Minimum speed ( h = 0.16 m):

\(N_{\text{min}} = \sqrt{\frac{90 (5 + 25/2)}{5 \times 0.16}} = \sqrt{\frac{90 \times 17.5}{0.8}} = \sqrt{1968.75} \approx 141 \, \text{rpm}\)

Maximum speed (h = 0.12 m):

\(N_{\text{max}} = \sqrt{\frac{90 (5 + 25/2)}{5 \times 0.12}} = \sqrt{\frac{90 \times 17.5}{0.6}} = \sqrt{2625} \approx 183 \, \text{rpm}\)

The approximate maximum and minimum speeds of the governor are:

Maximum speed = 183 rpm

Minimum speed = 141 rpm