To calculate the minimum speed, maximum speed, and range of the governor, we need to use the concept of the Porter governor and the given parameters.
5.1 Minimum Speed of the Governor:
The minimum speed of the governor corresponds to the position when the ball path is at its maximum, which is 120 mm.
Let's denote the length of the equal arms of the governor as L (150 mm) and the initial position of the ball path as x1 (100 mm).
From the geometry of the Porter governor, we can use the concept of similar triangles to determine the minimum speed. The ratio of the initial position of the ball path to the maximum position is equal to the ratio of the governor speed at the initial position to the minimum speed. Mathematically, we can express this as:
x1 / x_max = v1 / v_min
Substituting the given values, we have:
100 mm / 120 mm = v1 / v_min
Simplifying and solving for v_min:
v_min = (120 mm * v1) / 100 mm
v_min = (6/5) * v1
Therefore, the minimum speed of the governor is (6/5) times the speed at the initial position.
5.2 Maximum Speed of the Governor:
The maximum speed of the governor corresponds to the position when the ball path is at its minimum, which is 100 mm.
Using the same concept of similar triangles as above, we can set up the following equation:
x_min / x_max = v_max / v1
Substituting the given values:
100 mm / 120 mm = v_max / v1
Simplifying and solving for v_max:
v_max = (100 mm * v1) / 120 mm
v_max = (5/6) * v1
Therefore, the maximum speed of the governor is (5/6) times the speed at the initial position.
5.3 Range of the Speed:
The range of the speed is the difference between the maximum and minimum speeds of the governor.
Range = v_max - v_min
Range = (5/6) * v1 - (6/5) * v1
To simplify this expression, we need to find a common denominator:
Range = (5v1 - 6v1) / (6 * 5)
Range = -v1 / 30
Therefore, the range of the speed is -v1/30.
Please note that the speed values are relative to the initial position (v1).