Problem Statement
If the bubble tube of a level has a sensitiveness of 30′ per 2 mm division, find the error in staff reading on a vertically held staff at a distance of 150 m, caused by the bubble 2.5 divisions out of centre.
Solution
Define the variables and given information:
- Sensitiveness of bubble tube: 30′ (30 minutes) per 2 mm division
- Number of divisions bubble is out of center: \(n = 2.5\) divisions
- Distance to staff: \(D = 150\) m
- We need to find: Error in staff reading \(S\)
Use the relationship between the angular value and linear error:
Where:
- \(S\) = staff intercept (error in staff reading)
- \(\alpha’\) = angular value of one division in seconds
- \(n\) = number of divisions bubble is out of center
- \(D\) = distance to staff in meters
- 206,265 = number of seconds in one radian
Convert the angular value from minutes to seconds:
Substitute the values into the equation:
Therefore, the error in staff reading = 0.055 meters or 5.5 centimeters
Detailed Explanation
This problem deals with the error introduced in surveying measurements due to the bubble in a level instrument being off-center.
The key concepts involved are:
- Sensitiveness of a bubble tube: This is expressed as the angular deviation (30′ or 30 minutes of arc) that corresponds to a movement of the bubble by one division (2 mm).
- Angular error: When the bubble is 2.5 divisions out of center, it means the line of sight has an angular error of \(2.5 \times 30′ = 75’\) or \(75 \times 60 = 4500\) seconds.
- Linear error conversion: The formula \(S = \frac{\alpha’ \times n \times D}{206,265}\) converts an angular error to a linear measurement at a given distance. The constant 206,265 represents the number of seconds in one radian.
- Practical significance: The result shows that even a small angular error of 75 minutes can result in a significant measurement error of 5.5 cm at a distance of 150 meters, highlighting the importance of proper leveling in surveying operations.
In surveying practice, this underscores the importance of careful adjustment of the bubble tube before taking readings, especially at longer distances where small angular errors can magnify into substantial linear errors.