The reading taken on a staff held 100 m from the instrument with the bubble central is 1.786. The bubble is then moved 3 divisions out of centre and the staff reading is observed to be 1.817 m. Find the angular value of one division of bubble tube and the radius of curvature of the bubble tube, the length of one division being 2 mm.

Bubble Tube Radius of Curvature Problem and Solution

Problem Statement

Solution

Calculate the staff intercept (difference in readings) due to the bubble movement:

\begin{align} S &= \text{Staff reading with bubble out of center} – \text{Staff reading with bubble central} \\ &= 1.817 – 1.786 \\ &= 0.031 \text{ m} \end{align}

Note: A visual diagram showing the staff, instrument, and bubble tube would be displayed here.

Calculate the radius of curvature of the bubble tube using the formula:

\begin{align} R &= \frac{n \times d \times D}{1000 \times S} \\ \end{align}

Where:

\(R\) = radius of curvature in meters
\(n\) = number of divisions bubble is out of center = 3
\(d\) = length of one division = 2 mm
\(D\) = distance to staff = 100 m
\(S\) = staff intercept = 0.031 m

Substitute the values to find the radius of curvature:

\begin{align} R &= \frac{n \times d \times D}{1000 \times S} \\ &= \frac{3 \times 2 \times 100}{1000 \times 0.031} \\ &= \frac{600}{31} \\ &= 19.35 \text{ m} \end{align}

Calculate the angular value of one division using the formula:

\begin{align} \alpha’ &= \frac{S}{n \times D} \times 206,265 \text{ seconds} \\ \end{align}

Where:

\(\alpha’\) = angular value of one division in seconds
206,265 = number of seconds in one radian

Substitute the values to find the angular value:

\begin{align} \alpha’ &= \frac{S}{n \times D} \times 206,265 \text{ seconds} \\ &= \frac{0.031}{3 \times 100} \times 206,265 \\ &= \frac{0.031 \times 206,265}{300} \\ &= \frac{6,394.2}{300} \\ &= 21.31 \text{ seconds} \end{align}

Therefore, the radius of curvature of the bubble tube = 19.35 m

The angular value of one division = 21.31 seconds

Detailed Explanation

This problem demonstrates how to determine two important properties of a surveying instrument’s bubble tube:

The radius of curvature (R): This is a physical property of the bubble tube that determines its sensitivity. The larger the radius of curvature, the more sensitive the bubble tube is to tilting.
The angular value of one division: This represents how much angular deviation occurs when the bubble moves by one division mark on the tube. This is an important calibration parameter for surveying instruments.

The relationship between these properties is governed by the following principles:

When the bubble moves away from center, the line of sight tilts by a corresponding angle.
This tilt creates a measurable difference in staff readings (staff intercept).
The staff intercept (S), the number of divisions (n), and the distance to the staff (D) together allow us to calculate both the radius of curvature and the angular value.

The radius of curvature formula \(R = \frac{n \times d \times D}{1000 \times S}\) comes from geometric relationships between the curvature of the tube and the resulting angular displacement when the bubble moves.

The angular value formula \(\alpha’ = \frac{S}{n \times D} \times 206,265\) converts the observed linear measurement difference to an angular value, using the standard conversion factor of 206,265 seconds per radian.

In practical surveying, knowing these values helps surveyors understand the precision limitations of their instruments and make appropriate adjustments to their measurements.

Problem Statement

Solution

Detailed Explanation

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