Bubble sensitivity from staff readings at a known distance At a sight distance of 80 m, the staff reading with the level bubble centered is 1.31 m. After displacing the bubble by 5 divisions from center, the staff reading becomes 1.39 m. Compute the angular value (in arc-seconds) corresponding to one division of the bubble.

Introduction / Context:
The sensitivity of a level tube (bubble) is defined as the angular tilt of the line of sight per division of bubble movement. By comparing the change in staff reading at a known sight distance when the bubble is intentionally displaced a known number of divisions, we can compute arc-seconds per division. This is a standard field calibration task for levelling instruments.

Given Data / Assumptions:

• Sight distance to staff, D = 80 m.
• Reading with bubble centered = 1.31 m.
• Reading with bubble displaced 5 divisions = 1.39 m.
• Change in reading, Δs = 1.39 − 1.31 = 0.08 m.
• Small-angle assumption: tan φ ≈ φ in radians when φ is small.

Concept / Approach:

Tilting the telescope by a small angle φ shifts the line of sight vertically. Over a horizontal distance D, the change in intercept on the staff is approximately Δs ≈ D * tan φ ≈ D * φ (radians). If 5 divisions correspond to total tilt φ_total, then angle per division = φ_total / 5. Convert radians to arc-seconds using 1 rad ≈ 206265 seconds.

Step-by-Step Solution:

Verification / Alternative check:

Using exact tan instead of small-angle: tan φ_total = 0.08 / 80 = 0.001 → φ_total ≈ 0.001 rad to within 0.0000005, giving essentially the same result.

Why Other Options Are Wrong:

28.8 s and 25.05 s underestimate the computed sensitivity.

14.52 s is far too small for the given displacement and distance.

“None” is invalid because a consistent value (≈41.25 s) results from the data.

Common Pitfalls:

Using degrees instead of radians in the conversion; forgetting to divide by the number of divisions; not keeping the staff exactly vertical for the readings.

Final Answer:

41.25 seconds