Levelling instruments – bubble-tube sensitivity per division For a spirit-level bubble tube: if the length of one graduation (division) on the tube is x and R is the radius of curvature of the internal surface, what is the angular value φ (in seconds) corresponding to one division of bubble movement?

Introduction / Context:
The sensitivity of a level is the angle through which the line of collimation tilts when the bubble moves by one division on the bubble tube. Designers and field engineers use this relation to evaluate instrument precision and to compute staff intercept versus bubble displacement during adjustments. This question asks for the angular value per division in seconds as a function of the division length and the tube’s radius of curvature.

Given Data / Assumptions:

• Length of one division on the bubble tube arc = x (linear measure along the arc).
• Radius of curvature of the internal surface = R.
• Small-angle approximation holds: arc length s ≈ R * θ for θ in radians.
• We want φ in arc-seconds for one-division movement.

Concept / Approach:

When the bubble moves by one division, the arc length swept equals x. For small angles, θ (in radians) = s / R = x / R. To convert to seconds, multiply radians by 206265 (since 1 radian = 206265 arc-seconds). Therefore, φ = 206265 * (x / R). This directly links physical tube geometry to angular sensitivity per division.

Step-by-Step Solution:

Verification / Alternative check:

Instrument adjustment notes use the same formula to relate bubble movement, target displacement on the staff, and the length of sight, confirming the dependence on x/R and the conversion factor 206265 for seconds.

Why Other Options Are Wrong:

Option B: Gives φ in degrees, not seconds.

Option C: Inverts the correct proportionality; sensitivity increases with larger x and decreases with larger R.

Option D: Has wrong dimensional dependence (product x*R) and lacks the seconds-per-radian factor placement.

Common Pitfalls:

Confusing division length x with total tube length; forgetting to convert from radians to seconds; assuming linear displacement along the chord instead of the arc (for small angles, arc and chord are nearly equal, which justifies the approximation).

Final Answer:

φ = (206265 * x) / R