Spirit level sensitivity and tilt

(a) Spirit level

A spirit (bubble) level is a sealed glass vial, slightly curved, partly filled with alcohol or ether so that a bubble remains. The bubble always rises to the highest point. If the base is horizontal the bubble is at the centre; if the base tilts by a small angle the bubble moves along the graduated scale. It is used to check the levelness of machine tools, surface plates and slideways.

Sensitivity is the tilt of the base per division of bubble movement. It is expressed as an angle (seconds of arc per division) or as slope (mm per metre per division). For a vial of radius $R$ and division length $p$:

$$\theta = \frac{p}{R} \;\Rightarrow\; R = \frac{p}{\theta}$$

A larger $R$ gives greater sensitivity (small tilt moves the bubble far) but a longer settling time.

(b) Radius

$\theta = 10'' = 10 \times 4.848\times10^{-6} = 4.848\times10^{-5}$ rad

$$R = \frac{2\ \text{mm}}{4.848\times10^{-5}} = 41\,253\ \text{mm} \approx 41.3\ \text{m}$$

(c) Height difference

Total tilt $= 3 \times 10'' = 30'' = 1.454\times10^{-4}$ rad.

$$\Delta h = 500 \times 1.454\times10^{-4} = 0.0727\ \text{mm}$$

(a slope of 0.145 mm per metre).

Answer: $R = 41.3$ m; end-to-end height difference = 0.073 mm over 500 mm.