Measuring Pressure — Question 6

Question 6

Tilting a barometer changes length but not vertical head

An ideal barometer with negligible headspace pressure requires a vertical liquid head of $0.750\,\mathrm{m}$ in the current atmosphere. Its straight tube is tilted to $60^\circ$ above horizontal. The reservoir is large enough that its level does not change appreciably.

1. Find the liquid-column length along the tilted tube. What fraction of the true pressure would be reported if that length were mistaken for a vertical height?

2. The usable tube length above the reservoir is $0.90\,\mathrm{m}$. Find the minimum angle above horizontal at which this barometer can retain a headspace under the ideal assumptions.

Question 6 - Solution

Tilting a barometer changes length but not vertical head

(a) Derivation and calculation

$$h=\ell\sin\theta\Rightarrow\boxed{\ell=.750/\sin60^\circ=.866025\,\mathrm m}.$$$$\boxed{\frac{p_{\mathrm{indicated}}}{p_{\mathrm{true}}}=\frac{\rho g\ell}{\rho gh}=\frac1{\sin60^\circ}=1.15470}.$$Mistaking length for vertical head overstates pressure by $15.47\%$.

(b) Result and interpretation

For usable length $L=.900\,\mathrm m$,$$L\sin\theta\ge h\Rightarrow\theta\ge\sin^{-1}(.750/.900).$$$$\boxed{\theta_{\mathrm{limiting}}=56.4427^\circ}.$$A finite headspace requires $\theta>56.4427^\circ$; equality places the meniscus at the end of the usable tube.

Physical check

At $90^\circ$, the liquid length equals the vertical head, as required.