Measuring Pressure — Question 9

Question 9

A density calibration error survives perfect height readings

A liquid-column gauge is labeled using density $\rho_c=1000\,\mathrm{kg/m^3}$. Its actual liquid density at use is $\rho_a=960\,\mathrm{kg/m^3}$. The measured vertical separation is $0.250\,\mathrm{m}$ and $g=10\,\mathrm{m/s^2}$. Height and gravity are exact.

1. Find the indicated and actual pressure differences and the indicated error relative to the true value.

2. If the true pressure difference is held fixed while the liquid density decreases further, does the observed height increase or decrease? Derive the fractional relation for small density changes.

Question 9 - Solution

A density calibration error survives perfect height readings

(a) Derivation and calculation

$$\boxed{\Delta p_{\mathrm{indicated}}=\rho_cgh=(1000)(10)(.250)=2500\,\mathrm{Pa}}.$$$$\boxed{\Delta p_{\mathrm{actual}}=\rho_agh=(960)(10)(.250)=2400\,\mathrm{Pa}}.$$$$\boxed{\frac{\Delta p_{\mathrm{indicated}}-\Delta p_{\mathrm{actual}}}{\Delta p_{\mathrm{actual}}}=\frac{100}{2400}=4.1667\%}.$$

(b) Result and interpretation

At fixed true pressure difference and gravity,$$h=\frac{\Delta p}{\rho g},\quad\rho h=\text{constant}.$$$$\rho\,dh+h\,d\rho=0\Rightarrow\boxed{\frac{dh}{h}=-\frac{d\rho}{\rho}}.$$A density decrease increases the observed height; exact geometric measurements cannot correct a wrong density calibration.

Physical check

The small-change relation follows from a differential; for finite changes use $h_2/h_1=\rho_1/\rho_2$.