From Narciki

**Hydrometers** are used to measure water density, particularly for inclining tests. When placed into a water sample, i.e., in a bucket, the hydrometer will sink until it displaces its own set weight of water. A calibrated graduation on the instrument can be read at the point of water contact to determine how deep the hydrometer has sank, and thus the density of the water.

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The type of hydrometer typically used for determing water density is a *variable immersion hydrometer*. This hydrometer accepts mercury or lead pellets in the base, and has a uniform cylinder above, graduated between a specific gravity of 1.000 and 1.025. The denser the water, the higher the hydrometer will float in the sample.

To calibrate a variable immersion hydrometer, it is first allowed to float upright in fresh water (S.G. 1.000 or 1,000kg/m^{3}) with the waterlevel at the lowest graduation (i.e., just above the base). The hydrometer is then removed and weighed, and this weight recorded (M_{Initial}). The hydrometer is returned to the water, and weighted with lead or mercury until the stem is immersed to the highest graduation (i.e., just below the top of the cylinder). The hydrometer is again removed and weighed, and this weight recorded (M_{Final}). The difference between these two masses can be used to determine the stems displaced mass of freshwater between the graduations:

- M
_{Initial}- M_{Final}= M_{Stems displacement}

This displaced mass can then be used to determine the sectional volume of the hydrometer;

*where*Stem Volume = M_{stems displacement}x (1kg/1000m^{3}); and*length*is the distance between graduations;- Stem Cross-sectional Area (A
_{X}) = ΔVolume ÷ ΔLength

Our overall mass remains as M_{final}; and we recall, density = mass ÷ volume. Using this relationship, we can determine the bottom graduation;

- Volume
_{top grad.}= M_{Final}÷ 1000 kg/m^{3}(FW density) - Volume
_{btm grad.}= Volume_{top grad.}- (A_{X}x ΔLength) - Density
_{btm grad.}= M_{Final}÷ Volume_{btm grad.}

We now know the density at the bottom graduation, while the top graduation corresponds to a specific gravity of 1.000; to determine the scale values, a couple methods can be used.

Densities at various points can be manually determined mathematically, simularly to the determination of density at the bottom graduation above, and marked on the hydrometer.

Draw a straight line, MM_{0}, with an arbitrary length, L. Mark this line at lengths corresponding to the inverse of the desired graduations: i.e., for a cylinder whose top graduation is S.G. 1.000, bottom graduation is 1.025, and whom we'd like to be marked at specific gravities of 1.000, 1.005, 1.010, 1.015, 1.020 and 1.025;

- MM
_{1}= (1/1.005) * L - MM
_{2}= (1/1.010) * L - MM
_{3}= (1/1.015) * L - MM
_{4}= (1/1.020) * L - MM
_{5}= (1/1.025) * L

Taking an arbitrary point, O, to either side of MM_{0}, we can draw rays from O to each of M_{1}, M_{2}, M_{3}, etc. (OM_{0} - OM_{5}). Draw a line parrellel to MM_{0}, NN_{0}, where the hydrometers length between top and bottom graduations is equal to the distance from the intersections of (NN_{0} & OM_{5}) and (NN_{0} & OM_{0}). The intersections of the rays with this line, NN_{0}, form the corresponding density scale for the hydrometer.