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This answer to How are barometric pressure measurements traceable over centuries to 100 parts per million accuracy? describes a floating mercury barometer, and mentions that some employed a Vernier scale for additional precision.

Since mercury is so dense (about 13.5 g/cm3) that many metals will float in it, but I'm having a hard time imagining how to make a barometer using his principle, and how a scale (Vernier or otherwise) is used to read the height of an object floating in mercury in order to determine barometric pressure.

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    $\begingroup$ This article is a good place to start. $\endgroup$
    – Fred
    Jan 22 '20 at 3:50
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    $\begingroup$ Everything you wanted to know about mercury barometers & manometers from the National Bureau of Standards, 1960. All 68 pages of it. $\endgroup$
    – Fred
    Jan 22 '20 at 4:33
  • $\begingroup$ @Fred These are really interesting end enjoyable resources; excellent×2! $\endgroup$
    – uhoh
    Jan 23 '20 at 0:44
  • $\begingroup$ @Fred I've added a bounty $\endgroup$
    – uhoh
    Jul 26 at 10:18
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A good place to start would be with Fortin barometers. They were "commonly used at meteorological stations to measure atmospheric pressure".

The advantages of this type of barometer are its portability (inverted), and that it permits the inspection of both free surfaces of mercury whose difference in level have to be measured. The major disadvantage is the cistern and the mercury it contains require frequent cleaning to maintain the instrument's accuracy.

A portable mercury barometer that was invented in 1800 by the French instrument maker Jean Nicholas Fortin (1750–1831). In other mercury barometers air pressure is calculated from the distance between the level of the mercury in the reservoir and in the tube. In the Fortin barometer the bottom of the reservoir is flexible (originally it was made from leather) and can be raised or lowered by means of a screw, allowing the surface of the mercury to be adjusted to a predetermined level. A vernier scale on the side of the barometer tube is then lowered until its base touches the top of the mercury in the reservoir and a reading is taken from the fixed scale marked on the tube. Before traveling, the base of the reservoir is raised using the adjusting screw until both reservoir and tube are filled with mercury. The Fortin barometer makes no correction for changes in temperature, which affect the volume of mercury, or for capillarity, which introduces inaccuracies. The instrument is nevertheless sufficiently accurate for most purposes.

Another type of mercury based barometer is the Kew Pattern. Unlike a Fortin barometer the mercury system is fixed in position. To compensate for the effect this has on pressure readings the scale used by Kew Pattern barometers is contracted.

The diagram below illustrates the features of a Fortin barometer.

enter image description here

Immediately prior to taking a pressure reading, the air temperature needs to be measured so the reading from the barometer can be corrected. It also needs to be measured as soon as possible, prior to reading the pressure so the barometer is not affect by the body temperature of the person taking the measurement.

The height of the mercury cistern in the barometer is then adjusted to just touch the ivory pointer.The floating vernier scale is then lowered to just touch the meniscus of the mercury in the tube.

The scale on the side of the barometer gives the main part of the pressure reading and the vernier gives the decimal fraction.

Referring back to the question, the barometer doesn't float, the vernier scale is the item that "floats" by being adjustable.

The picture below illustrates the position of the vernier scale when taking a reading from a Fortin barometer.

enter image description here

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