Measurement

A measurement (from Latin metiri "to measure" + -mentum, the result of an action) is a relation between a quantity of one thing to a standard quantity of another thing (a "metric" or "standard"). That is, the way we measure a quantity of any kind is to compare it with another quantity of the same kind which we employ as a unit of measurement. Thus we measure a mass by determining how many times greater it is than some standard mass; we measure a length by the number of multiples it contains of a standard length; and an interval of time by the multiples of some standard time interval. [1]

Every measurement consists of two parts: a number and unit. A unit specifies the character of a physical quantity and represents the measure of the reference chosen for comparison. In contrast, a standard is an artifact or reproducible phenomenon which is the physical embodiment of the unit. [2]

The choice of unit for each measurable quantity defines a measurement system. There are two main measurements systems in use today: the metric system and the Imperial system. The modern metric system is known as the International System of units, abbreviated SI in all languages. SI is administered by the International Bureau of Weights and Measure (BIPM), whose headquarters is located in Sevres, France, across the Seine River from Paris. [2]

The science of measurement is called metrology.

Contents

1   Function

Measurement produces data.

2   Base units

There are two systems of measurement in use today: the Imperial system and the Metric system. Both systems are defined around three fundamental units of measurements: mass, length, and time.

2.1   Mass

static/images/kilogram_prototype.png

International prototype kilogram, protected by triple bell jars. In the foreground are special calibers used in handling the kilogram.

Mass is a property of physical bodies which causes them to have inertia. [1] The standard of mass is a cubic meter of water, and the standard unit is the gram which is equal to the mass of a cubic centimeter of water at the temperature of its greatest density (4 degrees Celsius, thought to be 0 degrees). [2]

The unit for mass is that of a platinum-iridium weight kept at the Bureau of Weights and Measures and defined to have a mass of 1 kilogram. [1] The standard is preserved at the International Bureau and is stored under triple bell jars in a vault 8 meters below the ground.

The international prototype is made from the same platinum-iridium alloy as the meter bar of 1889. It has the shape of a cylinder with a height equal to its diameter, 3.9 cm, with slightly rounded edges. For a cylinder, these dimensions present the smallest surface to volume ratio. [2]

Until 2014, the kilogram was the only unit defined in terms of an arbitrary artifact instead of a natural phenomenon. [2] The mass of 1 gram was defined to be that of a cubic centimeter of water at 4 degrees Centigrade (the temperature at which water has its greatest density). Hence the kilogram is very nearly the mass of 1,000 cubic centimeters of water. [1]

In 2014, at the quadrennial General Conference on Weights and Measures, the scientific community resolved to redefine the kilogram based on Planck's constant. [3]

There is an inevitable loss of material whenever it is used. A reduction of only 0.4nm, one layer of atoms, over the base area of 12cm^2 would be sufficient to cause a loss of 0.01mg. Therefore, the international prototype has only been used on three occasions. [2] However, the standard kilogram is still losing mass. [3] It now is ever-so-slightly lighter than the once-identical "witness" cylinders stored in labs around the world. Scientists don't know whether the BIPM prototype is losing mass, perhaps because of loss of impurities in the metals, or if the witnesses are gaining mass by accumulating contaminants. [3] Several years ago, NIST had to reissue certificates for its kilograms because they were 45 micrograms off the French prototype - about the weight of an eyelash. [3]

As a matter of quick comparison, here are some common masses.

Sun 10^33kg
Earth 10^28kg
Moon 10^26
Mountain 10^17kg
Bridge 10^11kg
Sequoia tree 10^9kg
Whale 10^8kg
Elephant 10^7kg
Human 100kg
Dog 10kg
Bottle of wine 1kg
Mouse 100g
Peanut 1g
Bee 0.1g
Ant 0.001g
Human cell 10^-9g
Virus 10^-15g
Atom 10^-23g

2.2   Length

static/images/meter_bar.png

International prototype meter bar of 1889.

The X-shaped Tresca cross-section is inscribable in a square 20 mm on a side. The meter was defined as the distance between two lines ruled on a polished region of the bridge at each end of the bar when the bar was a 0 degrees Celsius, standard atmmospheric pressure, and support by two cylinders placed symmetrically 571 mm apart.

Length is a measurement of the distance between two points. The standard for length is the circumference of the Earth, and the unit is the meter.

The standard of length is the meter bar, a bar composed a platinum-iridium alloy with a a special X-shaped cross section designed by the French physicist Henri Tresca was adopted for maximum rigidity. [2] Near each end of this bar there are engraved transversely three fine parallel lines. The distance from the middle line at one end of the bar to the middle line at the other end when the is at the temperature of melting ice (0 degrees Celsius) is defined to be 1 meter. [1] (Engraved lines are used instead of measuring the ends of the bar because older prototypes were observed to lose length.) The standard is preserved at the Bureau of Weights and Measures near Paris. [1]

Exact copies of this bar made by direct comparison have been constructed and distributed to other government of the various countries of the world. In the United States, this duplicate is kept at the Bureau of Standards in Washington. [1]

1 meter = 1000 mm 1mm = 1000 microns (= micrometer)

Common lengths:

Width of football field 70m
Height of average US man 1.75m
Diameter of golf ball 40mm
Diameter of pinhead 1mm
Diameter of human hair 100 microns
Thickness of paper 100 microns
Thickness of coat of paint 100 microns
Length of a dust particle 100 microns
Thickness of credit card 750 microns

2.3   Time

The standard for time is the mean time for the Sun to appear in the same meridian (the mean solar day), and the standard unit is the second.

See time.

3   Derived units

The units of mass, length, and time are said to be fundamental. By means of these we can also measure a large number of other secondary quantities which are accordingly said to be derived quantities. For example, "area" is a derived quantity depending upon length. Less obvious derived units are speed and velocity.

Quantities like velocity which have both magnitudes and directions are called vector quantities.

3.1   Position

The position of a point particle is defined with respect to an arbitrary fixed reference point, \(O\), in space, usually accompanied by a coordinate system, with the reference point located at the origin of the coordinate system. It is defined as the vector \(r\) from \(O\) to the particle.

3.2   Velocity

The velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time:

3.3   Acceleration

The acceleration, or rate of change of velocity, is the derivative of the velocity with respect to time (the second derivative of the position with respect to time).

3.4   Force

Force is any interaction that, when unopposed, will change the motion of an object.

\begin{equation*} F = ma \end{equation*}

3.5   Work

Work is said to be done when a force displaces a physical body. [1] A unit of work is that which is done when unit of force cause its point of application to move some distance in the direction in which the force acts. [1]

The work \(W\) done by a constance force of magnitude \(F\) on a point that moves a displacement \(s\) in a straight line in the direction of the force is the product \(W = Fs\).

3.6   Power

Power is the time rate of doing work. In the Metric system, when work is performed at the rate of 1 joule per second, the power is defined to be 1 watt. In the English system, the unit of work is the horsepower. The unit was defined by James Watt, who attempted to determine the rate at which a draft horse could do work so that he could use this for rating the power of his steam engines. The result he achieved was that 1 horsepower is a rate of doing work 33,000 foot-pounds per minute. [1]

3.7   Angular momentum

Conservation of angular momentum: \(L = Iw\). If the moment of inertia (\(I\)) decreases, the angular speed increases to that \(L\) remains constant.

4   Properties

The accuracy of a measurement system is the degree of closeness of measurements of a quantity to the quantity's true value.

The precision of a measurement system (related to reproducibility and repeatability) is the degree to which repeated under unchained conditions yield the same results.

5   History

The abbreviation for pound (lb) comes from an ancient Roman unit of measurement called libra pondo (a pound by weight). "Libra" meant balance or scales, and is also why the symbol for the British pound is a stylized "L" with a line through it.(Also why the astrological sign for Libra is a pair of scales.)


Prior to the metric system, there existed in France a multiplicity of names and methods of subdivision for the units of weights and measures. [2] Although many proposals were made to bring order to the disparate system of weights and measures, they were opposed by the guilds and nobles who benefited from the confusion. [2]

The advocates of reform sought to guarantee the permanence of the units by basing them on properties derived from nature. The occasion that made reform possible was the `French Revolution`_ of 1789. [2] In 1790, the National Assembly empowered a committee of the French Academy of Sciences to devise a new system of units. [2]

The proposal to define the meter as one ten millionth of the quadrant of the earth along a meridian passing through Paris was accepted by the National Assembly on March 26, 1791, and was enacted into law by King Louis XVI four days later. [2] The fieldwork necessary to established the length of the meter began in June 1792. [2] THe survey to determine the meter was completed in 1798.

Among the many changes in society during the revolutionary period was the adoption of a new calendar consisting of twelve months of thirty days each, concluded by a five or six day holiday. Each month was divided into three ten-day weeks. Furthermore, the day itself was divided into ten hours. Each hours was divided into 100 minutes, each minute into 110 seconds. [2] The calendar reform went into effect for twelve years, but the new method of keeping time was never truly adopted. [2]

On April 7, 1795, the revolutionary government issued a decree formalizing the adoption of the metric units and the terms that are in use today. Permanent standards made from platinum were constructed to serve as the legal representations of the metric units. On June 22, 1799 they were deposited int the Archives of the Republic. They became official by an act of December 10, 1799. [2]

The Netherlands adopted the metric system in 1816, the first of several countries to follow the French lead.

The Metre Convention of 1875 officially established the meter as an international measurement unit. It also established an international organization, the Bureau international des poids et mesures (BIPM) to conserve the prototypes and to carry out regular comparisons with national standards.

From its found in 1901 until 1959, the National Bureau of Standards used the relations 1 yard = 3600/3937 meters and 1 pound-mass = 0.4535924277 kg. On July 1, 1959 the definitions were fixed by international agreement to be 1 yard = 0.9144 m and 1 pound-mass = 0.45359237 kg exactly. The metric system had thus become the ultimate basis for all legal units of measure in the civilized world. [2]


An alloy of 90% platinum_ and 10% iridium_ was selected because of its inalterability (the material is not subject to oxidation_), hardness (the alloy is significantly harder than pure platinum), density (almost twice that of lead), luster, high coefficient of elasticity, and low coefficient of expansion. [2]

5.1   Meter

The original definition of the meter was given by the English cleric, John Wilkins, in 1667. The final version of the meter was defined in a slightly different way, but starting from Wilkins' proposal.

Wilkins proposed in an essay a decimal-based unit of length, on a pendulum with a two-second period. For a pendulum with a same angle at the surface of the earth, the period of oscillation is \(T = 2\pi\sqrt{\frac{L}{g}}\). If we set the period \(T\) to 2 second, then:

\begin{equation*} 2 = 2\pi\sqrt{\frac{L}{g}} \implies L = \frac{g}{\pi^2} \end{equation*}

Now, if we call \(L\) "1 meter", we have \(\pi^2 = g\).

However, French astronomer Jean Richer discovered that the length of a seconds pendulum varies from place to place: Richer had measured a 0.3% difference in length between Cayenne and Paris. (This is because the period of a pendulum depends on the local gravitational field strength.)

After the `French Revolution`_, the French Academy of Sciences decided that the new measure should be equal to one ten-millionth of the distance from the North Pole to the Equator, measured along the meridian passing through Paris. (This result was not achieved exactly so that by later measurements the Earth's quadrant is found to be 10,000,856 meters.) Still, we can say with considerable exactness that the circumference of the Earth is 40,000 kilometers. [1]

5.2   Imperial system

The Imperial system (= English system) is ...

[1]

The unit of length in the English system of measurement is the distance between the center of two transverse lines in two gold plugs in a bronze bar deposited at the Office of the Exchequer, when the bar is at a tempature of 62 degrees Fahrenheit. The distance is the standard "yard".

The unit of mass in the English system is that of a certain piece of platinum marked 'P.S., 1844, 1lb.,' which is deposited at the same place as the standard yard. This is known as the standard "pound avoirdupois".

The unit of time in the English system is the same as in the Metric.


'lbs' comes form the latin word 'libra' which means pound.


Here's a good mnemonic to convert Miles to Kilometers using the Fibonacci numbers (0,1,1,2,3,5,8,13...): 3 mi = 5 km, 5 mi = 8 km, 8 mi = 13 km. Also the ratio between 1 mi and 1 km is very close to the Golden Ratio, 1.61

6   Acquisition

A gauge is a device used to make measurement or in order to display certain information. For example, a speedometer, odometer, pedometer.

7   References

[1](1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) Technocracy Inc. 1945. Technocracy Study Course. Lesson 2. https://ia600204.us.archive.org/10/items/TechnocracyStudyCourseUnabridged/TechnocracyStudyCourse-NewOpened.pdf
[2](1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18) Robert Nelson. Dec 1981. Foundations of the international system of units. http://www.physics.umd.edu/deptinfo/facilities/lecdem/services/refs/refsa/Nelson-FoundationsSI.pdf
[3](1, 2, 3, 4) Sarah Kaplan. July 5, 2017. Scientists are about to change what a kilogram is. That's massive.. https://www.washingtonpost.com/news/speaking-of-science/wp/2017/07/05/scientists-are-about-to-change-what-a-kilogram-is-thats-massive/