Physics Bootcamp

Length is a measure of the distance between two points in space. Before the French Revolution (1790), different standards of length were used in different countries, and even different localities in the same country; for instance, a Greek foot was approximately \(1.012\) times the English foot, a Roman foot was approximately \(0.97\) of an English foot, etc. Furthermore, to make matters worse, the multipliers between different units in common use were not uniform; for instance, there are three feet in a yard, and twelve inches in a foot.

In order to create a uniform system of units, Government of France, in 1792, adopted a new system of weights and measures. They chose meter as the fundamental unit of length. The name meter comes from the Latin word metrum and the Greek word metron, both meaning “measure”. One meter was defined to be \(10^{-7}\) times or one ten-millionth of the distance on the meridian from the North Pole to the equator passing through Paris. The factor was chosen to get a size close to the “human scale”. Later it was found that prototypes based on earth-based definition were about 0.2 mm too short due to the flattening of earth from its rotational motion.

In 1889, the first meeting of International Committee for Weights and Measures (CIPM) replaced the earth-based meter to the distance between two fine markings on a Platinum-Iridium rod kept at zero degrees Celsius temperature and under a standard pressure at the International Bureau of Weights and Measures (BIPM) near Paris, France. Accurate copies of the original rod were made and distributed to other standard-keeping laboratories throughout the world. These secondary standards were used to produce more accessible copies such as meter rulers for the general public.

With the advancement in optical technology, it is now possible to measure lengths more precisely than the fine markings on the standard meter rod. In 1960 a new standard for meter based on the wavelength of orange-red light emitted by the atoms of Krypton-86 in a gas discharge tube was adopted. The meter was redefined to be equal to \(1,650,763.73\) wavelengths of this light.

To reduce further uncertainty in measurements, in 1983 General Conference on Weights and Measures (CGPM) replaced the definition of meter based on Krypton-86 to the one based on the measurement of the speed of light. An exact value of \(299,792,458\, \text{m/s}\) for the speed of light is assumed which is then used to define the meter.

One meter is the length of the path travelled by light in vacuum during a time interval of \(\frac{1}{299,792,458}\) of a second.

Note that this way of defining a meter uses the unit of time (second) to define the unit of length. With this definition of the unit of length, the precision of length is now tied with the precision of measurements of time. We will see below that measurements of time have become extremely accurate and therefore it is a better base for defining other units.

Time is a measure of rate of change. To define time empirically, we think of either a natural phenomenon that repeats itself or a device that performs a periodic activity. For instance, earth’s rotation about its axis provides a measure of a day of time and one full revolution gives a measure of year. On the other hand, a pendulum is a simple device in which a bob performs a repeating motion; by choosing a particular length of the pendulum, we get a time measuring device from the time it takes to make a period of the pendulum.

The standard unit of time chosen by the international community is one second. How one second came to be a standard of time is a fascinating story unto itself. It is known that ancient civilizations used the apparent motion of celestial bodies across the sky - Sun, Moon, planets and stars - for keeping track of the passage of time and seasons.

The current division of a year into 365 days seems to have come from Egyptian calendar as far back as 3100 BCE (Before Common Era) based on the rising of the Dog Star in Canis Major, now called Sirius, next to the sun every 365 days, which coincided with the flooding of Nile. Egyptians had built obelisks (slender, tapering, four-sided monuments) as far back as 3500 BCE whose shadow was used to determine time during the day. The obelisks were like primitive sundials and had markings at the base to indicate the shadows corresponding to the shortest and longest days of the year.

Egyptians also invented water clock or clepsydra before 1500 BCE, the earliest time keeping device not dependent on the motion of celestial objects. Greeks started using water clocks around 325 BCE and built even more impressive and elaborate water clocks. The complexities were added to make the flow of water as steady as possible. Despite these efforts it was very difficult to control the water flow with high accuracy and new mechanical clocks were needed.

Little progress after the Egyptian inventions seems to have been made in time keeping until Galileo Galilei (1564 - 1642 ) who suggested using the natural period of a pendulum. Although Galileo sketched a design of a pendulum clock, he never constructed it. The first pendulum clock was built by Christian Huygens of Netherlands in 1656. It had an error of less than 1 minute a day and was the most accurate clock to date. Christian Huygens also invented the balance wheel and spring assembly in 1675, which led to the construction of more accurate clocks. The oscillations of the balance wheel, which oscillates at around 5 cycles per second, provide the time standard for the mechanical watch. In 1889 Sigmund Riefler’s clock was made that kept time with an error of less than a hundredth’s of a second a day and became a standard fixture for astronomers.

With the discovery of piezoelectricity in 1880 by Pierre and Jacques Curie, the Curie brothers, it was found that piezo-crystals of quartz vibrate at a definite frequency when one applies voltage upon them. A vibrating quartz crystal generates an oscillating current of constant frequency that can be determined quite accurately with appropriate electrical circuitry.

Despite a better performance of quartz crystals, they are no match for atomic clocks developed in 1940’s and 50’s. The possibility of atomic clock based on atomic beam magnetic resonance was first suggested in 1945 by I. Rabi of Columbia University (New York). In 1949 the National Bureau of Standards of United States (now called the National Institute of Standards and Technology or NIST) developed the first atomic clock using ammonia molecule. However, the atomic clock based on ammonia molecule was not much better than the existing quartz clocks.

In 1955 Louis Essen at the National Physical Laboratory in United Kingdom constructed the world’s first atomic clock based on the atomic transitions of cesium atoms that had an accuracy of 1 sec in 300 years. The measured time using the atomic clock was compared with the time based on the rotation of earth and found to be much more accurate and stable. Therefore, in 1967 the 13th General Conference of Weights and Measures decided to replace the definition of a second by the following.

One second is the duration of \(9,192,631,770\) periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the Cesium-133 atom.

Although there have been improvements in atomic clocks, the definition of a second today is the one adopted in 1967. Atomic clocks are getting better everyday, and in 2010 the reported uncertainty for NIST-F1 clock shown in Figure 1.5 was merely \(3\times10^{-16}\) second in one second, i.e. good up to 1 second in 3 million years - a fantastic precision!

Mass is a measure of the mechanical inertia of an object. Two objects of equal mass, regardless of their chemical content, shape or size, are accelerated equally when subjected to the same force; two objects of different masses have different accelerations when subjected to the same force, the one with larger mass accelerates less than the one with the smaller mass.

The internationally agreed unit of mass is called kilogram (kg); it is the only base quantity now that is still defined by a physical artifact. The original sample adopted by the international committee was an alloy made up of \(90\%\) platinum and \(10\%\) iridium by mass in the shape of a cylinder of height 39 mm and diameter \(39\text{ mm}\) in 1879 by George Matthey of Johnson Matthey and stored at the atmospheric pressure in a special triple-bell jar at BIPM, the International Bureau of Weights and Measures near Paris. The alloy was chosen for its non-corrosive properties and the shape was chosen to correspond to the minimum area for a given volume of a cylinder; the spherical shape would be better for minimizing the surface exposed, but since spheres roll off easily it was decided that a cylinder would serve better. The definition of a kilogram can be given as follows.

One kilogram is the mass of the prototype of the kilogram kept at the International Bureau of Weights and Measures.

Forty copies of the original prototype were made in 1882 and distributed to various countries. Copies of these secondary standards were made widely available to the tradesmen and general public. Thus all 1-kg samples are traceable to the international prototype kept at BIPM.

The unit of kilogram defined by an artefact has some intrinsic problems. For instance, the prototype may be damaged or corrode due to oxidation or other wear and tear. As a result of these problems, the prototype is said to be gaining approximately 1 micro-gram per year. Therefore, it is hard to keep the standard kilogram constant to a very high degree of precision. Presently, several new methods for defining kilogram are being investigated. A particularly attractive possibility is to define the kilogram based on the mass of a fixed number of molecules of a substance that can be made with high purity. In another method developed at National Institute of Standards and Technology (NIST), force of gravity on a standard kilogram is balanced by magnetic force between two coils in a Watt balance (Figure 1.6).