Chapter 4

The International System of Units

The International System (SI) is a units system created in 1960 in the 11th. General Conference on Weights and Measures [3]. Its purpose was to establish a standard unit system that would be adopted by all countries such that science could become truly universal. An experiment performed in any country could be easily reproduced in any other country.

4.1 The SI base units

The International System was built from 7 base units [4]:

Physical quantity SI unit Name Symbol Name Symbol length l, x, y, z, r,... meter m mass m, M kilogram kg time, duration t, t second s electric current intensity I,i ampere A temperature T kelvin K luminous intensity IV candela cd quantity n mole mol

Table 4.1: Base SI units

4.2 The SI derived units

All physical quantities have units which are either one base unit or a combination of base units. If a unit results from combining more than one base unit it is call a derived unit. Some examples:

. m2 - the “square meter” is the SI unit of area. It is considered a derived unit because it results from combining the same base unit more than once.

. Hz - the “hertz” is the SI unit of frequency. It also derives from a single SI unit and is considered derived because it is not equal to the base unit. This unit pays homage to Heinrich Rudolf Hertz (1857 1894). 1 Hz = (4.1) s . N - the “newton” is the SI unit of force. It is derived from 3 SI base units: kg m N = · (4.2) s2 It pays homage to Sir Isaac Newton (1642 1727). . V - the “volt” is the SI unit of electric potential. It is derived from 4 SI base units:

19 4.3. SPECIAL SI DERIVED UNITS CHAPTER 4. THE INTERNATIONAL SYSTEM OF UNITS

kg m2 V = · (4.3) A s3 · It pays homage to Alessandro Giuseppe Antonio Anastasio Volta (1745 1827). . T- the “tesla” is the SI unit of magnetic flux density (sometimes loosely called magnetic field). It is derived from 3 SI base units: kg T = (4.4) A s2 · It pays homage to Nikola Tesla (1856 1943). . ⌦ - the “ohm” is the SI unit of electric resistance. It is derived from 4 SI base units: kg m2 ⌦ = · (4.5) A2 s3 · It pays homage to Georg Simon Ohm (1789 1854). . lm - the “lumen” is the SI unit of luminous flux. It is derived from 2 SI base units: lm = cd sr (4.6) · 4.3 Special SI derived units

In the International System the physical quantities angle and solid angle are dimensionless (just plain numbers) however they have units which are considered to be derived.

4.3.1 Angle or planar angle In order to understand what is an angle let us consider semi-straight line with its origin on point O. If one rotates the semi-straight line about O along a plane P , the angle will be a measure of the amplitude of the rotation. In the SI, this measure is given by the length of an arc. The arc is defined as such: . consider a unit radius circle centered in O. . the arc is the portion of the circle that is swiped by the semi-straight line as it rotates about O along the plane P In other words, if one wishes to measure an angle, one will draw a unit circle (e.g. 1 m) centered in O. The angle will be given by measuring the length (in the same units as the unit radius) of the arc (circle section) drawn in red in Figure 4.1. The definition is based on the following formula: s ✓ = (4.7) r where s is the length of the arc belonging to the r radius circle.

1

O

Figure 4.1: Planar angle

20 CHAPTER 4. THE INTERNATIONAL SYSTEM OF UNITS 4.3. SPECIAL SI DERIVED UNITS

The SI unit of an angle is the radian (rad). Since the perimeter P of a circle of radius r is given by:

P =2⇡r

it becomes apparent that if r =1m, then P =2⇡ m. In other words a full angle s = P will have 2⇡ rad. Some examples:

. right angle - represents 1/4 of a full angle hence it is equal to ⇡/2 rad.

. the inner angle of an equilateral triangle - represents 1/6 of a full angle hence it is equal to ⇡/3 rad.

4.3.2 Solid angle The solid angle can be regarded as a generalization to three dimensions of the planar angle. Let us consider a semi-straight line starting from origin point O such as that described for the planar angle (see 4.3.1). However, this time instead of rotating along a plane one rotates the semi-straight line freely about point O in 3D. The only requirement is that the starting point must match exactly the ending point such that a closed loop is defined. The solid angle will be given by the area of the surface obtained by intersecting the semi-straight line with a unit radius sphere centered in O. In other words, if one wants to measure a solid angle, one draws a unit radius sphere centered in O. The solid angle is equal to the area of the red section in Figure 4.2. The definition is based in the following formula:

A ⌦ = (4.8) r2 where A is the area of the radius r spherical section.

1 O

Figure 4.2: Solid angle

The SI unit of solid angle is the steradian (sr). Since the total surface area of a unit sphere is equal to 4⇡ one can conclude that a full solid angle has 4⇡ sr. Some examples:

. The solid angle defined by the interior of a cube vertex - represents 1/8 of a full solid angle therefore it is equal to ⇡/2 sr.

. The solid angle defined by the interior of a regular tetrahedron vertex (equal sides triangular pyramid) - is equal to 3 arccos 1 ⇡ 7⇡/40 sr. 3 ' . The solid angle defined by the interior of a platonic solid vertex - is equal to q✓ (q 2) ⇡ sr where q is the number of faces that form the solid angle and ✓ is the dihedral angle.

21 4.4. THE UNIT PREFIXES CHAPTER 4. THE INTERNATIONAL SYSTEM OF UNITS

4.4 The unit prefixes

In nature a physical quantity my vary many orders of magnitude. Having in mind effort economization, a set of unit prefixes was accorded so as to represent certain unit multiples and submultiples:

Multiples Name Symbol Factor yotta Y 1024 zetta Z 1021 exa E 1018 peta P 1015 tera T 1012 giga G 109 mega M 106 kilo k 103 hecto h 102 deca da 101

Table 4.2: Prefixes of unit multiples

Submultiples Name Symbol Factor 1 deci d 10 2 centi c 10 3 milli m 10 6 micro µ 10 9 nano n 10 12 pico p 10 15 femto f 10 18 atto a 10 21 zepto z 10 24 yocto y 10

Table 4.3: Prefixes of unit submultiples

4.5 Calculations with units and their advantages

In this book all calculations involving physical quantities will be executed while explicitly displaying their units at all steps. The intent is to emphasize the utility of such an exercise and hopefully to help create a habit which will endure well beyond this book. One could ask: what can one possibly gain by going through the extra work of carrying around all the units in all calculations? In order to understand such benefits let us consider the following example: An automobile is moving on a freeway with a constant velocity of 36 km/h. What distance will it have covered in 20 minutes? Some students may be wondering: what was that velocity formula? One can find the answer by simply reading the given data units. The given velocity units are km/h. In other words, just by looking at the velocity units one becomes aware that it must be given by a ratio between a distance (in km) and a time (in h). Therefore one can guess that the formula for the velocity v will be given by:

x v = t

where x is the distance between two points and t is the time it took to cover the distance between such two points.

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