# Physics with Calculus/Mechanics/The SI Unit System

## Fundamental SI Units

SI unit system defines the following seven fundamental units in terms of how they can be measured. All other units in SI unit system can be derived from these. Also note that from physical point of view, only the first three, length, mass, and time are fundamental. Mole and Kelvin are actually arbitrary choice of scaling factor that has proven useful in various fields, and candela is also defined in terms of existing quantities. Note that derived unit means that it can be determined from the fundamental units, and not that it can be expressed in terms of them.

• Meter (m), the unit of length. It is defined as being the length of the path traveled by light in vacuum in 1/299 792 458 of a second.
• Kilogram (kg), the unit of mass. Note that the gram is not the fundamental unit for mass. It is defined by an actual platinum-iridium bar, though there is discussion of defining it by a more universal constant. It's worth noting in advance that a kilogram is not a unit of weight, which is a type of force.
• Second (s), the unit of time. The 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. NIST(1).
• Ampere (A), the unit of electric current. The ampere is defined as that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per meter of length. NIST(2)
• Mole (mol), the unit of quantity. One mole contains as many items as 12 grams of Carbon-12 contains of atoms. That number is known as Avogadro's Number, which is approximately $6.02214199\times 10^{23}$.
• Kelvin (K), the unit of temperature. It is defined as the fraction 1/273.16 of the thermodynamic temperature of the triple point of waterNIST(3). The 'easy' way to find the temperature in Kelvin is to add 273.15 to the temperature in Celsius.
• Candela (cd), the unit of luminous intensity or brightness. It is defined as the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency $540\times 10^{12}$ hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. NIST(4).

## Derived SI Units

In physics there are many quantities that cannot be expressed by a single base unit. For example, speed, the rate at which a position changes with time, is expressed as a combination of the unit for distance (meters) and the unit for time (seconds). As we will soon learn, the speed is equal to the distance divided by the time. Therefore, the unit of speed is the meter per second, or m/s. The unit meter per second is called a derived unit, meaning that it is derived from the seven SI base units.

Units can be combined together in many possible combinations, and any physically-significant quantity will have its own units. Some frequently-used combinations get their own names. Here are a sample of some of the more common ones:

• Newton (N), the unit of force.
• $\mathrm{N} = \mathrm{kg} \cdot \mathrm{m}/\mathrm{s}^2$
• Joule (J), the unit of energy.
• $\mathrm{J} = \mathrm{N} \cdot \mathrm{m} = \mathrm{kg} \cdot \mathrm{m}^2/\mathrm{s}^2$
• Watt (W), the unit of power.
• $\mathrm{W} = \mathrm{J}/\mathrm{s} \$
• Pascal (Pa), the unit of pressure.
• $\mathrm{Pa} = \mathrm{N}/\mathrm{m}^2 \$
• Hertz (Hz), the unit of frequency.
• $\mathrm{Hz} = 1/\mathrm{s} = \mathrm{s}^{-1} \$
• Coulomb (C), the unit of electric charge.
• $\mathrm{C} = \mathrm{A} \cdot \mathrm{s}$
• Volt (V), the unit of electric potential or voltage.
• $\mathrm{V} = \mathrm{J}/\mathrm{C} = \mathrm{W}/\mathrm{A} \$
• Ohm ($\Omega$), the unit of electric resistance.
• $\Omega = \mathrm{V}/\mathrm{A} \$
• Farad (F), the unit of electric capacitance.
• $\mathrm{F} = \mathrm{C}/\mathrm{V} \$
• Tesla (T), the unit of magnetic field strength.
• $\mathrm{T} = \mathrm{N}/\mathrm{A} \cdot \mathrm{m}$
• Weber (Wb), the unit of magnetic flux.
• $\mathrm{Wb} = \mathrm{T} \cdot \mathrm{m}^2$

## SI Prefixes

Physics spans the very small to the very large. Particle physicists are interested in distances as small as the radius of a proton, 10-15 m or smaller. Meanwhile, astronomers may be interested in measuring distances of a parsec, around 1016 m, or greater.

The "English" system used different units to indicate the scale of measurement. For example, one might say that one's house has a length of 53 feet. However, perhaps New York City is 150 miles away. While it would be perfectly accurate to do so, no person would state that their house was 0.010 miles long or that New York City was 792 000 feet away. In both cases, the unit doesn't accurately represent the scale of the measurement.

In SI, prefixes are available to adjust the size of a unit so as to keep the number of those units reasonable. Prefixes may be added to either base units or derived units.

Here are the accepted SI prefixes:

SI prefixes
1000n 10n Prefix Symbol Short scale Decimal equivalent in SI writing style
10008 1024 yotta Y Septillion 1 000 000 000 000 000 000 000 000
10007 1021 zetta Z Sextillion 1 000 000 000 000 000 000 000
10006 1018 exa E Quintillion 1 000 000 000 000 000 000
10005 1015 peta P Quadrillion 1 000 000 000 000 000
10004 1012 tera T Trillion 1 000 000 000 000
10003 109 giga G Billion 1 000 000 000
10002 106 mega M Million 1 000 000
10001 103 kilo k Thousand 1 000
10002/3 102 hecto h Hundred 100
10001/3 101 deka da Ten 10
10000 100 (none) (none) One 1
1000−1/3 10−1 deci d Tenth 0.1
1000−2/3 10−2 centi c Hundredth 0.01
1000−1 10−3 milli m Thousandth 0.001
1000−2 10−6 micro µ (u, mc) Millionth 0.000 001
1000−3 10−9 nano n Billionth 0.000 000 001
1000−4 10−12 pico p Trillionth 0.000 000 000 001
1000−5 10−15 femto f Quadrillionth 0.000 000 000 000 001
1000−6 10−18 atto a Quintillionth 0.000 000 000 000 000 001
1000−7 10−21 zepto z Sextillionth 0.000 000 000 000 000 000 001
1000−8 10−24 yocto y Septillionth 0.000 000 000 000 000 000 000 001

For example, it would be awkward to speak about the capacitance of a particular device as "0.000 000 000 010 farads." Using the unit "10 picofarads" is a lot more sensible.

For some reason, however, not all prefixes are frequently used. Distances are rarely given in units larger than kilometers and masses usually aren't given in units larger than kilograms. Experience will indicate when certain implied conventions apply, but it's never incorrect to refer to a distance of 5 400 000 000 m as 5.4 Gm. The point will be made and the value will be understood.

## Calculations with Units

When performing calculations in physics, it is very helpful to keep track of the units of each numerical quantity. The unit of the result can often lend insight into the nature of what has been calculated.

When adding or subtracting values with units, each quantity must have the same unit. The result has the same unit as the operands. For example:

5 m + 3 m = 8 m
2.50 kg + 3.712 kg = 6.21 kg
57 feet + 192 inches = ?????

If quantities have different units, one must first convert to a common unit before the calculation is completed:

57 feet + 192 inches = 57 feet + 16 feet = 73 feet.

When multiplying or dividing values with units, the units are multiplied or divided too.

$(20\ \mathrm{m}) \times (5\ \mathrm{m}) = 100\ \mathrm{m}^2$
$(1000\ \mathrm{N}) / (40\ \mathrm{m}) = 25\ \mathrm{N}/\mathrm{m}$

Sometimes the resulting unit has a special name:

$(400\ \mathrm{J}) / (5\ \mathrm{s}) = 80\ \mathrm{J}/\mathrm{s} = 80\ \mathrm{W}$