As a US high school student, you probably need to know how to convert between metric and USC units.

Knowing how to convert between units and perform **dimensional analysis** is also extremely useful for checking whether your working to a problem checks out. It can assist you if you can't quite remember a formula, as an equation must be **dimensionally consistent**.

**Dimensional analysis,** which you may have met as the 'factor-label method', is the process of treating units as you would algebraic symbols to check that an equation or formula is **dimensionally consistent** or to check a unit conversion. That means it is balanced with regards to the units of measurement that make up the equation.

Universal constants in physics can have quite complicated and difficult to conceptualize units. But they can help you remember a formula when the constants are provided for you in a test or exam, but not the formulae! If you need to use Newton's Law of Gravitation, recall it contains the gravitational constant, \( G\) which has units \(\text{N m$^2$/kg$^2$}\). This reminds us that there were the product of two masses in the numerator, and that this equation obeys the inverse square law and therefore we have \((\text{distance in m})^2\) in the denominator.

In high school physics, you are encouraged to carry units through a calculation, showing them at every step.

## What is a Unit?

Units are used to describe **physical quantities**. A unit gives a quantity its physical meaning. Some of these units have as their basis something tangible. Take the kilogram for instance, which was previously based on the mass of a solid cylinder of platinum-iridium alloy. We call such units that can be described only by measurement **operational units**^{1}. Other units need to be given in terms of others, for example \(\text{m/s}\), the units of speed or velocity, tells us that a meter is traversed for every second in time. We call these **derived units**.

In physics, we also use vectors to describe physical quantities, which gives the measure of a quantity a **direction** in addition to a numerical or symbolic value, or **magnitude**. This leads to the concept of a **unit vector**, which is also covered in Vectors. The unit vector is indivisible and, in a similar way to units, we obtain \(\text{number}\times\text{unit vector}\) to fully describe a vector. The number can be unitless (dimensionless) or it can be a physical quantity and need a unit of measurement, and we would write, for example, \(1.8 \;\hat{\textbf{i}}\text{ m/s } \) for a vector representing a velocity of magnitude \(1.8\text{ m/s}\) in the Cartesian x-direction.

## A Brief History of Units

Some units carry a lot of ancient history. The 'league', which you may have heard of, was derived from an ancient Celtic unit, and adopted by the ancient Romans. The actual measurement varied throughout the world, as did many old units. Even the inch varied. The basis for the inch, the barleycorn, was exactly that - a grain of barley. It must have seemed a fairly standard unit of measure at the time, and this definition lasted till the mid-19th century.

There was a time in France, when the system used varied not only across countries but from town-to-town. This made things complicated and expensive for outsiders. Similar to how we may have paid more for goods when using a different currency abroad, there was a lot of swindling! The metric system was adopted after the French revolution (1789), as the enlightenment swept Europe. The US was expected to follow suit, as the second country after France. There are some interesting stories about why it did not - one even involving pirates!^{2} Indeed, there were advantages to the old ways, such as having a familiar physical meaning to people.

In the US, as well as in Britain, we tend to think in British units. We know our 'weight' (actually our mass) in stones and ounces, and we know that 2 pounds of sugar is roughly a kilogram. We can think of *masses* in both systems, and also talk in terms of yards and miles in the US and Britain. Everywhere in the UK and US is signposted in miles. However, the UK and much of the rest of the world has converted to the metric system^{3}.

In 2019, the basic units of the SI system were redefined in terms of physical constants. The kilogram that we touched on in the previous section was redefined in terms of Planck's constant^{4}. The idea is that these constants of nature are so unchanging over space and time that they can better define the units used to express them than a physical artifact.

We have gone from cubits to cosmic measurement!

### Some old systems of units

The CGS system (Centimeter, Gram, Second) has the gram as the fundamental unit of mass, and the same for the centimeter and length. The unit of force is the \(\text{dyne}\) (In the next section you will see how to find the SI unit of force using Newton's Second Law, and in the flashcards you will revisit the dyne). The CGS unit of work is the dyne centimeter, and is called an \(\text{erg}\).

The MKS mass system, denotes 'Meter, Kilogram, Second' and was the early form of the SI system used today (see the next section). It introduced the watt as a measure of power, where \(1\text{ W}=1\text{ J/s}\), where the joule is the unit of work.

Confusingly, the MKS force system regarded the kilogram as a unit of force. A 'kilogram force' was defined to be the weight of a kilogram under normal gravity conditions. The terms 'mass' and 'force' are so often confused in real life. Just think about when we shop at a market and when asked what *weight* of something we want, we reply in pounds or ounces and grams or kilograms. We don't reply in newtons!

In the MKS force system the unit of force had the lengthier name of 'kilogram second squared per meter' (\(\text{kg$^2$/m}\)), for consistency with Newton's Second Law. Similarly, the American Engineering system (AE) treated the pound as a unit of force and had as its unit of force the 'pound second squared per foot' (\(\text{lb s$^2$/ft}\)) called the \(\text{slug}\). The slug was also used as a unit of mass, and it is still used so today.

So, you see, when the name of a unit can be used to describe two or more different things, we must take care in our conversions!

## The Main Modern Systems of Units & their Unit Conversion Factors

The actual units of measure in older systems could vary around the world, even those by the same name, like the league and the mile. It was desired to have a system where the units can be easily and intuitively linked to one another and a system where the units of measurement are universally agreed upon and familiar to all. This is what has happened with the SI system. The dedicated article SI Units will relate to you an especially expensive and disastrous mistake because of engineers using different systems of measurement to one another. Therefore, as burgeoning scientists, you will have to learn to use two main systems of measurement (and stick to them!) which we look at here.

### The International System (SI)

The SI (Système Internationale d’Unités) or what is more commonly known as "the metric system" can be broken down into 7 fundamental units. This system is preferred by the NIST (National Institute of Standards and Technology) in the US, and most other scientific institutions around the world.

**Length - meter (\(\text{m}\))****Time - second (\(\text{s}\))****Amount of substance - mole (\(\text{mol}\))****Electric current - ampere (\(\text{A}\))****Temperature - kelvin (\(\text{K}\))****Luminous intensity - candela (\(\text{cd}\))****Mass - kilogram (\(\text{kg}\))**

These units can be used to derive other units.

We know the SI unit of force is the newton. How might we break down the newton into the above basic units?

*Answer:*

Newton's Second Law is commonly expressed

\[\vec{F} = m\vec{a}\]

Where \(m\) is the mass in \(\text{kg}\)** **and \(\vec{a}\) (a vector)** **is the acceleration of the body in \(\text{m/s$^2$}\)**.**^{ }All we need do is substitute in the units to obtain the **newton **expressed in fundamental units

\[1\text{N} = 1 \text{kg m/s$^2$}\]

Similarly, we can find the basic units for other quantities like energy in joules (\(1\text{ J} = 1\text{ N}\cdot\text{m} = 1 \text{ kg m$^2$/s$^2$} \)) from the basic equations for work (\(W\,=\,F\cdot \, s\)) or potential energy (\(E_g\,=\,m\,g\,\Delta h\)). There is more detail on this system in the article SI Units.

### The American/Imperial System of Units: Unit Conversion Chart

The measurement units of the British Imperial System were officially used in the United Kingdom until the metric system was adopted. The US stuck with the British system, but it has different versions of the units and is now called the US Customary (USC) system!

The following units are used

**Length is measured in inches ("), feet (') and yards (\( \text{yd} \))****Mass in ounces (\(\text{oz}\)), pounds (\(\text{lbm}\)), and tons (\(\text{ton}\))****Volume in****cubic inches (\(\text{in}^3\)), cubic feet (\(\text{ft}^3\)) and fluid gallons (\(\text{gal}\))**

There are rational/whole number conversions between these units. Inches are further subdivided into \(\frac{1}{2}\)s, \(\frac{1}{4}\)s, \(\frac{1}{8}^{\text{th}}\)s, \(\frac{1}{16}^{\text{th}}\)s and other powers of \(\frac{1}{2}\), but is also split into hundredths (\(\text{caliber}\)) or thousandths (\(\text{mil}\)) so that \(\text{1 mil} = \frac{1}{1000}^{\prime\prime}\).

inch ( \( {}^{\prime\prime} \) ) | foot ( \( {}^{\prime} \) ) | yard (\(\text{ yd}\)) | mile (\(\text{mi}\)) | |

US unit conversion | \({\frac{1}{12}^{\text{th}}}^\prime\) / \(\frac{1}{36}^{\text{th}} \text{ yd}\) | \(12^{\prime\prime}\) \(\frac{1}{3}^{\text{rd}}\text{ yd}\) | \(3^\prime\) \(\frac{1}{1760}\text{ mi}\) (UK/US) | \(1760\text{ yd}\) \(5280^{\prime}\) |

US to SI unit conversion | \(0.0254\text{ m}\) (exactly) | \(\sim0.3148\text{ m}\) | \(\sim0.9144\text{ m}\) | \(1,609\text{ m}\) (approx.)\(1\text{ km}=0.6214\text{ mi}\) |

US length unit conversions^{5,6}

ounce (\(\text{oz}\)) | pound (\(\text{lbm}\)) | stone (\(\text{st}\)) | \(\text{ slug}\) | short hundredweight (\(\text{cwt}\)) | short ton (\(\text{ton}\)) | |

US unit conversion | 16 drams | \(\frac{1}{14}^\text{th}\text{ st}\)\(16\text{ oz}\) | \(\frac{7}{1000}\text{ ton}\)\(14\text{ lb}\)\( 224 \text{ oz}\) | \(\sim 32.17\text{ lbm}\) | \(100\text{ lb}\) | \( 20 \text{ cwt} \) \(2000 \text{ lb}\)\(32000 \text{ oz}\) |

US to SI unit conversion | \(\sim 0.0283 \text{ kg}\) | \(\sim 2.205\text{ kg}\) | \(\sim 6.350\text{ kg}\) | \(\sim14.59 \text{ kg}\) | \(\sim45.36\text{ kg}\) | \(\sim907.4\text{ kg}\) |

US mass unit conversions^{5,6}

cubic inch (\(\text{in}^3\)) | cubic foot (\(\text{ft}^3\)) | fluid gallon (\(\text{gal}\)) | dry gallon | |

US unit conversion | \(\frac{1}{1728}\text{ ft$^3$}\)\(\frac{1}{231}\text{ gal}\) | \(1728\text{ in$^3$}\)\(7.481\text{ gal}\) | \(231 \text{ in$^3$}\)\(0.1337 \text{ ft$^3$}\) | \(\sim1.164\) fluid gallons |

US to SI unit conversion | \( 1.639\times10^{-5}\text{ m$^3$}\) | \(0.02832\text{ m$^3$}\) | \(3.785\times10^{-3}\text{ m$^{-3}$}\) | \(4.406\times10^{-3}\text{ m$^{-3}$}\) |

US volume unit conversions^{5,6}

As you can see, the conversion to metric (SI) units is rather messy! The metric system is the most widely followed system around the world. However, it is important to understand how to convert between the two - and it is good to hone your mathematical skill!

Other non-SI units that occur in physics

Some non-SI units appear worldwide in colleges and universities, perhaps because of laboratory equipment that dates back to the 1960s which is still functional, and with which you get hands-on with many physical phenomena. Some of these are:

Unit | SI conversion |

\(\mathring{A}\) (Angstrom) | \( 10^{-10} \text{ m}\) |

\(\text{u}\), \(\text{amu}\) or \(\text{Da}\) (atomic mass unit, or Dalton) | \(\sim 1.6606\times10^{-27}\text{kg}\) defined as 1/12 the mass of a carbon-12 atom |

\({}^{\circ}\) (degree) | \(\frac{\pi}{180}\text{ rad}\) |

\(\text{atm}\) (Earth atmosphere) | \(101.325\text{ kPa}\) |

\(\text{Gs}\) (gauss) | \(100 \mu\text{T}\) (100 microtesla) |

\( \text{mmHg}\) (millimeters mercury) | \(133.32 \text{ Pa}\) |

\(\text{mbar}\) (millibar) | \(100\text{ Pa}\) |

\(\text{L}\) (liter) | \(1\times10^{-3}\text{ m$^3$}\) |

Some examples of approved non-SI units |

#### Temperature units

The Fahrenheit temperature scale is used every day in the US. Using it in thermodynamics entails using some other non-SI units, like the \(\text{Btu}\). Celsius temperature (which indicates the number of degrees above freezing) is commonly used in science alongside metric units and converts to the SI unit, kelvin, by the equation\[T_K = T_C + 273.15\]Fahrenheit and Celsius are related by\[T_F = \frac{9}{5}T_C+32^{\circ}\]\[T_C = \frac{5}{9}T_F-32^{\circ}\]#### Angle units

The conversion from angles in degrees (\({}^\circ\)) to the SI unit radians (\(\text{rad}\) or \({}^\text{c}\)) is\[\text{angle in degrees}\times\frac{\pi}{180^{\circ}}=\text{angle in radians}\]## Unit Prefixes and Powers of Ten

The metric unit has been widely accepted as it makes calculations very easy as they're all factors of \(10\). The unit conversion factor can sometimes also depend on the prefix that is placed before the base unit. Here we look into common prefixes that are used

- Milli means one-thousandth (or \(1/1000^{\text{th}}\))
- Centi means one hundredth (or \(1/100^{\text{th}}\))
- Kilo means one thousand (\(1000\times\))

These are the most common prefixes that are added in front of a unit. A kilometer would be \(1000\text{ m}\) and a millimeter would be \(1/1000^{\text{th}}\) of a meter. The same logic can be applied to other units such as those of mass, but recall that the fundamental unit of mass is already a kilo-gram!

In physics, magnitudes can span from very small values to large numbers. To make it easier to remember we use prefixes that are powers of 10.

Name/Symbol | Value | Power of 10 |

Peta, P | \(1\times10^{15}\) | \(10^{15}\) |

Tera, T | \(1\times10^{12}\) | \(10^{12}\) |

Giga, G | \(1,000,000,000\) | \(10^9\) |

Mega, M | \(1,000,000\) | \(10^6\) |

Kilo, k | \(1,000\) | \(10^3\) |

Centi, c | \(1/100\) | \(10^{-2}\) |

Milli, m | \(1/1,000\) | \(10^{-3}\) |

Micro, µ | \(1/1,000,000\) | \(10^{-6}\) |

Nano, n | \(1/1,000,000,000\) | \(10^{-9}\) |

Pico, p | \(1/10^{12}\) | \(10^{-12}\) |

Femto, f | \(1/10^{15}\) | \(10^{-15}\) |

Most commonly used prefixes and their powers |

Take a look at the following quantities, you will see how using prefixes reduces time and effort and makes dealing with large numbers much more efficient.

\(10.23\text{ GPa} = 10.23\text{ gigapascals} = 10.23\times10^9\text{ Pa (10 230 000 000 Pa)}\)

\(5\text{ kA} = 5\text{ kiloAmps} = 5\times10^3\text{ A (5000 A)}\)

\(0.1\,\mu\text{C} = 0.1\text{ microcoulombs} = 1\times10^{-7}\text{ C (0.0000001 C)}\)

\(8\text{ nm} = 8\text{ nanometers} = 8\times10^{-9}\text{ m (0.000000008 m)}\)

Now, let's look at how to convert between different units using a few examples.

## Customary Units Conversion

Below we have listed the unit conversions for some of the most common imperial units that are still in use today. Take a look at the unit conversion chart below for reference (two 2 decimal places) and see the examples below

\[\begin{align}1\text{ kg (kilogram)} &= 2.20\text{ lbm (pound-mass)}\\1\text{ lbm (pound-mass)} &= 0.45\text{ kg (kilogram)}\\1\text{ lbm (pound-force)} &= 4.45\text{ N (newtons)}\\1\text{ N (newton)} &= 0.22\text{ lbm (pound-force)}\\1\text{ L (liter)} &= 0.22\text{ gal (gallon) }\\1\text{ gal (gallon)} & = 4.55\text{ L (liters)}\\1^{\prime\prime}\text{ (inch)} &= 2.54\text{ cm (centimeters)}\\1\text{ cm (centimeter)} &= 0.39^{\prime\prime}\text{ (inch)}\\1\text{ km (kilometer)} &= 0.62\text{ mi (mile)}\\1\text{ mi (mile)} &= 1.61\text{ km (kilometers)}\end{align}\]

These are only the fundamental USC units. But what if we had to convert \(\text{m/s$^2$}\) into \(\text{km/hr$^2$}\)? We look into how we can do this for such complex units in the next section.

## Unit conversion examples

Convert \(25.0\text{ lb}\) (pounds) into \(\text{kg}\).

Step 1 Write down the conversion factor between the two units

\[ 1\text{ kg } =\, 2.20\text{ lb}\]

Step 2 Rearrange to obtain the number of kilograms in one pound

\[1\text{ lb } =\,\frac{1}{2.20}\text{ kg}\]

Step 3 Multiply this conversion formula by the number of pounds to complete the conversion

\[\begin{align}25.0\text{ lb } &=\,25\times\frac{1}{2.20}\text{ kg}\\25.0\text{ lb } &=\,11.4\text{ kg}\end{align}\]

This method can be used for any unit conversions. All we need to know is the conversion factor! Let's look at another simple example.

Convert \(60.0\text{ mph}\) into \(\text{kmph}\).

Here the unit is measuring velocity. It has two fundamental units in it, of distance and time. We need to consider both of them.

Step 1 Write down the conversion factor between the two units:

\[1\text{ mi} = 1.61 \text{ km}\]

Step 2 Multiply the conversion factor with the number of miles:

\[60.0\text{ mi} = 60.0\times1.61 \text{ km}\]

\[60.0\text{ mi} = 96.6 \text{ km}\]

Step 3 Final conversion from mph to kmph.

Since the unit of time is the same in both the units, there is no need for any further unit conversion

\[1\text{ mph} = 1.61 \text{ kmph}\]

\[60.0\text{ mph} = 96.6 \text{ km}\]

Convert \(50\text{ mph}\) into \(\text{m/s}\).

For a slightly different approach, we can do the conversion is as many stages as we like. For example:

\[50\text{ mph} = \left(\frac{50\text{ mi}}{1\text{ hr}}\frac{1\text{ hr}}{3600\text{ s}}\right)\cdot\left(\frac{1\text{ km}}{0.62\text{ mi}}\frac{10^3\text{ m}}{1\text{ km}}\right)\simeq 22\text{ m/s}\]

In the first stage in parentheses, we convert from per hour to per second, using the fact that there are \(3600 \text{ s}\) in an hour. The second parentheses then converts from kilometers to meters using the above conversion factor. Notice how we can cancel out the units we are no longer using, leaving us with our desired units. This is how we can check we have done the conversion properly!

$$50\text{ mph}=\left(\frac{50\text{ }\cancel{\text{mi}}}{1\text{ }\cancel{\text{hr}}}\frac{1\text{ }\cancel{\text{hr}}}{3600\text{ s}}\right)\cdot\left(\frac{1\text{ }\cancel{\text{km}}}{0.62\text{ }\cancel{\text{mi}}}\frac{10^3\text{ m}}{1\text{ }\cancel{\text{km}}}\right)\simeq22\text{ m/s}$$

We could, of course, have converted directly from miles to metres, if we had had the conversion factor for that.

Convert \(10\text{ m/s$^2$}\) into \(\text{km/hr$^2$}\).

Here the unit is measuring acceleration. It has two fundamental units in it, distance and time. We will have to convert both of them. For this, we will need to estimate the conversion factor between \(\text{ m/s$^2$}\) and \(\text{km/hr$^2$}\).

Step 1 Write down the conversion factor between kilometers and meters:

\[1\text{ km} = 1000\text{ m}\]

\[10\text{ m} = 10/1000\text{ km}\]

Step 2 Write down the conversion factor between hours and seconds:

\[ 1\text{ hr} = 60\text{ mins} = 60\times 60\text{ s} =3600\text{ s}\]

Now we need to convert \(\text{hr}^2\) into \(\text{s}^2\). To do so we need to square the conversion factor between them.

\[1\text{ hr}^2 = (3600\times3600)\text{ s}^2=12960000\text{ s}^2\]

Now we have all the data to calculate the conversion factor between \(\text{ m/s$^2$}\) and \(\text{km/hr$^2$}\).

Step 3 conversion factor from \(\text{ m/s$^2$}\) to \(\text{km/hr$^2$}\).\[\begin{align}1\text{ m/s$^2$} &= \frac{\frac{1}{1000}}{\frac{1}{12.96\text{ Ms}}}\text{ km/hr$^2$ ($1$ Ms $=10^6$ s)}\\&= 1.296\times 10^4\text{ km/hr$^2$}\end{align}\]

Step 4 Final Conversion:

\[\begin{equation}\begin{split}10\text{ m/s$^2$} & = 10\times1.296\times10^4\text{ km/hr$^2$}\\&= 1.296\times10^5\text{ km/hr$^2$}\\&=1.3\times10^5\text{ km/hr$^2$ (2 s.f.)}\end{split}\end{equation}\]

(We round to 2 significant figures (2 s.f.) because we were given a minimum of 2 s.f. in the question.)

And there you have it! Using the conversions factors of fundamental units we can derive the factors for more complex units.

## Unit Conversion - Key Takeaways

- Units are used to describe physical quantities and a unit gives a quantity its physical meaning.
- These units are non-divisible, and represent a singular unit of some physical quantity. But where a unit is non-fundamental, it can be broken down into its fundamental units.
- Units can be treated algebraically to carry out conversions. This is sometimes called the 'factor-label method', or 'dimensional analysis
- The 2 most-used standards of measurement are the United States Customary (USC) and metric systems, also known as the SI syste
- To make it easier for dealing with large numbers in SI, we use prefixes and powers of 1
- You must be careful when using the non-SI units that you don't confuse same-name units (e.g. pound-force, pound-mass) as these carry different meanings.

## References

- Hugh D. Young and Richard A. Freedman, Sears and Zemansky's University Physics, 2013
- Why doesn't the US use the metric system? | Live Science
- International Measuring System of Units by Country (chartsbin.com)
- The redefinition of the SI units - NPL
- British-American System of Units – The Physics Hypertextbook
- Unit Conversion Factors | MechaniCalc

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##### Frequently Asked Questions about Unit Conversion

What are the types of unit conversion ?

There are conversions from like-for-like units of one system of measurement to another system, for example centimeters to inches, and also bigger units into smaller units, such as hours to seconds. Some units of the same quantity can be expressed in different units, for example one ampere is one coulomb per second.

What is the unit conversion of length?

The SI unit conversions for length are 1 kilometer would be 1000 meters, and 1 millimeter would be 1/1000th of a meter. SI to American unit conversions for length are 1 inch would be 0.0254 meters.

What is an example of a unit conversion?

Converting seconds into hours, meters into kilometers, and grams into kilograms are examples of unit conversions. Or converting centimeters into inches, grams into ounces.

What are unit conversions?

Unit conversion is the process of converting units in one system of measurement into those of another system (both measuring the same quantity) - or different units of the same system of measurement - using a conversion factor. For example, converting centimeters into inches, for SI units to American units. Or hours to seconds (SI unit to SI unit).

What are unit conversions used for?

Unit conversions are used to represent units of the same quantity in the units of a different system of measurement, or into different units in the same system of measurement.

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