### Tools (click to expand)

Contents

- I. Introduction
- I.1 Physics
- I.2 Units
- I.3 Proportionality
- I.4 Measuring Devices

- M. Mechanics
- M.1 Linear Motion
- M.2 Vectors and Scalars
- M.3 Newton's Laws
- M.4 Momentum
- M.5 Circular Motion
- M.6 Gravity
- M.7 Density and Pressure
- M.8 Moments and Equilibrium
- M.9 Simple Harmonic Motion
- M.10 Work
- M.11 Energy
- M.12 Power

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## Abstract

### Learning Outcomes

- Recognise SI units.
- Recognise standard derived units.
- Recognise scientific prefixes.
- Recognice scientific notation.

### Keywords

- Unit
- Significant Figures

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Physics

This section will contain some information which is useful for understanding the rest of the course, but is not directly examinable. You will never be asked *"What is the name of the unit used to measure length?"*, or *"What is meant by proportionality?"*. Instead, you will need to recognise the unit of length and use it in calculations, or recognise a proportional relationship when you see one.

## Units

Units are used in all sciences as a way to quantify numerical results. The sentence: *"The distance between Dublin City and Cork City is 220."* is meaningless without further information. How far is 220 here? We could just as easily say *"The width of an A4 page is 220."*. Clearly, the two lengths are not the same. There must be some sort of definition for how long '1' is in each case. The international standard for measuring length is called the *metre*. When we refer to something as being ten metres in length, what we mean is that the object is ten times as long as a metre. When we say a soccer pitch is 100 metres long and 65 metres wide, we mean that it is 100 times as long as a metre, and 65 times as wide as a metre. In the above cases, the distance between the two cities is 220 *thousand* metres, and the width of the page is 220 *thousandths* of a metre. Below is a list of the various international standards we'll be using:

Parameter | Unit Name | Symbol |
---|---|---|

Length | metre | m |

Time | second | s |

Mass | kilogram | kg |

Electric Current | ampere | A |

Temperature | kelvin | K |

There are two more international standard base units - these are the candela (cd) which measures luminous intensity, and the mole (mol) which measures the amount of substance. These two are not examinable on our course.

This list of units is called the *Système International d'Unitès* (International System of Units), or SI units for short. Some units are named after famous scientists (e.g. André-Marie Ampère and Lord Kelvin): the unit names do *not* have capital letters, but their symbols do. Ask your English teacher about common nouns!

### Extra Credit: Unit Definitions (click to expand)

To make sure everyone means the same thing when they use one of these units, the definitions have to be very exact. Below, see how each unit is defined:

Unit Name | Definition |
---|---|

metre | The distance travelled by light in vacuum in \(\frac{1}{299792458}\) second. |

second | The duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom. |

kilogram | The mass of the international prototype kilogram. |

ampere | The constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed \(1m\) apart in vacuum, would produce between these conductors a force equal to \(2\times 10^{−7}\) newtons per metre of length. |

kelvin | \(\frac{1}{273.16}\) of the thermodynamic temperature of the triple point of water |

mole | The amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12. |

candela | 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 \(\frac{1}{683}\) watt per steradian. |

Notice how most of these units are defined in terms of some measurable quantity. For instance, if you have some caesium-133 and the right equipment, you can find out how long a second is as precicely as you like. If you have light in a vacuum and knowledge of how long a second is, you can calculate the length of a metre. This is absolutely intentional!

Each of these units used to be defined less accurately. For instance the second used to be defined as \(\frac{1}{86400}\) of a day. As it turns out, the day is not always the same length: it can vary based on how you measure it and the time of year for example. The original definitions for the other units had similar problems, so it was important to create these new definitions for each of them.

You might notice that the kilogram stands out among the others since it isn't measurable by anyone with the right equipment. The international prototype kilogram is a specific object in Paris made from a platinum-iridium alloy. This one cylinder is *the* kilogram by definition. All measurements of mass are made with respect to it and if it were to ever change, the kilogram would automatically change its definition also. One little scrape or a little erosion from air and all our measurements become technically wrong. You can see why this is a problem! There are a number of projects underway to find a better definition for the kilogram that is not dependent on a particular object. Derek explains further:

## Derived Units

Not every quantity gets measured using these 7 units. For instance, area and power both require the introduction of a *derived unit*. These are products of the base units. Area is measured using the square metre (\(m^2\)), while power is measured in watts \(\left(\frac{kg\cdot m^2}{s^{-3}}\right)\). Many of these units are either base units to a power, or are named after a famous scientist. We will come across tens of derived units on our course, but here is a small sample:

Parameter | Unit Name | Symbol | Base SI Equivalent |
---|---|---|---|

Frequency | hertz | Hz | \(s^{-1}\) |

Force | newton | N | \(kg\cdot m \cdot s^{-3}\) |

Charge | coulomb | C | \(A\cdot s\) |

Voltage | volt | V | \(kg \cdot m^2 \cdot s^{-3} \cdot A^{-1} |

## Unit Prefixes and Scientific Notation

Many of the ideas and laws we discuss in physics apply on a huge variety of scales. We can talk about the speed of rockets and snails, or the size of planets and atoms using the same units. Representing numbers with tens of digits can make them difficult to read by humans, so scientists use one of two shortcuts to show the scale of numbers.

The first is scientific prefixes. You've probably heard of the *centi*metre, *milli*metre, *kilo*metre etc. They are also used in computing, so your smartphone might have 16 *giga*bytes of capacity for example. These are scaled versions of a metre, and the prefix tells you what the scale is. For instance centi as a prefix means hundreth, so centimetre means hundreth of a metre. Milli means thousandth, so millimetres are thousandths of a metre. Each prefix has a symbol which can be attached to the unit symbol also. The symbol for *kilo* is \(k\), so we can represent kilometres as \(km\) or kilograms as \(kg\) (Extra credit: kilogram is the standard base unit of mass instead of gram for history reasons. Ask your history teacher!).

Alternatively, scientists use scientific notation to represent large or small numbers. Essentially it means looking at which power of ten is below the number (known as its order of magnitude) and combining that with the important digits of the number. For instance \(2340\) is just above \(1000\), so we represent it as \(2.340\times 1000\) or \(2.340\times 10^3\). See the full list of prefixes and notation we may use on our course in the table below:

Prefix | Symbol | Factor | Prefix | Symbol | Factor |
---|---|---|---|---|---|

peta | P | \(10^{15}\) | deci | d | \(10^{-1}\) |

tera | T | \(10^{12}\) | centi | c | \(10^{-2}\) |

giga | G | \(10^9\) | milli | m | \(10^{-3}\) |

mega | M | \(10^6\) | micro | \(\mu\) | \(10^{-6}\) |

kilo | k | \(10^3\) | nano | n | \(10^{-9}\) |

hecto | h | \(10^2\) | pico | p | \(10^{-12}\) |

deka | da | \(10^1\) | femto | f | \(10^{-15}\) |

### Extra Credit: Accuracy, Decimals, and Significant Figures (click to expand)

When perfoming calculations, we need to figure out how accurate our measurements and our answer are. If we use a metre stick to measure the length of an object, then we can only make our measurement to the closest notch on the ruler (usually these are millimetres). If the length of the object falls between two notch, we must round the length to the nearest notch. Think of it as the physical equivalent of rounding numbers.

On the other hand, say the length of a pencil is measured by someone else to be \(10.2cm\). When we are told that measurement, we must understand that the actual length may be not be exactly that number, but can be anywhere from \(10.15cm\) to \(10.25cm\), since those would round to \(10.2cm\). We can also usually say that the first few digits are *certain*, but the last digit (or two in some cases) is *uncertain*. This means for example, that there is a difference between the lengths \(10cm\) and \(10.0cm\) in physics. \(10cm\) means that the length is actually somewhere between \(9.5cm\) and \(10.5cm\), and that result is rounded, whereas \(10.0cm\) means that the length is actually somewhere between \(9.95cm\) and \(10.05cm\). The latter is a more accurate measurement because it has more *significant figures*.

Significant figures (affectionately known as sigfigs) are the amount of digits a measurement has starting with the first non-zero digit. This is different to *decimal places* which start counting from the first digit after a decimal point.

Number | Significant Figures | Decimal Places |
---|---|---|

10 | 2 | 0 |

10.0 | 3 | 1 |

1 250 | 4 | 0 |

6 832 714 | 7 | 0 |

6 832 714.000 | 10 | 3 |

0.325 | 3 | 0 |

0.000004128 | 4 | 9 |

**Last modified:**2017-12-06, 20:58:19