##### PHYSICS

# Measurement Units

Measurement Units: Units of measurement are **established models for measuring different magnitudes** , such as length, capacity, mass, time and volume.

The international system of units (SI) defines the reference unit of each measure. Based on the metric system, the SI arose from the need to standardize the units that are used in most of the countries.

## Lenght Measures

There are several measures of length, such as, for example, the yard, the inch and the foot.

In the SI, the reference unit of the length is **the meter (m)** . Currently, it is defined as the distance traveled by light in vacuum during a time interval of 1 / 299,792,558 of a second.

The multiples of the meter are the kilometer (km), the hectometer (hm) and the decámetro (dam). The submultiples are the decimeter (dm), the centimeter (cm) and the millimeter (mm).

## Capacity Measures

The unit of measurement of capacity most used is **the liter (l)** . The gallon, the barrel, the room, among others are still used.

The multiples of the liter are kiloliter (kl), hectoliter (hl), decalitro (dal). The submultiples are the deciliter (dl), the centiliter (cl) and the milliliter (ml).

## Mass Measurements

In the SI, the mass measurement is **the kilogram (kg)** . A cylinder of platinum and iridium is used as a universal reference for the kilogram.

The units of mass are kilogram (kg), hectogram (hm), decagram (dag), gram (g), decigram (dg), centigram and milligram (mg).

## Volume Measurements

In the SI, the unit of volume is **the cubic meter (m ^{3} )** . The multiples of the cubic meter are the cubic kilometer (km

^{3}), the cubic hectometer (hm

^{3}), the cubic dekameter (dam

^{3}). The submultiples are the cubic decimeter (dm

^{3}), the cubic centimeter (cm

^{3}) and the cubic millimeter (mm

^{3}).

We can transform a measure of capacity into volume, since liquids take the shape of the container that contains them. For that, we use the following relationship:

1l = 1dm ^{3}

## Measurement Conversion Table

We can use the same method for different measurements. First, we design a table and place in the center the base units of measurement that we want to convert, for example:

- Capacity: liter (l)
- Length: meter (m)
- mass: gram (g)
- volume: cubic meter (m
^{3})

All that is on the right side of the base measure are the submultiples. The deci, centi and milli prefixes correspond respectively to the tenth, hundredth and thousandth part of the fundamental unit.

Multiples | Base measure | Submultiples | ||||
---|---|---|---|---|---|---|

kilo | hecto | deca | deci | centi | milli | |

kl | hl | dal | liter | dl | cl | ml |

km | hm | dam | subway | dm | cm | mm |

kg | hg | dag | gram | dg | cg | mg |

km ^{3} | hm ^{3} | dam ^{3} | cubic meter | dm ^{3} | cm ^{3} | mm ^{3} |

### Examples

#### 1) How many milliliters correspond to 35 liters?

a) To make this conversion, we write the number in the table of capacity measures. Recall that the measure can be written as 35.0 liters. The comma and the digit before the comma are placed in the place of the corresponding measurement, in this case in the liter.

kl | hl | dal | l | dl | cl | ml |
---|---|---|---|---|---|---|

3 | 5, | 0 |

b) Next, we complete the places to the right with zeros until we reach the requested unit and we run the comma behind zero.

kl | hl | dal | l | dl | cl | ml |
---|---|---|---|---|---|---|

3 | 5 | 0 | 0 | 0, |

Thus, 35 liters correspond to 35,000 ml.

#### 2) Transform 700 grams in kilograms.

a) Write the value as 700.0 g. We place the comma and the number in front of it in the position of the corresponding measure, that is, the 0 in grams. The numbers ahead go in the previous positions.

kg | hg | dag | g | dg | cg | mg |
---|---|---|---|---|---|---|

7 | 0 | 0, | 0 |

b) After, we complete with zeros until we reach the requested unit, which in this case is the kilogram. The comma runs after the zero in kilogram.

kg | hg | dag | g | dg | cg | mg |
---|---|---|---|---|---|---|

0, | 7 | 0 | 0 | 0 |

Thus, 700 g correspond to 0.7 kg.

#### 3) How many cubic meters does a parallelepiped of 4,500 cm ^{3 have} ?

a) When we transform volume measurements, we proceed in a similar way to the previous cases, but placing three digits in each box.

We write the measurement as 4.500,0 cm ^{3} .

km ^{3} | hm ^{3} | dam ^{3} | m ^{3} | dm ^{3} | cm ^{3} | mm ^{3} |
---|---|---|---|---|---|---|

4 | 500, | 0 |

b) Now we complete with three digits in each box until we reach the requested unit.

km ^{3} | hm ^{3} | dam ^{3} | m ^{3} | dm ^{3} | cm ^{3} | mm ^{3} |
---|---|---|---|---|---|---|

0, | 004 | 500 |

Thus, 4,500 cm ^{3} corresponds to 0.0045 m ^{3} .

## What happens with time?

The base unit of measurement of time in the SI is **the second** (s). The definition of the second is the duration of 9,192,631,770 vibrations of the radiation emitted by the electronic transition between hyperfine levels of the ground state of the cesium atom 133.

The multiples of the second are the minute, the hour and the day. These measurements are not decimals, so the following relationships are used:

1 minute (min) = 60 seconds (s)

1 hour (h) = 3,600 seconds (s)

60 minutes = 1 hour (h)

24 hours (h) = 1 day (d)

The submultiples of the second are:

Tenth of a second = 0.1s or 1/10 s

Hundredth of a second = 0.01 s or 1/100 s

Thousands of a second = 0.001 s or 1/1000 s

Additionally, there is a unit of measurement used in astronomy to indicate huge distances. It is called the light year.

## Metric system

### Units of length

kilometer | km | 1000 m |

hectometer | hm | 100 m |

decameter | dam | 10 m |

subway | m | 1 m |

decimeter | dm | 0.1 m |

centimeter | cm | 0.01 m |

millimeter | mm | 0.001 m |

### Mass units

kilogram | kg | 1000 g |

hectogram | hg | 100 g |

decagram | dag | 10 g |

gram | g | 1 g |

decigram | dg | 0.1 g |

centigram | cg | 0.01 g |

milligram | mg | 0.001 g |

### Other units of mass

**Metric ton**

**1 t = 1000 kg**

**Quintal metric**

**1 q = 100 kg**

### Capacity Measurement Units

kiloliter | kl | 1000 l |

hectolitre | hl | 100 l |

decaliter | dal | 10 l |

liter | l | 1 l |

deciliter | dl | 0.1 l |

centiliter | cl | 0.01 l |

milliliter | ml | 0.001 l |

### Surface Measurement Units

square kilometer | km ^{2} | 1 000 000 m ^{2} |

square hectometer | hm ^{2} | 10,000 m ^{2} |

square decameter | dam ^{2} | 100 m ^{2} |

square meter | m ^{2} | 1 m ^{2} |

square decimeter | dm ^{2} | 0.01 m ^{2} |

square centimeter | cm ^{2} | 0.0001 m ^{2} |

square millimeter | mm ^{2} | 0.000001 m ^{2} |

### Agrarian surface Measurement Units

#### Hectare

**1 Ha = 1 Hm ^{2} = 10 000 m²**

#### Area

**1 a = 1 dam ^{2} = 100 m²**

#### Centiarea

**1 ca = 1 m²**

### Volume Measurement Units

cubic kilometer | km ^{3} | 1 000 000 000 m ^{3} |

cubic hectometer | hm ^{3} | 1 000 000m ^{3} |

cubic decameter | dam ^{3} | 1 000 m ^{3} |

subway | m ^{3} | 1 m ^{3} |

cubic decimeter | dm ^{3} | 0.001 m ^{3} |

cubic centimeter | cm ^{3} | 0.000001 m ^{3} |

cubic millimeter | mm ^{3} | 0.000000001 m ^{3} |

### Relationship between units of capacity, volume and mass

Capacity | Volume | Mass (of water) |
---|---|---|

1 kl | 1 m³ | 1 t |

1 l | 1 dm ^{3} | 1 kg |

1 ml | 1 cm³ | 1 g |

## English system

### Units of length

**Inch** = 2.54 cm.

**Pie** = 12 inches = **30.48 cm.**

**Yard** = 3 feet = 91.44 cm.

**Breaststroke** = two yards = 1. 829 m.

**Land mile** = 880 fathoms = **1,609 kilometers.**

**Nautical mile = 1,853 m.**

### Mass units

**Ounce** = 28.3 g.

**Libra** = 454 g.

### Capacity units

**Pinta** (Great Britain) = 0.568 l.

**Pinta** (USA) = 0.473 l.

**Barrel = 159 l.**

### Surface units

**Acre** = 4 047 m².

## Traditional units

### Units of length

The fundamental unit was the **rod** , its most used value was 83.6 cm.

Other measures were:

**Inch** : approximately 2.3 cm

**Span** = 9 inches, approximately 20.9 cm.

**Pie** = 12 inches, approximately 27.9 cm.

**Rod** = 3 feet = 4 spans, approximately 83.6 cm.

**Step** = 5 feet, approximately 1.39 m.

**Mile** = 1000 steps, approximately 1.39 km.

**Legua** = 4 miles, approximately 5.58 km.

### Capacity units

#### Liquid

**Cántara** = 16.13 l

#### For solids

**Bushel** = 55.5 l

### Mass units

The fundamental unit was the **pound** , its most used value was 460 g.

Other measures were:

**Ounce** = ¼ pound, approximately 115 g.

**Pound** = 460 g

**Arroba** = 25 pounds, approximately 11.5 kg.

### Surface units

**Fanega de tierra** = 65 areas = 6 500 m².