Standard and Compound Units for the ESAT

Standard Units

Standard units provide a consistent language for measurement. In the ESAT, you must be familiar with the following commonly used units across different physical dimensions.

Mass and Force

Mass measures the amount of matter in an object, while force measures the interaction that changes the motion of an object. The standard units are:

• Mass: Milligrams ($mg$), Grams ($g$), Kilograms ($kg$), and Tonnes ($t$).
• Force: Newtons ($N$).

Length and Area

Length measures distance, while area measures the extent of a two dimensional surface. Area units are derived by squaring length units.

• Length: Millimetres ($mm$), Centimetres ($cm$), Metres ($m$), and Kilometres ($km$).
• Area: Square millimetres ($mm^2$), Square centimetres ($cm^2$), Square metres ($m^2$), and Square kilometres ($km^2$).

Capacity and Volume

Volume and capacity describe the amount of space an object occupies or can hold. These are closely related, with specific conversion factors:

• Capacity: Millilitres ($ml$) and Litres ($l$).
• Volume: Cubic millimetres ($mm^3$), Cubic centimetres ($cm^3$), Cubic metres ($m^3$), as well as millilitres and litres.

Key relationships to remember:

• $1\text{ ml} = 1\text{ cm}^3$
• $1\text{ l} = 1\text{ dm}^3 = 1000\text{ cm}^3$
• $1000\text{ l} = 1\text{ m}^3$

Small quantities of liquid, such as medicine or drinks, are typically measured in $ml$ and $l$. Larger quantities, such as reservoirs or swimming pools, are measured in $m^3$.

Time

Time is measured using seconds, minutes, hours, days, weeks, months, and years. Note the following specific durations:

• A year consists of 12 months.
• A normal year has 365 days, while a leap year has 366 days. Leap years occur nearly every 4 years.
• A century is 100 years and a millennium is 1000 years.

Exercise: Identifying Units

1. What unit measures the volume of water in a swimming pool? Answer: $m^3$ (to avoid excessively large numbers).
2. What unit measures a kitchen floor area? Answer: $m^2$.
3. What unit measures the volume of a cola can? Answer: $ml$.

Compound Units

Compound units are formed when two different quantitative measurements are combined. For example, average speed is found by dividing the distance travelled (in $km$) by the time taken (in hours). This results in the unit $km/h$, which can also be written as $km\text{ h}^{-1}$.

Common compound units include:

• Density: $mass \div volume$, often expressed in $g/cm^3$.
• Average Speed: $total\ distance \div total\ time$, expressed in $m/s$ or $km/h$.
• Rate of Pay: $pay \div time$, expressed in $\pounds/h$.
• Pressure: $force \div area$, expressed in $N/m^2$.

Unit Cost

If $x$ items cost $\pounds y$ in total, the unit cost is the price of exactly one item. This is calculated by dividing the total cost by the number of items: $\text{Unit Cost} = \pounds \frac{y}{x}$.

Worked Example: Unit Cost If 50 boxes of sweets cost $\pounds 215$, find the unit cost of one box.

1. Total cost = $\pounds 215$.
2. Number of items = 50.
3. Unit cost = $215 \div 50 = \pounds 4.30$.

Changing Between Units

To convert between units, you must apply the correct conversion factor.

Length, Area, and Volume Conversions

Linear conversions are straightforward: $1\text{ km} = 1000\text{ m}$, $1\text{ m} = 100\text{ cm}$, and $1\text{ cm} = 10\text{ mm}$.

For area and volume, the conversion factor must be squared or cubed:

• Area: $1\text{ m}^2 = 100^2\text{ cm}^2 = 10,000\text{ cm}^2$.
• Volume: $1\text{ m}^3 = 100^3\text{ cm}^3 = 1,000,000\text{ cm}^3$.

Worked Example: Area Conversion Convert $2.65\text{ m}^2$ into $cm^2$. Since $1\text{ m}^2 = 10,000\text{ cm}^2$, we calculate $2.65 \times 10,000 = 26,500\text{ cm}^2$.

Changing Between Compound Units

To convert compound units, convert the numerator and denominator separately.

Worked Example: Density Conversion Convert a density of $20\text{ g cm}^{-3}$ into $kg\text{ m}^{-3}$.

1. Express as a fraction: $\frac{20\text{ g}}{1\text{ cm}^3}$.
2. Convert grams to kilograms: $20\text{ g} = \frac{20}{1000}\text{ kg} = 0.02\text{ kg}$.
3. Convert $cm^3$ to $m^3$: $1\text{ cm}^3 = \frac{1}{1,000,000}\text{ m}^3$.
4. Combine: $\frac{0.02}{1/1,000,000} = 0.02 \times 1,000,000 = 20,000\text{ kg m}^{-3}$.

Worked Example: Average Speed A car travels $35,000\text{ m}$ in $30\text{ minutes}$. Calculate the speed in $km/h$.

1. Convert distance to $km$: $35,000 \div 1000 = 35\text{ km}$.
2. Convert time to hours: $30 \div 60 = 0.5\text{ hours}$.
3. Calculate speed: $speed = \frac{35}{0.5} = 70\text{ km/h}$.