Using standard units
Length, time and volume are all examples of that can be measured.
A unit of measurement is one unit of a quantity: for example, one second.
Standard units of measurement are the units most typically used to measure a quantity.
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Units of length and time
Units of length
Kilometres (km) and miles are the standard units used to measure long distances.
Smaller lengths are measured in metres (m), centimetres (cm) or millimetres (mm).
Shorter distances can be measured with a ruler or tape measure. Longer distances can be measured using a . Each time the wheel rotates it travels one metre.
Units of time
Time has various units of measurement, eg seconds, minutes, hours, days and years. Clocks or stopwatches can be used to measure the amount of time something lasts - for example, how long it takes an athlete to complete a race in an event or training.
Clocks and timers have different levels of accuracy:
- Sun dials cast a shadow to give a rough indication of the time of day
- pendulum clocks rely upon the swinging of a mass and are more accurate than sundials
- atomic clocks are the most accurate
Average values
Some distances and times are short, such as the swing of a pendulum of a clock or the test for reaction times. When measuring these it is more precise to take multiple readings and calculate an average.
The mean is a measure of average. To find the mean of a list of numbers, add them all together and divide by how many numbers there are:
\(\text{mean} = \frac{\text{sum of all the numbers}}{\text{amount of numbers}}\)
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Units of area and volume
The area of a 2D shape is the amount of space inside it.
Area can be measured in square kilometres (km2), square metres (m2), square centimetres (cm2) and square millimetres (mm2).
Volume measures the space inside a object. The standard units of volume are cubic metres (m3), cubic centimetres (cm3) and cubic millimetres (mm3).
Capacity measures the amount that a 3D object can hold. The standard units of capacity are litres (l) and millilitres (ml).
Measuring cylinders can be used to measure the volume of liquids. To ensure an accurate result, use a measuring cylinder only a little larger than the volume.
They can also be used to measure the volume of solids when used with a . The solid is lowered into the can and the volume of water that is pushed out into the measuring cylinder is the same as the volume of the object.
Example
A gold bar is a cuboid measuring 5 cm by 10 cm by 8 cm. It is melted down and made into cubes with edges of length 2 cm. How many cubes can be made?
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Converting units of area
The unit conversions for length can be used to calculate areas in different units.
The two squares have the same area.
Square 1
Area = \(1~\text{m} \times 1~\text{m}\)
Area = 1 m2
Square 2
Area = \(100~\text{cm} \times 100~\text{cm}\)
Area = 10,000 cm2
Since square 1 and square 2 have the same area, \(1~m^2 = 10,000~cm^2\)
Use the same method to convert cm2 into mm2.
Square 3
Area = \(1~\text{cm} \times 1~\text{cm}\)
Area = 1 cm2
Square 4
Area = \(10~\text{mm} \times 10~\text{mm}\)
Area = 100 mm2
Since square 3 and square 4 have the same area, \(1~cm^2 = 100~mm^2\)
Example
Convert 5.2 m2 into cm2.
1 m2 = 10,000 cm2
So, \(5.2~\text{m}^2 = 5.2 \times 10,000 = 52,000~\text{cm}^2\)
Converting units of volume
The two cubes have the same volume.
Cube 1
Volume = \(1~\text{m} \times 1~\text{m} \times 1~\text{m}\)
Volume = 1 m3
Cube 2
Volume = \(100~\text{cm} \times 100~\text{cm} \times 100~\text{cm}\)
Volume = 1,000,000 cm3
Since cube 1 and cube 2 have the same volume, \(1~\text{m}^3 = 1,000,000 ~\text{cm}^3\)
The same method can be used to convert cm3 into mm3.
Cube 3
Volume = \(1~\text{cm} \times 1~\text{cm} \times 1~\text{cm}\)
Volume = 1 cm3
Cube 4
Volume = \(10~\text{mm} \times 10~\text{mm} \times 10~\text{mm}\)
Volume = 1,000 mm3
Since cube 3 and cube 4 have the same volume, \(1~\text{cm}^3 = 1,000 ~\text{mm}^3\)
Some example metric unit conversions for volume are:
- 1 m3 = 1,000,000 cm3
- 1 cm3 = 1,000 mm3
- 1 litre = 1,000 ml
Example
Convert 25,000 cm3 into m3.
\(1~\text{m}^\text{3} = 1,000,000~\text{cm}^\text{3}\)
So, \(25,000~\text{cm}^\text{3} = 25,000 \div 1,000,000~\text{m}^\text{3} = 0.025 ~\text{m}^\text{3}\)
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Scalar quantities
Extended syllabus content: Scalar quantities
If you are studying the Extended syllabus, you will also need to know about scalar quantities. Click 'show more' for this content:
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Vector quantities
If you are studying the Extended syllabus, you will also need to know about vector quantities and calculations. If you are studying the Core syllabus, go straight to the quiz:
Vector quantities have both magnitude and an associated direction. This makes them different from scalar quantities, which just have magnitude.
Vector examples
Some examples of vector quantities include:
- force, eg 20 newtons (N) to the left
- weight, eg 600 Newtons downwards
- velocity, eg 11 metres per second (m/s) upwards
- acceleration, eg 9.8 metres per second squared (m/s²) downwards
- momentum, eg 250 kilogram metres per second (kg m/s) south west
- electric field strength, eg 7 V/m from positive to negative
- gravitational field strength, eg 9.8 N/kg downwards.
The direction of a vector can be given in a written description, or drawn as an arrow. The length of an arrow represents the magnitude of the quantity. The diagrams show three examples of vectors, drawn to different scales.
Calculations involving forces and velocities
The is a single force that has the same effect as two or more forces acting together. You can easily calculate the resultant force of two forces that act in a straight line.
Two forces in the same direction
Two forces that act in the same direction produce a resultant force that is greater than either individual force. Simply add the magnitudes of the two forces together.
Example
Two forces, 3 newtons (N) and 2 N, act to the right. Calculate the resultant force.
3 N + 2 N = 5 N to the right
Two forces in opposite directions
Two forces that act in opposite directions produce a resultant force that is smaller than the combined forces. It is often easiest to subtract the magnitude of the smaller force from the magnitude of the larger force.
Example
A force of 5 N acts to the right, and a force of 3 N act to the left. Calculate the resultant force.
5 N - 3 N = 2 N to the right
Resultant velocity
The same principle applies with velocities. The is a single velocity that has the same effect as two or more velocities acting together. You can easily calculate the resultant velocity of two velocities that act in a straight line or opposite to each other.
Example
Calculate the resultant velocity from +5 m/s and -1 m/s.
5 -1 = +4 m/s
Free body diagrams and vector diagrams
Free body diagrams are used to describe situations where several forces act on an object. Vector diagrams are used to resolve (break down) a single force into two forces acting at right angles to each other.
Forces and velocities at right angles
In the following diagram of a toy trailer, when a child pulls on the handle, some of the 5 Newton (N) force pulls the trailer upwards away from the ground and some of the force pulls it to the right.
Vector diagrams can be used to resolve the pulling force into a horizontal component acting to the right and a vertical component acting upwards.
Vector diagrams
Draw a right-angled triangle to scale, in which each side represents a force. Try to choose a simple scale, for example 1 cm = 1 N. For the toy trailer example above, draw:
- a line representing the 5 N force at 37°
- a horizontal line ending directly below the end the first line
- a vertical line between ends of the two lines
- arrow heads to show the direction in which each force acts
Measure the lengths of the horizontal and vertical lines. Use the scale for the first line to convert these lengths to the corresponding forces.
The two component forces together have the same effect as the single force (in this example, the child's pulling force).
The same applies with velocities. You can draw vector diagrams for these too.
Podcast: Scalar and vector quantities, contact and non-contact forces
Measurements can be split into two groups - scalar quantities and vector quantities. In this episode, James Stewart and Ellie Hurer break down the key facts about quantities and contact and non-contact forces.
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Quiz
Test your knowledge on scalar and vector quantities with this quiz.
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