T-Level Unit 5 Session 15 Recap Lesson

1

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

5. Essential science for engineering and
manufacturing

5.1 Units of measurement used in engineering

PowerPoint 1: SI Units

2

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

5. Essential science for engineering and
manufacturing

5.2 Vector and Coordinate Measuring Systems

PowerPoint 2: Vectors and coordinates

3

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Objectives

By the end of this session, learners should be able to:

•State the units of measurement used in engineering and what they
represent.

•Convert between SI units and imperial units

•Convert between different multiples and submultiples.

4

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

What are SI units ?

The units that scientists use all over the world are standardised in the
Système Internationale d’Unités: SI units. (There is one exception –
North America uses imperial units.) Working in SI units means
measurements are consistent and recognised globally.

There are seven base units:
•kilogram (kg)

•metre (m)

•second (s)

•ampere (A)

•kelvin (K)

•mole (mol)

•candela (cd).

5

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

SI units in engineering

Here are some of the base units you are mostly likely to encounter within
the engineering sector

Property

SI unit

Abbreviation

Measures

Mass

kilogram

kg

mass

Time

second

s

time

Length
(distance)

metre

m

length

Note that when using SI units there should be a space between the value
and the unit. When using the full word it should not be capitalized unless
at the start of a sentence. For example, 3 m = 3 metres,

6

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Mass and weight

Mass is the amount of matter in an object and is measured in kilograms.
An object cannot have a mass of zero.

Weight is a force due to the pull of gravity on the mass of the object and
is measured in newtons (N). Weight = force x gravity’s acceleration.

We often confuse the terms and weight is often quoted in kilograms.

Pressure is defined as force/area. The SI unit for force is newtons (N)
and the SI unit for area is square meters (m²). One newton per square
metre (N/m²) equals 1 pascal.

Example: when a liquid or a gas is in contact with a solid surface such as
water in a tank it produces a force on that surface. The pressure of the
fluid is defined as the force per unit area.

7

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Using SI units

Note that when using SI units there should be a space between the value
and the unit. When using the full word it should not be capitalized unless
at the start of a sentence. For example, 3 m = 3 metres,

•Area is measured in m2(eg, a room 10 m long x 10 m wide = 100 m2).

•Volume is measured in m3 (eg, a room 10 m long x 10 m wide x 10 m
high = 1000 m3).

•One litre is a volume of 0.001 m3 or 1000 cm3.

8

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

SI units in engineering

Derived units come from the seven base units. For example, a newton
is a unit of force, which is the same as kg m s-2

Here some of the derived units you are mostly likely to encounter within
the engineering sector:

Property

SI unit

Abbreviation

Equivalent to

Density

kg m-3

Velocity

m s-1

Acceleration

m s-2

Force

Newton

N

kg m s-2

Pressure

Pascal

Pa

N m-2

9

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

You will encounter the following in other units in this qualification:

•Specific heat capacity is measured in kJ/kg/ºC.

When working with electricity, as well as the base SI unit of amperes (A):

• Volt (V) is the derived unit for electric potential, electric potential

difference (voltage) and electromotive force

• Ohm (Ω) is the derived unit of electrical resistance

• Coulomb (C) is the derived unit of electric charge

• Watt (w) is an SI unit of power used to measure the rate of energy

transfer.

Other derived units used in the engineering sector

10

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Multiples and sub-multiples

When working with small areas, volumes, distances, or power supplies
the units you calculate are comparatively small and therefore not difficult
to record.

For example, the UK mains power supply is an alternating current with a
frequency of 50 hertz (Hz) and 230 volts (V).

In large engineering projects, the forces, distances, speeds or volumes
involved can be much larger, meaning the numbers become
inconvenient to record in full. In precision engineering, there will also be
very tiny and precise measurements.

In engineering notation or engineering form the exponent of 10 must
be divisible by 3 (i.e. powers of a thousand, but written as, for example,
106instead of 10002).

11

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Multiples and sub-multiples

Multiples and sub-multiples allow orders of magnitude to be recorded in
shortened form.

Multiple size

Power of 10 (x10²)

Prefix
Prefix

abbreviation

1,000,000,000,000

12

tera-

T

1,000,000,000

9

giga-

G

1,000,000

6

mega-

M

1,000

3

kilo-

k

0.01

-2

centi-

c

0.001

-3

milli-

m

0.000001

-6

micro-

μ

0.000000001

-9

nano-

n

0.000000000001

-12

pico-

p

12

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T Level Technical Qualification in
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300 Engineering Common Core Content

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Worked example: Engineering standard form

The distance from the Sun to Pluto is 4,400,000,000,000 m

This can also be written as:

4.4 × 1012 m

It can be shortened again with the prefix Tera to:

4.4 Terametres

Or abbreviated to:

4.4 Tm

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T Level Technical Qualification in
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300 Engineering Common Core Content

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Which is greater?

Which is greater:

230 kV or 2.3 MV?

230 kV is equivalent to 230 kV = 230 x 1,000= 230,000 V

2.3 MV is equivalent to 2.3 MV = 2.3 x 1,000,000 = 2,300,000 V

So, 2.3 MV is the greater amount.

14

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Metric and imperial units

Within Engineering you might encounter units that follow the imperial
system rather than the SI units, for example when using American
Equipment. The table below shows how to convert between metric (SI)
and imperial measurements:

Property

SI unit

Imperial unit

Conversion

length

metre

Inch (in)

1 in = 0.0254 m = 25.4 mm

length

metre

Foot (ft)

1 ft = 0.3048 m = 304.8 mm

length

metre

Yard (yd)

1 yd = 0.9144 m = 914.4 mm

mass

kilogramme

Ounce (oz)

1 oz = 0.02835 kg = 28.35 g

volume

litre

Gallon (gal)

1 gal = 4.546 litres

15

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T Level Technical Qualification in
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300 Engineering Common Core Content

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Other SI units used in the engineering sector

SI unit

Abbreviation

Measures

Watt

W

Power

Bar

bar

Pressure

Joule

J

Energy

Candela

cd

Luminous intensity

Illuminance

Lux

Illumination

Kelvin

k

Temperature

Here are some of the other units you might encounter within the
engineering sector

16

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

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5. Essential science for engineering and
manufacturing

5.2 Vector and Coordinate Measuring Systems

PowerPoint 2: Vectors and coordinates

17

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Objectives

By the end of this session, learners should
be able to:

•Explain the difference between scalar
and vector quantities.

•State examples of scalar and vector
quantities.

•Convert between Cartesian and polar
coordinates.

18

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T Level Technical Qualification in
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300 Engineering Common Core Content

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Scalars and vectors
•A quantity that has only magnitude is called a scalar quantity.

•Examples of scalar quantities include distance, speed and
acceleration.

•Other scalar quantities include time and temperature.

•A quantity that has magnitude and direction is called a vector
quantity and can be represented by vectors.

•Examples of vector quantities include displacement and velocity.

19

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T Level Technical Qualification in
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300 Engineering Common Core Content

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Scalars and vectors: speed and velocity example

•Speed is a scalar quantity
as it only has magnitude.

•Velocity is a vector quantity
as it has both magnitude and
direction.

20

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T Level Technical Qualification in
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300 Engineering Common Core Content

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Co-ordinates

•Co-ordinates specify positions – for example, GPS systems use
them to determine where you are.

•One common use in engineering is to instruct the tool where to
move in CAM machines.

•Types of co-ordinates include Cartesian (x,y) and polar (r,Θ).

21

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Objectives

By the end of this session, learners should be able to:

•State the units of measurement used in engineering and what they
represent.

•Convert between SI units and imperial units

•Convert between different multiples and submultiples.

22

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Cartesian co-ordinates

•Cartesian coordinates indicate the location of a point relative to
a fixed reference point (the origin)

•They are read across first, then up (or down).

•The coordinates are:

23

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

What are SI units ?

The units that scientists use all over the world are standardised in the
Système Internationale d’Unités: SI units. (There is one exception –
North America uses imperial units.) Working in SI units means
measurements are consistent and recognised globally.

There are seven base units:
•kilogram (kg)

•metre (m)

•second (s)

•ampere (A)

•kelvin (K)

•mole (mol)

•candela (cd).

24

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
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Polar representation

r

z

2

y

x

r

z





Using polar coordinates we mark a point by how far away, and what
angle it is such as (14, 36⁰).

The polar form of a coordinate may be written in the form

The term r is called the modulus

This is denoted by mod

or

To find the modulus from a cartesian coordinate, use Pythagoras’
theorem:

(r, θ)

or

25

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

SI units in engineering

Here are some of the base units you are mostly likely to encounter within
the engineering sector

Property

SI unit

Abbreviation

Measures

Mass

kilogram

kg

mass

Time

second

s

time

Length
(distance)

metre

m

length

Note that when using SI units there should be a space between the value
and the unit. When using the full word it should not be capitalized unless
at the start of a sentence. For example, 3 m = 3 metres,

26

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Polar argument



z



z

x
y




tan

The angle

is called the argument (or amplitude) of the

It is denoted by arg

or amp

Hence

arg

polar coordinate

To find the argument of a polar coordinate from a cartesian coordinate,
use trigonometry:

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T Level Technical Qualification in
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300 Engineering Common Core Content

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Mass and weight

Mass is the amount of matter in an object and is measured in kilograms.
An object cannot have a mass of zero.

Weight is a force due to the pull of gravity on the mass of the object and
is measured in newtons (N). Weight = force x gravity’s acceleration.

We often confuse the terms and weight is often quoted in kilograms.

Pressure is defined as force/area. The SI unit for force is newtons (N)
and the SI unit for area is square meters (m²). One newton per square
metre (N/m²) equals 1 pascal.

Example: when a liquid or a gas is in contact with a solid surface such as
water in a tank it produces a force on that surface. The pressure of the
fluid is defined as the force per unit area.

28

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T Level Technical Qualification in
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300 Engineering Common Core Content

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Converting between Cartesian and polar co-ordinates

•Convert the Cartesian co-ordinate (2,4) into polar form.

Using Pythagoras’ theorem and trigonometry:

r = √ (22+ 42) = √ 20 = 4.47

θ = tan-1(4/2) = tan-12 = 63.4o

•Convert the polar co-ordinate (4.47, 63.4o) into Cartesian form

Using trigonometry:

x = r cos θ = 4.47 cos 63.4o = 2

y = r sin θ = 4.47 sin 63.4o = 4

29

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Using SI units

Note that when using SI units there should be a space between the value
and the unit. When using the full word it should not be capitalized unless
at the start of a sentence. For example, 3 m = 3 metres,

•Area is measured in m2(eg, a room 10 m long x 10 m wide = 100 m2).

•Volume is measured in m3 (eg, a room 10 m long x 10 m wide x 10 m
high = 1000 m3).

•One litre is a volume of 0.001 m3 or 1000 cm3.

30

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T Level Technical Qualification in
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300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Conversion from Cartesian to polar

To convert from Cartesian to polar:

When we know a point in Cartesian coordinates (x,y) and we want it in
polar coordinates (r,θ) we solve a right triangle with two known sides.

Example: What is (12,5) in polar coordinates?

31

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

SI units in engineering

Derived units come from the seven base units. For example, a newton
is a unit of force, which is the same as kg m s-2

Here some of the derived units you are mostly likely to encounter within
the engineering sector:

Property

SI unit

Abbreviation

Equivalent to

Density

kg m-3

Velocity

m s-1

Acceleration

m s-2

Force

Newton

N

kg m s-2

Pressure

Pascal

Pa

N m-2

32

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T Level Technical Qualification in
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300 Engineering Common Core Content

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Use Pythagoras’ theorem to find the long side of the triangle (the
hypotenuse (r)):

r2= 122+ 52

r = √ (122+ 52)

r = √ (144 + 25)

r = √ (169) = 13

Use the tan function to find the angle:

tan ( θ ) = 5 / 12

θ = tan-1( 5 / 12 ) = 22.6° (to one decimal place)

Answer: (12,5) is equal to (13, 22.6o) in polar coordinates.

33

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T Level Technical Qualification in
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300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
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You will encounter the following in other units in this qualification:

•Specific heat capacity is measured in kJ/kg/ºC.

When working with electricity, as well as the base SI unit of amperes (A):

• Volt (V) is the derived unit for electric potential, electric potential

difference (voltage) and electromotive force

• Ohm (Ω) is the derived unit of electrical resistance

• Coulomb (C) is the derived unit of electric charge

• Watt (w) is an SI unit of power used to measure the rate of energy

transfer.

Other derived units used in the engineering sector

34

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Conversion from polar to Cartesian

To convert from polar to Cartesian:

When we know a point in polar coordinates (r, θ), and we want it in
Cartesian coordinates (x,y) we solve a right-angled triangle with a known
long side and angle:

Example: What is (13, 22.6o) in Cartesian coordinates?

35

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Multiples and sub-multiples

When working with small areas, volumes, distances, or power supplies
the units you calculate are comparatively small and therefore not difficult
to record.

For example, the UK mains power supply is an alternating current with a
frequency of 50 hertz (Hz) and 230 volts (V).

In large engineering projects, the forces, distances, speeds or volumes
involved can be much larger, meaning the numbers become
inconvenient to record in full. In precision engineering, there will also be
very tiny and precise measurements.

In engineering notation or engineering form the exponent of 10 must
be divisible by 3 (i.e. powers of a thousand, but written as, for example,
106instead of 10002).

36

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
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Use the Cosine function for x:

cos 22.6°= x / 13

Rearranging and solving:

x = 13 × cos 22.6°

x = 13 × 0.923

x = 12.0 to one decimal place

Use the Sine function for y:

sin 22.6°= y / 13

Rearranging and solving:

y = 13 × sin 22.6°

y = 13 × 0.391

y = 5.0 to one decimal place

Hence (13, 22.6o) is (12, 5) in Cartesian coordinates.

37

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T Level Technical Qualification in
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300 Engineering Common Core Content

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Multiples and sub-multiples

Multiples and sub-multiples allow orders of magnitude to be recorded in
shortened form.

Multiple size

Power of 10 (x10²)

Prefix
Prefix

abbreviation

1,000,000,000,000

12

tera-

T

1,000,000,000

9

giga-

G

1,000,000

6

mega-

M

1,000

3

kilo-

k

0.01

-2

centi-

c

0.001

-3

milli-

m

0.000001

-6

micro-

μ

0.000000001

-9

nano-

n

0.000000000001

-12

pico-

p

38

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T Level Technical Qualification in
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Example: velocity of an aircraft

Velocity can also be represented
by a vector.

Imagine an aircraft flying north at a
speed of 60 m/s.

The arrow represents the velocity
of the aircraft; the length
represents the speed, and the
direction represents the direction of
travel.

60 m/s N

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Worked example: Engineering standard form

The distance from the Sun to Pluto is 4,400,000,000,000 m

This can also be written as:

4.4 × 1012 m

It can be shortened again with the prefix Tera to:

4.4 Terametres

Or abbreviated to:

4.4 Tm

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T Level Technical Qualification in
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300 Engineering Common Core Content

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Velocity example (continued)

Imagine the same aircraft
flying north at a speed of 60
m/s, but, in this case, there is
a wind blowing that gives the
aircraft a velocity of 20 m/s to
the east.

An additional vector can be
drawn to represent the actual
velocity of the aircraft taking
into account the wind.

60 m/s N

20 m/s E

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T Level Technical Qualification in
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Which is greater?

Which is greater:

230 kV or 2.3 MV?

230 kV is equivalent to 230 kV = 230 x 1,000= 230,000 V

2.3 MV is equivalent to 2.3 MV = 2.3 x 1,000,000 = 2,300,000 V

So, 2.3 MV is the greater amount.

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Velocity example (continued)

The actual velocity of the
aircraft, resulting from its
northward motion and the
eastward wind, is
represented by another
vector; the dotted arrow.

60 m/s N

20 m/s E

?

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T Level Technical Qualification in
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300 Engineering Common Core Content

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Metric and imperial units

Within Engineering you might encounter units that follow the imperial
system rather than the SI units, for example when using American
Equipment. The table below shows how to convert between metric (SI)
and imperial measurements:

Property

SI unit

Imperial unit

Conversion

length

metre

Inch (in)

1 in = 0.0254 m = 25.4 mm

length

metre

Foot (ft)

1 ft = 0.3048 m = 304.8 mm

length

metre

Yard (yd)

1 yd = 0.9144 m = 914.4 mm

mass

kilogramme

Ounce (oz)

1 oz = 0.02835 kg = 28.35 g

volume

litre

Gallon (gal)

1 gal = 4.546 litres

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Finding the actual velocity of the aircraft
can be found by adding the two vectors
together.

If two vectors are at right angles to
each other, you can use simple algebra
based on Pythagoras’ theorem.

Magnitude

c2= a2+ b2

c2= (60 × 60) + (20 × 20) = 4,000

c = 63.24 m/s

a2+ b2= c2

a

b

c

Velocity example (continued)

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Other SI units used in the engineering sector

SI unit

Abbreviation

Measures

Watt

W

Power

Bar

bar

Pressure

Joule

J

Energy

Candela

cd

Luminous intensity

Illuminance

Lux

Illumination

Kelvin

k

Temperature

Here are some of the other units you might encounter within the
engineering sector

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The resultant direction of the plane can
also be found using simple algebra
based on Pythagoras’ theorem.

Direction

Angle = tan-1

= tan-120

= 18.4°

a
b

60

60 m/s N

20 m/s E

Solution: The velocity can therefore be represented as either the
Cartesian (20, 60) or the polar (63.24, 18.4o).

Velocity example (continued)

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5. Essential science for engineering and
manufacturing

5.3 Scientific methods and approaches to scientific

inquiry and research

PowerPoint 3: Scientific method

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Objectives

By the end of this session, learners should be able to:

•Explain the concept of the scientific method.

•State the steps involved in a scientific method.

•Explain the characteristics required of data to allow valid conclusions
to be made.

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Introduction

Science is defined as:

‘The use of evidence to construct testable explanations and predictions
of natural phenomena, as well as the knowledge generated through this
process.’

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The scientific method

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Concept of the scientific method

•The scientific method is a systematic process to acquire knowledge.

•It reduces cognitive assumptions that distort interpretations of what is
happening in the observed situation.

•It involves:
•Observation

•Questioning

•Making (or refining) a hypothesis
(by predicting/ simulating the
outcome)

•Testing (experimentation)

•Come to a conclusion/result

•Iteration – repeating the above
steps until the hypothesis is proved

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Hypothesis

A hypothesis is a TENTATIVE explanation for an
observation, phenomenon or scientific problem
that can be tested with further investigation.

Scientists do not randomly produce hypotheses.
They base them upon observation, knowledge
and data from other studies.

A simple hypothesis:

‘Water MAY help plants to grow.’

The statement must be tentative and must be
testable.

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Example of a formal hypothesis

•Eating sweets may lead to tooth decay.

•Ultraviolet light from the Sun may cause skin cancer.

In scientific investigation scientists combine the educated guess with a
calculated prediction of what will happen during the process of the
experimentation or period of observation.

This could be phrased as an “if…then” statement, such as:

‘IF skin cancer is related to UV rays, THEN people who expose
themselves to the Sun (i.e. sunbathing) will have a higher frequency of
being diagnosed with skin cancer.’

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Hypothesis – degree of uncertainty

The hypothesis is not a prediction, but it allows predictions to be made
that can be tested.

A scientific hypothesis is never proven to be absolutely correct.

Scientists will always talk about the increasing probability that something
is correct as they add more data from experimentation or observation.

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Non scientific hypothesis

For a hypothesis to be considered
scientific, it must be possible to be
prove it is either correct or incorrect.

An example is that people believe that
we are not the only intelligent life within
our galaxy.

This cannot be classed as a scientific
hypothesis as currently we do not have
the ability to search to the limits of the
galaxy. As this cannot be proven or
disproven, it is not a scientific
hypothesis.

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Directional hypothesis

When scientists create a formal hypothesis to start the experimentation
or investigative process, they will often include a directional prediction.

Examples:

‘Affect the speed’ is classed as a non-directional statement

‘Make the object faster’ or ‘Make the object slower’ are classed as
directional statements.

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Testing hypotheses

Typically a hypothesis is devised around
observation and data.

The scientist then sets out to prove that it is
almost certainly true via experiments or
observations.

Before the scientific hypothesis becomes
established, it is typically reviewed by other
scientists who will attempt to disprove the
hypothesis by conducting the same
experiments again, or running similar
experiments using different methods.

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Characteristics of useful data

•Accuracy – the set of measurements
should be close to their true value.

•Precision – the recorded data values
should be close to each other (avoids
scatter due to random fluctuations or errors)

•Reliability – the data should be collected in
a way that is appropriate for the situation it
represents

•Replication – it should be possible to carry
out the same test and get the same result
(repeatability).

Ideally any data collected when testing a hypothesis should have four
characteristics:

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Collection of data

To ensure that data is collected in an
accurate manner by doing the following

•Repeating the readings and calculating
the average (mean).

•Repeat the measurement with another
identical instrument to ascertain if the
readings are the same.

•Use equipment that is of a high quality.

•Take the readings carefully in a timely
manner.

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Accurate measurements

In order to gain an accurate measurement this can
be achieved by:

•Selecting and using a measurement instrument
that has the range and scale to cover the values
under measurement.

•Calibrating the equipment to manufacturers
specifications.

•Check measurements twice prior to recording
them to eliminate errors.

•Repeat the exercise to confirm multiple readings
are grouped close to one another.

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Hypothesis and theory

In science, if a hypothesis is proved
correct after testing, then it can
become established as a recognised
theory or in some cases a scientific
law.

One such example is Sir Isaac Newton
and his ‘Gravitational Law’.

He proved mathematically that the
force acting upon an apple to pull it
from the tree is in exact scale to the
force acting upon the Moon to keep it
in Earth’s orbit.

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Development of scientific theories

•One scientific theory can be replaced by another
to explain the same observations. For example,
our understanding of the size of the Universe has
evolved as scientific advancements allow us to
observe more of it.

•Theories may change as a result of new
technologies, e.g. the ability to accurately measure
the differences in wavelength of light (red shift).

•Theories may also change as a result of new
areas of science being developed, such as the
discovery of helium and hydrogen in the
observable universe.

NASA’s James Webb
telescope

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5. Essential science for engineering and
manufacturing

5.4

Measurement equipment, techniques and
principles

PowerPoint 4: Measurement

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Objectives

By the end of this session, learners should be able to:

•Explain the principles of measurement.

•Describe a variety of equipment used for measurement and what
they are used to measure.

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Principles of Measurement

•Precision – measurements should be close to their true value

•Accuracy – the values of measurements should be close to each
other (low scatter)

•Uncertainty - the statistical dispersion of the
values attributed to a measured quantity
should be small

•Resolution – ability to detect small changes
in the characteristic of the measurement

•Calibration – measurements using a device
should be compared to a calibration
standard of known accuracy before use

•Tolerance – the maximum permissible
variations in a dimension.

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Measurement devices

Engineering Rule

•Used to measure linear dimensions

•Accurate to approximately 0.5 mm

Vernier Calliper

Not as accurate as a micrometer but can be used to measure:

•Outside diameter or length

•Inside diameter or length

•Depth of slot or pocket

•High accuracy to 0.02mm

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Vernier calliper

Internal jaws
Locking screw

Imperial scale

Metric scale

External jaws

Depth
measuring
blade

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Example Vernier measurement

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Measurement devices

Outside Micrometer

•Highly accurate to within 0.01mm

•Typically used to measure outside
diameters and outside lengths blocks

•Come in various sizes but always with a
25mm range - for example: 0-25mm, 25-
50mm, 50-75mm

Inside Micrometer

•Highly accurate to within 0.01mm

•Used to measure inside dimensions like
the inside diameter of a hole or tube

•Two types: caliper-type and tubular and rod inside micrometers.

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Outside micrometer

Anvil
Spindle

Thimble

Sleeve

Ratchet
stop

Lock
lever

Frame

Thermal insulator

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Measurement Devices

Depth Micrometer

•Used to measure the depth of
holes, slots, counterbores and
recesses

•Different size rods can be
attached to alter the
measurement range

•Highly accurate to 0.01mm

•Typical range of 25mm

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Measurement Devices

Dial Test Indicator (DTI)

•Extremely accurate to within 0.005mm.

Uses include:

•To check for runout when holding a billet in a lathe.

•In a quality environment to check for flatness of a
particular surface.

•On the workshop floor, to initially set up or calibrate a
machine, prior to a production run.

•By toolmakers (mould makers) in the process of
manufacturing precision tooling.

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Measurement devices

Slip Gauges/Gauge Blocks

•Precision ground and lapped measuring
blocks.

•Used as references for the setting of
measuring equipment such as micrometres,
or dial test indicators (when used in an
inspection role).

•Used to measure slots and grooves with
high accuracy (0.005mm).

•Slip gauges are wrung together to give a
stack of the required dimension.

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Using slip gauges correctly

1.

Use the minimum number of blocks possible to measure each
dimension

2.

Wring the individual blocks together

3.

Wipe the measuring faces clean using soft clean cloth

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Measurement devices

Angle gauges

•Precision ground and lapped measuring blocks of hardened steel,
typically approximately 75mm long and 16mm wide.

•They differ from slip gauges in that their wringing surfaces are not
parallel to each other, but are at an angle (which is engraved on
them).

•They can facilitate measurement of any angle between 0 degrees
and 360 degrees, with an accuracy of 0.5 to 0.25 second (1 second
= 1/60 of a minute = 1/3600 of a degree).

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Go/no-go gauge design

The ‘Go’ gauge should always be so designed that it will cover the
Maximum Metal Condition (MMC – smallest hole, largest shaft), whereas
a ‘NO-GO’ gauge will cover the minimum (Least) Metal Condition (LMC
largest hole, smallest shaft) of a feature, whether external or internal.

The ‘Go’ gauge should always be so designed that it will cover as many
dimensions as possible in a single operation, whereas the ‘NO-GO’
gauge will cover only one dimension.

For example, this means a Go plug gauge should have a full circular
section and be of full length of the hole being checked.

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Go/No Go Principle

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Go/No Go Principle

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Coordinate Measuring Machine (CMM)

A coordinate measuring machine (CMM) is a device that measures the
geometry of physical objects by sensing discrete points on the surface of
the object with a probe.

Various types of probes are used in CMMs, including mechanical, optical,
laser, and white light. Depending on the machine, the probe position may
be manually controlled by an operator or it may be computer controlled.

CMMs typically specify a probe's position in terms of its displacement from
a reference position in a three-dimensional Cartesian coordinate system
(i.e., with XYZ axes).

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5. Essential science for engineering and
manufacturing

5.5

Chemical Composition and Behaviours

PowerPoint 7: Chemical composition- Atomic structure

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Objectives

By the end of this session, learners should be able to:

•Describe the structure of an atom.

•Define the following terms: atom, nucleus, electron, proton, neutron,
valence, valence shell, ion, element, molecule.

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Atoms

An atom is the building block of all
things in the universe.

It is the smallest unit into which matter
can be divided into without there being
a release of electrically charged
particles.

It is the smallest unit of matter that has
the characteristic properties of what is
classified as a chemical element.

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The material of the atom consists of:

•a positively charged nucleus of
protons and neutrons

•This is surrounded
by electrons arranged in shells.

•The nucleus is small and dense which
compares with the electrons, which are
known as the lightest charged particles.

Electrons are attracted to the positive charge by their electric force
In an atom, electrons are bound to the nucleus by electric forces.

The structure of an atom

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Protons, neutrons and electrons
Each atom is made up of electrons and nuclei. The
particles in these atoms are electrically charged and it is
the subsequent electrical force that holds the atoms
together.

Most of the mass of the atom is a positively charged
nucleus.

This is composed of protons, that have a positive charge,
and neutrons, with no charge (neutral).

Protons, neutrons, and the electrons surrounding them
are long-lived particles present in all ordinary, naturally
occurring atoms.

Protons-
Positive

+

Neutrons-

Neutral

Electrons-
Negative

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Valence

Valence describes how easily an
atom or radical can combine with
other chemical species. This is
determined based on the number of
electrons that would be added, lost,
or shared if it reacts with other
atoms.

Valence is denoted using a positive
or negative integer, used to
represent this binding capacity.

For example, common valences of
copper are 1 and 2.

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Valence shell

•The valence shell is the outermost shell that is present in every
element.

•It consists of electrons which are known as the valence electrons.

•The electrons present in the outermost shell participate in bond
formation with the atoms of the other elements.

•These electrons can be lost, gained or shared among the valence
shells of different atoms.

•The properties of all the elements depend upon the number of
electrons that are present in the valence shell of an atom.

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Valence shell and electron

This diagram shows the single
valence electron of sodium
(chemical symbol Na).

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Covalent bonding

•A covalent bond is a chemical bond that involves the sharing of
electrons to form electron pairs between atoms.

•These electron pairs are known as shared pairs or bonding pairs.

•The stable balance of attractive and repulsive forces between atoms,
when they share electrons, is known as covalent bonding.

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Ions

An ion is defined as any atom or group
of atoms that bears one or more
positive or negative electrical charges.

•Positively charged ions are called
cations.

•Negatively charged ions are anions.

Ions are formed by the addition of electrons to, or the removal of
electrons from, neutral atoms, molecules or other ions.

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Elements

•An element is a substance that cannot be
broken down into any other substance.

•Each element is made up of its own
unique type of atom.

•Everything in the universe contains the
atoms of at least one or more elements.

•There are approximately 100 known
elements.

Gold, with the chemical
symbol Au, is a metallic
element solid at room
temperature.

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Elements: principles

•The periodic table lists all the elements.

•They are grouped together according to their similar properties.

•Most elements are metals but there are non-metals that are solid at
room temperature, such as carbon.

•Other elements, such as oxygen and hydrogen are gases.

•Each element has a chemical symbol.

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The periodic table

Elements are
grouped
according to
similar properties,
shown by colour
bands here.

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Molecule

A molecule is defined as a group of
two or more atoms.

For example, a water molecule (H2O)
has one atom of oxygen (O) and two of
hydrogen (H).

This is the smallest unit which a pure
substance can be divided into.

A single molecule will still retain the
composition and chemical properties of
that substance.

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5. Essential science for engineering and
manufacturing

5.5 Chemical composition and behaviours

PowerPoint 8: Chemical composition – Chemical
structure

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Objectives

By the end of this session, learners should be able to define the
following terms:

•Solutions

•Suspensions

•Solubility

•Compound

•Mixture

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Chemical structure

In engineering you will encounter different types of chemical structures.

The two we will consider are:

•Solutions

•Suspensions

There are others (such as colloids, dispersions, gels, sols, aerosols and
emulsions) but they are outside the content of this course.

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A solution is a homogeneous mixture of two or more components.

The dissolving agent is the solvent.

The substance that is dissolved is the solute.

The components of a solution are atoms, ions, or molecules, making
them 10-9m or smaller in diameter.

Example: Sugar and water

Solutions

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Solubility

Solubility can be defined as the degree
to which a substance dissolves in a
solvent to make a solution (usually
expressed as grams of solute per litre of
solvent).

Solubility of one fluid (liquid or gas) in
another may be complete (sometimes
termed miscible; e.g., methanol and
water) or partial (oil and water dissolve
only slightly).

A solution in which no more solute can
be dissolved at a given temperature and
pressure is said to be saturated.

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The particles in suspensions are larger than those found in solutions.

Components of a suspension can be evenly distributed by mechanical
means such as shaking the contents.

However, the components will eventually settle out.

Example: paints with dyes suspended in oil.

Suspensions

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Compounds

A compound is a pure substance that is
made from more than one element. In a
compound, elements are chemically
bonded together, which makes it very
difficult to separate them.

When a compound is made, the atoms of
the elements bond together in a fixed ratio.
This means that each compound can be
represented by a chemical formula.

For example, the formula of water is H O
and the formula of carbon dioxide is CO .

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To break apart the elements in a compound, a chemical reaction must
take place, for example, electrolysis or thermal decomposition.

However, when elements make a compound through a physical process
such as filtration or boiling, it is impossible to separate them.

Types of chemical reaction

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Mixtures

A mixture is formed when two or more elements or compounds are present
without being chemically bonded together.

The substances which have been mixed are not present in specific
amounts or ratios like they are in a compound, e.g. two hydrogen atoms for
each oxygen atom in water.

They can be in any combination, e.g. for a mixture of sand and water you
could have any amount of sand with any amount of water

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In a mixture, the two ingredients can be
separated using physical processes, without
chemical reactions. This is because they are
not chemically bonded together.

•A mixture of sand and water can be
separated using filtration.

•A solution of salt and water can be
separated using crystallisation or
distillation.

•A mixture of iron filings and sulphur
powder can be separated using a magnet.

Separating mixtures

The filtration process
is shown below.

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5. Essential science for engineering and
manufacturing

5.5 Chemical composition and behaviours

PowerPoint 7: Chemical behaviours

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Objectives

By the end of this session, learners should be able to:

•Describe the types of electrical cells and their characteristics

•Describe electrolysis

•Describe the reactions of metals and alloys with acids and alkalis

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Batteries and cells

Usually a battery is made up of
cells. The cell is what converts the
chemical energy into electrical
energy.

A simple cell contains two different
metals (electrodes) separated by a
liquid or paste called an electrolyte.

One metal is more reactive than the
other. Negatively-charged electrons
flow from the more reactive metal
through the wires to the less
reactive metal.

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Primary and secondary cells

Primary cell

Primary cells have high power density
and get discharged slowly. Since
there is no fluid inside these cells they
are also known as dry cells. The
internal resistance is high and the
chemical reaction is irreversible.

Secondary cell

Secondary cells have low energy
density and are made of molten salts
and wet cells. The internal resistance
is low and the chemical reaction is
reversible.

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Battery examples

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Primary versus secondary cells

Primary Cell

Secondary Cell

Have high energy density (cell capacity)

Smaller energy density (cell capacity)

Slow to discharge (output power
capacity)

Can be discharged slightly quicker
(higher output power capacity)

There are no fluids in the cells (hence
also called dry cells)

These are made up of wet cells (flooded
and liquid cells) and molten salt (liquid
cells with different composition)

High internal resistance

Low internal resistance

Irreversible chemical reaction

Has a reversible chemical reaction

Initial cost is low

Initial cost is high

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Electrolysis

Ionic substances contain charged
particles called ions. For example,
lead bromide contains positively
charged lead ions and negatively
charged bromide ions.

Electrolysis is the process by
which ionic substances are
disassociated from their electrode
when an electric current is passed
through them.

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For electrolysis to work, the compound must contain ions. Covalent
compounds cannot act as electrolytes because they contain neutral
atoms.

This is what happens during electrolysis:

•Positively charged ions (cations) move to the negative electrode
(cathode) during electrolysis. They receive electrons and are
reduced.

•Negatively charged ions (anions) move to the positive electrode
(anode) during electrolysis. They lose electrons and are oxidised.

•The cations and anions move through the electrolyte.

The process of electrolysis

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Electroplating is an electrolytic process in
which a metal is disassociated from an
electrode and deposited over any metallic or
non-metallic surface. Electroplating is usually
used to protect the metals from corrosion by
atmospheric air and moisture

Anodising

The electrolysis process of deposition of an
oxide film on a metal surface is known as
anodising. It is mainly used to increase the
thickness of the natural oxide layer on the
surface of the metal parts.

Plating

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Galvanic protection

Galvanic cathodic protection involves
protecting a metal surface of a piece of
equipment using another metal that is more
reactive.

The coating is the galvanic or sacrificial anode.
It has less negative electrochemical potential
compared to the metal component being
protected. Therefore, it undergoes oxidation
rather than the metal it is protecting.

Sometimes, steels are galvanized rather than connected to galvanic
anodes. Galvanized steels are steels that are coated with a protective
zinc layer. The zinc layer acts to cathodically protect steel against
corrosion.

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Metals and acids

Acids react with some metals to produce a salt and hydrogen gas.

Metal + acid → salt + hydrogen

•When a metal is put in acid, it gets smaller and smaller as it gets used
up during the chemical reaction which occurs.

•At the same time, bubbles of gas can be seen. These bubbles
produced in the reaction are hydrogen gas.

•Some metals are very reactive. This means they easily take part in
chemical reactions to make new substances.

•Other metals are very unreactive, and do not easily take part in
chemical reactions.

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Reactivity of metals

Sodium, Potassium

Calcium

Magnesium, Aluminum

Iron, Zinc

Tin, Lead

Copper, Silver, Gold, Platinum

Most reactive

Least reactive

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Reactivity series

Metal

Reaction with dilute acids

Potassium

Violent reaction

Sodium

Calcium

Rapid bubbling

Magnesium

Aluminum

Rapid bubbling with a slow

start

Zinc

Iron

Slow bubbling

Tin

Lead

Very slow bubbling

Copper

Silver

No reaction

Gold

Platinum

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Alloys and acids

An alloy will react to the presence of an acid
or alkali depending upon what the alloy is
made of.

If the alloy mixture contains an element that
is more reactive than the other element within
the alloy, then the reaction would be reduced
in comparison to it being a single element.

Example of alloys include brass, bronze,
stainless steel, nickel steel, titanium and
aluminum alloys.

Titanium alloy

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High risk operations

Being aware of how metals react with acids and alkalis is
important. Many acids react and corrode many metals to
produce hydrogen gas but whether a particular reaction will
occur or not depends on:

•The acid

•The metal

•Whether the acid or alkali is concentrated or dilute

The hydrogen produced during the reaction with acid is
lighter than air and likely to produce flammable/ explosive
atmospheres in confined spaces.

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Applications

These interactions are widely used in
engineering:

•Electroplating can be used to improve
aesthetics or reduce corrosion.

•Galvanic protection can be used to
protect metal parts from corrosion.

•Chemical etching by reaction with acids
or alkalis is used to make markings or
create features on metal surfaces by
reacting away unwanted material.

Chemical etching is used to
make copper traces to
connect different
components on a circuit
board.

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Chemical etching

•Chemical etching is also used to change the
surface texture – for example:

•increasing the roughness for surface finishing
(aesthetics or grip)

•improve the effectiveness of bonding

•to facilitate the retention of oils and
lubricants.

•Conversely these processes could be used to
make the surface less rough, to reduce friction
and lubrication requirements.

Chemically etched
surface to provide a
rough finish.

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5. Essential science for engineering and
manufacturing

5.6 Forces and motion in engineering

PowerPoint 8: Types of motion and forces

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Objectives

By the end of this session, learners should be able to:

•Describe the types of motion.

•Calculate force and pressure.

•Define moments about a force and torque.

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Types of Motion

Motion can be defined as being one of the following types;

•Linear

•Rotary

•Reciprocating

•Oscillating

Each type of motion makes use of an applied force (input) to make a
motion in a set pattern (output)

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Types of Motion

Linear motion moves something in a straight line, such a train moving
down a track.

Rotary motion is where something moves around an axis or pivot point,
such as a bicycle wheel

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Reciprocating motion has a repeated up and down motion or back-
and-forth motion, such as a piston:

Oscillating motion has a curved backwards and forwards movement
that swings on an axis or pivot point, such as a swing or a clock
pendulum:

Types of Motion

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Force

•A FORCE is a push or pull exerted by one object on another.

•Force is generally recognised and measured by its effect on a body.
Force on a static body tends to either move the body or deform it in
some way.

•The SI unit of force is the newton and is defined as:

1 Newton is the force that gives a mass of 1kg

an acceleration of 1ms-2

•As a force has magnitude and direction it is a VECTOR quantity.

•Weight force F = mg where g = 9.81ms-2

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Applied Forces

Any applied force is called an ACTION and the opposing force is called
a REACTION.

Magnitude (size)

Direction

Point of application

A force has 3 defining characteristics:

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Pressure

Next >

p =F

A

Applying a force to any object will create
pressure on it, given by:

Pressure

(Pa)

Area
(m2)

Force

(N)

where:

p = pressure in Pa
F = force in N
A = area in m2

Rearranging this, A = F p and F = p A

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A force of 400 N is applied to an area of 200
mm2. What is the pressure in bar?

Next >

p =
400 N

200 × (1 × 10-6) m2

p = 2 × 106Pa

p = 2 × 106N/m2

p = (2 × 106) × (1 × 10-5) bar

Answer: p = 20 bar

p in Pa = F in N

A in m2

Area = 200 mm2

Force = 400 N

Pressure = ?

Since 1 Pa = 1 × 10-5bar:

Example calculation 1

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Example calculation 2

This cylinder is powered by a
hydraulic pump producing pressure
of 260,000 Pa. What force will the
cylinder produce?

Next >

F = p × A

A = πr2 = 3.14 × (22.5 × 10-3m)2

A = 1.59 × 10-3m2

F = 260,000 Pa × 1.59 × 10-3m2

Force = 413 N

Bore diameter

= 45 mm
First, find the area:

Insert known values into the equation:

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Example calculation 3

Next >

A piece of industrial machinery stands on four
feet, one at each corner. Each foot has an area
of 900 mm2. The force pushing down on each
foot equates to 12,500 N.

What is the pressure on the ground under each
foot?

Give your answer in MPa to 1 d.p.

Pressure =
Force
Area

= 12,500 N

0.0009 m2

Pressure = 13,888,888 Pa ≈ 13.9 MPa

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Next >

A hydraulic system is used to raise a load. The pressure in the
system is 1.2 MPa and the piston has an area of 0.0014 m2.
What force is produced by the piston?

Give your answer in N.

Force = Pressure × Area

Force = 1.2 × 106Pa × 0.0014 m2

Force = 1,680 N

1.2 MPa = 1.2 × 106Pa

Example calculation 4

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For levers to balance, the MOMENTS must be the same.

5N

1.5 m

Moments about a force

•A moment is the tendency of a force to rotate a body.

•A moment is calculated by multiplying the load (Newton) by the
distance from the fulcrum (or pivot) (metre).

•The unit of a moment is Newton/metre (Nm)

•Moments are identified as either clockwise or anti-clockwise.

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Calculating moments: example 1

Anti-clockwise moment = Clockwise moment

6N

A

2.5 m

1.5 m

What force is required at A to balance this lever?

Rearranging 15 ÷ 1.5 = 10 N

6 N x 2.5 m = A N x 1.5 m

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Calculating moments: example 2

Anti-clockwise moment = Clockwise moment

Irrespective of the type of lever, the calculation method is the same.
Calculate the moment and force required at B to balance this lever.

Moment = 12 N x 8 M = 96 N/m

Force: B N x 2 m = 12 N x 8 m, rearranging B = 96 ÷ 2 = 48 N

12N

B

2m

8m

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Torque

• Torque is a moment that causes rotation about an axis.
• A common example of torque occurs when using a spanner.
• The torque being applied by the spanner to this nut is calculated in

a similar way as previous lever examples.

Torque applied = (0.3 x 20) + (0.6 x 10) = 12 N/m

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5. Essential science for engineering and
manufacturing

5.6

Forces and motion in engineering

PowerPoint 9: Vector representation of forces and
coplanar forces

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Objectives

By the end of this session, learners should be able to:

•Carry out calculations using forces represented by vectors.

•State the conditions for equilibrium.

•Carry out calculations for coplanar forces.

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A force has magnitude and direction and can be represented graphically
in two dimensions as a vector. The length of the line represents the
magnitude, with the angle indicating the direction.

y

x

F

θ

Vector representation of force

The force F acts at an angle θ to the
horizontal and as such, can be split into a
vertical and a horizontal component.

The vertical component
𝑽= F sin θ

The horizontal component
𝑯= F cos θ

Resultant magnitude R =
𝒗𝟐

𝑯
𝟐

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Resultant angle θ

•The angle in which the resultant force
is applied can be either measured from
the force diagram using a protractor or
calculated from the equation below

•Resultant angle

𝑽

𝑯

y

x

F

θ

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Sign convention

M +

M -

1. Upward forces are positive (+) and downward forces are negative(-).

2. Horizontal forces acting to the right are positive(+) and to the left are

negative(-).

3. Clockwise acting moments / couples are positive(+) and

anticlockwise ones are negative(-).

+

-

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Calculate the resultant of the two forces shown.

F1 = 7N

F2 = 5N

30°

40°

Force vector example

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FV = 7 sin 70° + 5 sin 30° = 9.08N

Resultant force (R) =
�
�

�
� = √(9.082+6.722) = 11.3N

Resultant angle (θ) =

Resolving vertical components:

��

�

= 53.5°

Force vector example

Resolving horizontal components:
FH = 7 cos 70° + 5 cos 30° = 6.72N

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Equilibrium

Equilibrium means that all forces acting on an object or system are
arranged in a way that there is an overall resultant force of zero. This
means that there can be no acceleration and so the object must either be
static or moving at a constant velocity.

There are several conditions that need to be met in order to bring an
object or system into equilibrium:

•The sum of the forces in the horizontal direction is zero.

•The sum of the forces in the vertical direction is zero.

•The sum of all moments is zero.

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Conditions for static equilibrium

•An object is in equilibrium if the net forces on it are zero.

•This does not mean there are no forces on it – it means that the
forces on the object balance out.

•Mathematically this is represented as:

•

Ʃ FV = 0

•

Ʃ FH = 0

•

Ʃ M = 0

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Equilibrant force

5N@30°

Resultant = 11.3N

7N@70°

53.5o

Equilibrant =
11.3N @ 233.5o

53.5o

This is the force that would need to be present in order to bring the
system into equilibrium. It would need to be equal in magnitude and
opposite in direction to the resultant as shown below.

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Equilibrium conditions

•For a system with two forces, it is in equilibrium when:

F1 + F2 = 0, i.e. F1 = - F2.

•For a system with 3 forces, it is in equilibrium when:

F1 + F2 + F3 = 0, i.e. F1 = - (F2 + F3).

Trigonometry, Pythagoras’ theorem and vector addition may be needed
to calculate the balancing forces.

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Trigonometry and Pythagoras’ Theorem

• Trigonometry may be required to calculate

the angles between force vectors

• Tan θ = O / A

• Rearranging, θ = tan-1(O / A)

• Pythagoras’ Theorem (a2+ b2= c2) may be

needed to calculate the magnitude
O

H

A

θ

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Example calculation

A motion-damping system in a machine is at equilibrium when three
forces are acting on it:

R N, (5i – 8j) N and (3i + 4j) N.

a)

Express R in the form ai + bj

Answer: Rearranging R + F1 + F2 = 0, R = - (F1 + F2)

therefore R = -((5i – 8j) + (3i + 4j)) = (-8i + 4j) N

b)

Calculate the magnitude of R

Answer: Magnitude = √ (i2+ j2) = √ (-82+ 42) = √ (64 + 16) = 8.94 N

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Example calculation

c) Calculate the angle that R makes to the i direction.

Answer: θ = tan-1(4 / -8) = tan-1-0.5 = 26.6o

However, the i value of the vector is negative.

Hence, relative to the i direction the angle of the cable, is

(180 + 26.6) o= 206.6o(or -153.4o)

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Coplanar forces

Forces that act within a two dimensional plane are referred to as
coplanar forces.

If all of the lines of action of these forces pass through the same point,
known as the point of concurrence, then we have a concurrent
coplanar force system.

F4

F1

F3

F2
Point of

concurrence

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Four co-planar forces act at a point O. The values and direction of the
forces are:

Calculate the resultant force and describe the magnitude and direction of
the force required to bring the system into equilibrium.

10N

Coplanar forces example

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Force (N)

Horizontal Component (N)

Vertical Component (N)

50

50 cos 0o= 50 x 1 = 50N

50 sin 0o= 50 x 0 = 0N

10

10 cos 90o= 10 x 0 = 0N

-10 sin 90o= 10 x 1 = -10N

20

-20 cos 30o= -20 x 0.866 = -17.32N

-20 sin 30o= -20 x 0.5 = -10N

30

-30 cos 60o= -30 x 0.5 = -15N

30 sin 60o= 30 x 0.866 = 25.98N

Resultant

50 – 17.32 – 15 = 17.68N

-10 -10 +25.98 = 5.98N

Resultant force R
OB2= OA2+ AB2
R2= (17.68)2+ (5.98)2
R = √312.58 + 35.76
R = 18.66N

��

𝐨

Coplanar Forces Example

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The following force is required to bring the system into equilibrium.

18.69o

198.69o

Resultant =
18.66N @ 18.69o

Equilibrant =
18.66N @ 198.69o

Coplanar Forces Example

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Non-concurrent coplanar force systems

Another type of coplanar system exists where the lines of action of the
forces do not pass through the same point. These type of systems are
called NON-CONCURRENT COPLANAR FORCE SYSTEMS.

In this type of system, there is not only a tendency for the force to
move the body in a certain linear direction, but also to make it rotate.

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Forces in the same plane that
do not meet at 1 point

Resultant force can be
calculated as concurrent forces.

However, a turning force occurs
as well.

This can be calculated using
moments.

F2
F1

F3
F4

Non-concurrent coplanar force systems

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O

Non-concurrent coplanar force system example

The members in this structure
are acted upon by the four
forces shown.

Determine the magnitude and
direction of the resultant force
and the perpendicular
distance of its line of action
from O.

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Force

Angle

FH
FV
MH
MV

4kN

40°

4cos40 =
3.06kN

4sin40 =
2.57kN

0

2.57 x 0.25 =

-0.64kNm

7kN

180°

7cos180 =

-7kN

7sin180 =

0kN

0

8kN

135°

8cos135 =

-5.66kN

8sin135 =

5.66kN

5.66 x 0.4

= -

2.26kNm

0

Total

-9.60kN

8.23kN

kNm

Anti-clockwise

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T-Level Unit 5 Session 15 Recap

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