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Engineering Principles
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,UNIT 1
Getting to know your unit
To make an effective contribution to the design and development of
Assessm ent engineered products and systems, you must be able to draw on the
This unit is extern ally principles laid down by the pioneers of engineering science. The
assesse d using an unseen theories developed by the likes of Newton and Ohm are at the heart of
paper-based examin ation the work carried out by today’s multi-skilled engineering workforce.
This unit covers a range of both mechanical and electrical principles and
that is marke d by
some of the necessary mathematics that underpins their application to
Pearson .
solve a range of engineering problems.
How you will be assessed
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This unit is externally assessed by an unseen paper-based examination. The examination is set and marked by Pearson.
Throughout this unit you will find practice activities that will help you to prepare for the examination. At the end of the
unit you will also find help and advice on how to prepare for and approach the examination. The examination must be
taken under examination conditions, so it is important that you are fully prepared and familiar with the application of
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the principles covered in the unit. You will also need to learn key formulae and be confident in carrying out calculations
accurately. A scientific calculator and knowledge of how to use it effectively will be essential.
The examination will be two hours long and will contain a number of short- and long-answer questions. Assessment
will focus on applying appropriate principles and techniques to solving problems. Questions may be focused on a
particular area of study or require the combined use of principles from across the unit. An Information Booklet of
Formulae and Constants will be available during the examination.
This table contains the skills that the examination will be designed to assess.
Assessment objectives
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AO1 Recall basic engineering principles and mathematical methods and formulae
AO2 Perform mathematical procedures to solve engineering problems
AO3 Demonstrate an understanding of electrical, electronic and mechanical principles to solve engineering problems
AO4 Analyse information and systems to solve engineering problems
AO5 Integrate and apply electrical, electronic and mechanical principles to develop an engineering solution
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This table contains the areas of essential content that learners must be familiar with prior to assessment.
Essential content
A1 Algebraic methods
A2 Trigonometric methods
B1 Static engineering systems
B2 Loaded components
C1 Dynamic engineering systems
D1 Fluid systems
D2 Thermodynamic systems
E1 Static and direct current electricity
E2 Direct current circuit theory
E3 Direct current networks
F1 Magnetism
G1 Single-phase alternating current theory
2 Engineering Principles
, Learning aim A UNIT
Getting started
To get started, have a quick look through each topic in this unit
and then in small groups discuss why you think these areas are
considered important enough to be studied by everyone taking a
BTEC National Engineering course. Pick one or more topics and
discuss how they might be relevant to a product, activity or industry
you are familiar with.
A Algebraic and trigonometric mathematical methods
Engineers have to be confident that the solutions they The laws of indices
devise to address practical problems are based on sound ▸▸ When dealing with equations that contain terms
scientific principles. In order to do this,, they often have to
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involving indices, there is a set of basic rules that you
solve complex mathematical problems, so they must be can apply to simplify and help solve them. These are the
comfortable and competent when working with algebra laws of indices. They are summarised in Table 1.1.
and trigonometry.
AF ▸▸ Table 1.1 The laws of indices
A1 Algebraic methods Operation Rule
Algebra allows relationships between variables to be Multiplication am × an = am + n
expressed in mathematical shorthand notation that can be a m
Division ____ = am − n
manipulated to solve problems. In this part of the unit, we an
will consider algebraic expressions involving indices and
Powers (am)n = am × n
logarithms.
Reciprocals 1
___ = a−n
Indices an
Even if you do not yet recognise the term, you will Index = 0 a0 = 1
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already be familiar with the use of indices in common Index = __12 or 0.5 __1
a 2 = √a
__
mathematical expressions. For example:
1 __
Index = __ __1
a n = √a
n
▸▸ 3 × 3 is otherwise known as ‘three squared’ or 32 in n
mathematical notation using indices. Index = 1 a1 = a
▸▸ 5 × 5 × 5 is otherwise known as ‘five cubed’ or 53 in
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mathematical notation using indices.
Key terms
The two parts of the notation used to describe indices are
Expression – a mathematical statement such as a2 + 3
called the base and the index. For example:
or 3a − t. Expressions can easily be recognised because
▸▸ In the expression 32 the base is 3 and the index is 2. they do not contain an equals sign.
Often in engineering mathematics we have to consider Base – the term that is raised to an index, power or
situations where we do not yet know values for the exponent. For example, in the expression 43 the base is 4.
numbers involved. We use algebra to represent unknown Index – the term to which the base is raised. For
numbers with letters or symbols. When applied to indices: example, in the expression 43 the index is 3. The index
▸▸ a × a = a2, where a is the base and 2 is the index. may also be called the power or exponent. The plural of
▸▸ b × b × b = b3, where b is the base and 3 is the index. ‘index’ is ‘indices’.
Where the index is also unknown, it can be represented by Equation – used to equate two expressions that have
a letter as well, for example: equal value, such as a2 + 3 = 19 or t − 1 = 3a + 12.
▸▸ an, where a is the base and n is the index. Equations can easily be recognised because they always
contain an equals sign.
▸▸ bm, where b is the base and m is the index.
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, ▸▸ When using common logarithms, there is no need to
Worked Example include the base 10 in the notation, so where N = 10x,
you can write simply log N = x.
You can use the laws of indices to simplify
This corresponds to the log function on your calculator.
expressions containing indices.
• a2a4a0 a−3.2 = a2 + 4 + 0 − 3.2 = a2.8 Natural logarithms
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• (√ a ) 3 a −1 = (a 0.5) 3 a −1 = a 1.5 a −1 = a 1.5−1 = a 0.5 In a similar way, when dealing with natural logarithms
__ with base e:
or √ a
__3 ▸▸ Where N = e x, then loge N = x.
a2b a 2 −1
• _________ = a1.5b2a−1b−1 = a1.5 − 1b2 – 1 = a0.5b1
b ▸▸ Natural logarithms with base e are so important in
__
or b√ a mathematics that they have their own special notation,
__1
where loge N is written as ln N. So where N = e x, then
a −1 b −1 a 2
• __________ = a−1b−1a0.5b2 = a−1 + 0.5b−1 + 2 = a−0.5b1 ln N = x.
b −2
b__ This corresponds to the ln function on your calculator.
or ___
√a
__
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√a
___ The laws of logarithms
• a (b ) a = a
−3 −2 2 3.5 −3 + 3.5 b = a b or 4
−4 0.5 −4
b There are a number of standard rules that can be used
to simplify and solve equations involving logarithms.
These are the laws of logarithms. They are summarised
Logarithms in Table 1.2.
Logarithms are very closely related to indices.
The logarithm of a number (N) is the power (x) to which a
given base (a) must be raised to give that number.
AF ▸▸ Table 1.2 The laws of logarithms
Operation Common logarithms Natural logarithms
▸▸ In general terms, where N = a x, then loga N = x. Multiplication log AB = log A + log B ln AB = ln A + ln B
In engineering, we encounter mainly common A A
Division log __ = log A − log B ln __ = ln A − ln B
logarithms, which use base 10, and natural logarithms, B B
which use base e. Powers log An = n log A ln An = n ln A
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The Euler number (e) is a mathematical constant that Logarithm of 0 log 0 = not defined ln 0 = not defined
approximates to 2.718. You will come across this again later in
Logarithm of 1 log 1 = 0 ln 1 = 0
the section dealing with natural exponential functions.
Common logarithms
Common logarithms are logarithms with base 10.
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▸▸ Where N = 10 x, then log10 N = x.
Use your calculator to practise finding the common and natural logarithms of
PAUsE POINT
a range of values. Be sure to include whole numbers and decimal fractions less
than 1.
What happens when you try to find the logarithm of 0 or a negative number using
your calculator?
Hint From the relationship between logarithms and indices you know that if log 0 = x,
then 0 = 10x. What value must x take if 10x = 0?
Extend Take a few moments to think about why log 0 and log −1 are not defined and will
produce an error on your calculator.
4 Engineering Principles
