level 3 cambridge technical in engineering formula bookletformula booklet unit 1 mathematics for...
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Version 5 Last Updated: Oct 2017
Level 3 Cambridge Technical in Engineering
05822/05823/05824/05825/05873
Formula Booklet
Unit 1 Mathematics for engineering
Unit 2 Science for engineering
Unit 3 Principles of mechanical engineering
Unit 4 Principles of electrical and electronic engineering
Unit 23 Applied mathematics for engineering
This booklet contains formulae which learners studying the above units and taking
associated examination papers may need to access.
Other relevant formulae may be provided in some questions within examination papers.
However, in most cases suitable formulae will need to be selected and applied by the
learner. Clean copies of this booklet will be supplied alongside examination papers to be
used for reference during examinations.
Formulae have been organised by topic rather than by unit as some may be suitable for use
in more than one unit or context.
Note for teachers
This booklet does not replace the taught content in the unit specifications or contain an
exhaustive list of required formulae. You should ensure all unit content is taught before
learners take associated examinations.
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2
1. Trigonometry and Geometry
1.1 Geometry of 2D and 3D shapes
1.1.1. Circles and arcs
Circle: radius r
Area of a circle = ๐๐2
Circumference of a circle = 2๐๐
Co-ordinate equation of a circle: radius r, centre (a, b)
(๐ฅ โ ๐)2 + (๐ฆ โ ๐)2 = ๐2
Arc and sector: radius r, angle ฮธ
Arc length = ๐๐, for ฮธ expressed in radians
Area of sector =1
2๐2๐, for ฮธ expressed in radians
Arc length =๐
180๐๐, for ฮธ expressed in degrees
Area of sector =๐
360๐๐2, for ฮธ expressed in degrees
Converting between radians and degrees
x radians =180๐ฅ
๐ degrees
x degrees =๐๐ฅ
180 radians
3
1.1.2 Triangles
Area =1
2๐โ or
1
2๐๐ sin ๐ด
1.2 Volume and Surface area of 3D shapes
Cuboid
Surface area = 2๐๐ค + 2๐คโ + 2โ๐
= 2(๐๐ค + ๐คโ + โ๐)
Volume = ๐๐คโ
Sphere
Surface area = 4๐๐2
Volume =4
3๐๐3
Cone
Surface area = ๐๐2 + ๐๐๐
Volume = 1
3๐๐2โ
4
Cylinder
Surface area = 2๐๐2 + 2๐๐๐
Volume = ๐๐2โ
Rectangular Pyramid
Volume =๐๐คโ
3
Prism
Volume = area of shaded cross-section ร l
Density
Density = mass
volume
5
1.3 Centroids of planar shapes
1.4 Trigonometry
sin ๐ =๐
๐
cos ๐ =๐
๐
tan ๐ =๐
๐
Pythagorasโ rule: ๐2 = ๐2 + ๐2
Sine rule: ๐
sin ๐ด=
๐
sin ๐ต=
๐
sin ๐ถ
Cosine rule: ๐2 = ๐2 + ๐2 โ 2๐๐ cos ๐ด
a
b
c
ฮธ
a
A
b
B
C
c
6
1.4.1 Trigonometric identities
Basic trigonometric values
sin 60ยฐ =โ3
2
cos 60ยฐ =1
2
tan 60ยฐ = โ3
sin 45ยฐ = cos 45ยฐ =1
โ2
tan 45ยฐ = 1
sin 30ยฐ =1
2
cos 30ยฐ =โ3
2
tan 30ยฐ =1
โ3
Trigonometric identities
sin ๐ด = cos(90ยฐ โ ๐ด) for angle A in degrees
cos ๐ด = sin(90ยฐ โ ๐ด) for angle A in degrees
sin ๐ด = cos (๐ด โ๐
2)
cos ๐ด = โsin (๐ด โ๐
2)
tan ๐ด =sin ๐ด
cos ๐ด
sin2 ๐ด + cos2 ๐ด = 1
sin(โ๐ด) = โsin ๐ด
cos(โ๐ด) = cos ๐ด
sin(A ยฑ B) = sinAcosB ยฑ cos AsinB
cos(A ยฑ B) = cos Acos B โ sinAsinB
sin 2๐ด = 2 sin ๐ด cos ๐ด
cos 2๐ด = cos2 ๐ด โ sin2 ๐ด
7
2. Calculus
2.1 Differentiation
2.1.1 Differentiation of the product of two functions
If ๐ฆ = ๐ข ร ๐ฃ d๐ฆ
d๐ฅ= ๐ข
d๐ฃ
d๐ฅ+ ๐ฃ
d๐ข
d๐ฅ
2.1.2 Differentiation of the quotient of two functions
If ๐ฆ =๐ข
๐ฃ
d๐ฆ
d๐ฅ=
๐ฃd๐ข
d๐ฅ โ ๐ข
d๐ฃ
d๐ฅ
๐ฃ2
2.1.3 Differentiation of a function of a function
If ๐ฆ = ๐ข(๐ฃ) d๐ฆ
d๐ฅ=
d๐ข
d๐ฃ
d๐ฃ
d๐ฅ
f x df
d
x
x
c 0
nx 1nnx
sin ax cosa ax
cos ax sina ax
tan ax 2seca ax
eax eaxa
ln ax 1
x
xa lnxa a
loga x 1
lnx a
8
2.2 Integration
2.2.1 Indefinite integrals
2.2.2 Definite integral
โซ f๐
๐
(๐ฅ) d๐ฅ = [F(๐ฅ)]๐
๐= F(๐) โ F(๐)
2.2.3 Integration by parts
โซ ๐ขd๐ฃ
d๐ฅd๐ฅ = ๐ข๐ฃ โ โซ ๐ฃ
d๐ข
d๐ฅd๐ฅ
f(๐ฅ) โซ f(๐ฅ) d๐ฅ (+๐)
a ax
xn
for n โ โ1 1
1
n
xn
1
๐ฅ xln
e๐๐ฅ e๐๐ฅ
๐
๐๐ฅ ๐๐ฅ
ln ๐
sin(๐๐ฅ) โ cos(๐๐ฅ)
๐
cos(๐๐ฅ) sin(๐๐ฅ)
๐
9
3. Algebraic formulae
3.1 Solution of quadratic equation
๐๐ฅ2 + ๐๐ฅ + ๐ = 0, ๐ โ 0
โ ๐ฅ =โ๐ ยฑ โ๐2 โ 4๐๐
2๐
3.2 Exponentials/Logarithms
๐ฆ = e๐๐ฅ โ ln ๐ฆ = ๐๐ฅ
4. Measurement
Absolute error = indicated value โ true value Relative error = absolute error
true value Absolute correction = true value โ indicated value Relative correction = absolute correction
true value
10
5. Statistics
For a sample, size N, ๐ฅ1, ๐ฅ2, ๐ฅ3, โฆ , ๐ฅ๐,
sample mean ๏ฟฝฬ ๏ฟฝ =๐ฅ1+๐ฅ2+๐ฅ3+โฏ+๐ฅ๐
๐
standard deviation
N
i
i
N
i
i xxN
xxN
S1
22
1
2 )(1
)(1
5.1 Probability For events A and B, with probabilities of occurrence P(A) and P(B),
P(A or B) = P(A) + P(B) โ P(A and B)
If A and B are mutually exclusive events,
P(A and B) = 0
P(A or B) = P(A) + P(B)
If A and B are independent events,
P(A and B) = P(A) x P(B)
11
6. Mechanical equations
6.1 Stress and strain equations
axial stress (ฯ) = axial force
cross sectional area
axial strain (ฮพ) = change in length
original length
shear stress (ฯ) = shear force
shear area
Youngโs modulus (E) = stress
strain
Working or allowable stress = ultimate stress Factor of Safety (FOS)
6.2 Mechanisms
Mechanical advantage (MA) = output force (or torque)
input force (or torque)
Velocity ratio (VR) = velocity of output from a mechanism
velocity of input to a mechanism
6.2.1 Levers
Class one lever
Class two lever
Class three lever
MA = ๐น0
๐นI=
๐
๐ ๐๐ =
๐0
๐I=
๐
๐
12
6.2.2 Gear systems
MA = Number of teeth on output gear Number of teeth on input gear
VR = Number of teeth on input gear Number of teeth on output gear
6.2.3 Belt and pulley systems
MA = Diameter of output pulley Diameter of input pulley
VR = Diameter of input pulley Diameter of output pulley
6.3 Dynamics Newtonโs equation force = mass x acceleration (๐น = ๐๐)
Gravitational potential energy (Wp) = mass x gravitational acceleration x height (mgh)
Kinetic energy (Wk) = ยฝ mass x velocity2 (1
2๐๐ฃ2)
Work done = force x distance (Fs)
Instantaneous power = force x velocity (Fv)
Average power = work done / time (๐
๐ก)
Friction Force โค coefficient of friction x normal contact force (๐น โค ๐๐)
Momentum of a body = mass x velocity (mv)
Pressure = force / area ( ๐น
๐ด )
13
6.4 Kinematics
Constant acceleration formulae
a โ acceleration
s โ distance
t โ time
u โ initial velocity
v โ final velocity
6.5 Fluid mechanics
Pressure due to a column of liquid
= height of column ร gravitational acceleration ร density of liquid (hgฯ)
Up-thrust force on a submerged body
= volume of submerged body ร gravitational acceleration ร density of liquid (Vgฯ)
6.5.1 Energy equations
Non-flow energy equation
U1 + Q = U2 + W so Q = (U2 โ U1) + W
where Q = energy entering the system
W = energy leaving the system
U1 = initial energy in the system
U2 = final energy in the system.
Steady flow energy equation
Q = (W2 โ W1) + W
where Q = heat energy supplied to the system
W1 = energy entering the system
W2 = energy leaving the system
W = work done by the system.
๐ฃ2 = ๐ข2 + 2๐๐
๐ = ๐ข๐ก +1
2๐๐ก2
๐ฃ = ๐ข + ๐๐ก
๐ =1
2(๐ข + ๐ฃ)๐ก
๐ = ๐ฃ๐ก โ1
2๐๐ก2
14
7. Thermal Physics p โ pressure
V โ volume
C โ constant
T โ absolute temperature
n โ number of moles of a gas
R โ the gas constant
Boyleโs law ๐๐ = ๐ถ ๐1๐1 = ๐2๐2
Charlesโ law ๐
๐= ๐ถ
๐1
๐1=
๐2
๐2
Pressure law ๐
๐= ๐ถ
๐1
๐1=
๐2
๐2
Combined gas law ๐1๐1
๐1=
๐2๐2
๐2
Ideal gas law ๐๐ = ๐๐ ๐
Characteristic gas law ๐๐ = ๐๐ ๐ where m = mass of specific gas and
R = specific gas constant
Efficiency ๐ =work output
work input
7.1 Heat formulae Latent heat formula
Heat absorbed or emitted during a change of state, Q = mL
where Q = Energy, L = latent heat of transformation, m = mass
Sensible heat formula
Heat energy, Q = mcโT
where Q = Energy, m = mass, c = specific heat capacity of substance,
โT is change in temperature
15
8. Electrical equations
Q = charge
V = voltage
I = current
R = resistance
ฯ = resistivity
P = power
E = electric field strength
(capacitors)
C = capacitance
L = inductance
t = time
l = length
ฯ = time constant
W = energy
A = cross sectional area
= magnetic flux
N = number of turns
ฦ = angle (in radians)
f = Frequency (in cycles per second)
ฯ = 2ฯf
XL, XC = inductive reactance, capacitive reactance
Z = impedance
ร = phase angle
E = emf (motors)
Ia = armature current
If = field current
Il = load current
Ra = armature resistance
Rf = field resistance
n = speed (motors)
T = torque
ษณ = efficiency
Charge and potential energy Q = It
V = W/Q
W = Pt
Drift velocity (current) I = nAve
Power P = V I
P = I 2R
P = V 2/R
Resistance and Ohms law Series resistance: R = R1 + R2 + R3 +โฆ
Parallel resistance: 1
๐ =
1
๐ 1+
1
๐ 2+
1
๐ 3 +โฆ
Ohms law: R = V/I V = IR I = V/R
Resistivity ฯ = RA/l
Electric field and capacitance E = V/d
C = Q/V
W = ยฝQV
Inductance and self-inductance L =N/ I
WL= ยฝLI 2
RC circuits ฯ = RC
v = v0e-t/RC
AC waveforms v = V sinฦ
i = I sinฦ
v = V sinฯt
i = I sinฯt
AC circuits โ resistance and reactance R = V/I
XL = V/I and XL = 2ฯfL
XC = V/I and XC = 1
2ฯfC
Series RL and RC circuits
Z = โ(R2 + XL
2)
and cosรธ = R/Z
Z = โ(R2 + XC
2)
and cosรธ = R/Z
16
Series RLC circuits
When XL > XC
Z = โ[R2 + (XL โ XC)
2] and cosรธ = R/Z
When XC>XL
Z = โ[R2 + (XC โ XL)
2] and cosรธ = R/Z
When XL= XC Z = R
DC motor V = E + IaRa
DC generator V = E โ IaRa
DC Series wound self-excited generator
V = E โ IaRt
Where Rt = Ra + Rf
DC Shunt wound self-excited generator
V = E โ IaRa
Where Ia = If + Il
If = V/Rf
Il = P/V DC Series wound motor
V = E + IaRt
Where Rt = Ra + Rf
E โ n DC Shunt wound motor - No-load conditions:
V = E1 + IaRa
Where Ia = Il โ If
If = V/Rf
DC Shunt wound motor - Full load conditions:
V = E2 + IaRa
Where Ia = Il โ If
E1/E2 = n1/n2
T1/T2 = (1Ia1)/(2Ia2)
Speed control of DC motors - Shunt motor V = E + IaRa
n = (V โ IaRa)/(k)
DC Machine efficiency
ษณ = output/input
ษณ = 1 โ (losses/input)
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