level 3 cambridge technical in engineering formula booklet
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Oxford Cambridge and RSA
Level 3 Cambridge Technical in Engineering05822/05823/05824/05825/05873
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.
Β© OCR 2021 [L/506/7266, R/506/7267, Y/506/7268, D/506/7269, R/506/7270]
C306/XXXX/1 OCR is an exempt Charity Version 6
C306--
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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 = ππππ2
Circumference of a circle = 2ππππ
Co-ordinate equation of a circle: radius r, centre (a, b)
(π₯π₯ β ππ)2 + (π¦π¦ β ππ)2 = ππ2
Arc and sector: radius r, angle ΞΈ
Arc length = ππππ, for ΞΈ expressed in radians
Area of sector = 12ππ2ππ, for ΞΈ expressed in radians
Arc length = ππ180
ππππ, for ΞΈ expressed in degrees
Area of sector = ππ360
ππππ2, for ΞΈ expressed in degrees
Converting between radians and degrees
x radians = 180π₯π₯ππ
degrees
x degrees = πππ₯π₯180
radians
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1.1.2 Triangles
Area = 1
2ππβ or 1
2ππππ sinπ΄π΄
1.2 Volume and Surface area of 3D shapes
Cuboid
Surface area = 2ππππ + 2ππβ + 2βππ
= 2(ππππ + ππβ + βππ)
Volume = ππππβ
Sphere
Surface area = 4ππππ2
Volume = 43ππππ3
Cone
Surface area = ππππ2 + ππππππ
Volume = 13ππππ2β
4
Cylinder
Surface area = 2ππππ2 + 2ππππβ
Volume = ππππ2β
Rectangular Pyramid
Volume = ππππβ3
Prism
Volume = area of shaded cross-section Γ l
Density
Density = massvolume
5
1.3 Centroids of planar shapes
1.4 Trigonometry
sinππ =ππππ
cosππ =ππππ
tanππ =ππππ
Pythagorasβ rule: ππ2 = ππ2 + ππ2
Sine rule: ππ
sinπ΄π΄= ππ
sinπ΅π΅= ππ
sinπΆπΆ
Cosine rule: ππ2 = ππ2 + ππ2 β 2ππππ cosπ΄π΄
a
b
c
ΞΈ
a
A
b
B
C
c
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1.4.1 Trigonometric identities
Basic trigonometric values
sin 60Β° =β32
cos 60Β° =12
tan 60Β° = β3
sin 45Β° = cos 45Β° =1β2
tan 45Β° = 1
sin 30Β° =12
cos 30Β° =β32
tan 30Β° =1β3
Trigonometric identities
sinπ΄π΄ = cos(90Β° β π΄π΄) for angle A in degrees
cosπ΄π΄ = sin(90Β° β π΄π΄) for angle A in degrees
sinπ΄π΄ = cos οΏ½π΄π΄ βππ2οΏ½
cosπ΄π΄ = βsin οΏ½π΄π΄ βππ2οΏ½
tanπ΄π΄ =sinπ΄π΄cosπ΄π΄
sin2 π΄π΄ + cos2 π΄π΄ = 1
sin(βπ΄π΄) = βsinπ΄π΄
cos(βπ΄π΄) = cosπ΄π΄
sin(A Β± B) = sinAcosB Β± cos AsinB
cos(A Β± B) = cos Acos B β sinAsinB
sin 2π΄π΄ = 2 sinπ΄π΄ cosπ΄π΄
cos 2π΄π΄ = cos2 π΄π΄ β sin2 π΄π΄
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2. Calculus
2.1 Differentiation
2.1.1 Differentiation of the product of two functions If π¦π¦ = π’π’ Γ π£π£
dπ¦π¦dπ₯π₯
= π’π’ dπ£π£dπ₯π₯
+ π£π£ dπ’π’dπ₯π₯
2.1.2 Differentiation of the quotient of two functions
If π¦π¦ = π’π’π£π£
dπ¦π¦dπ₯π₯
=π£π£dπ’π’dπ₯π₯ β π’π’dπ£π£dπ₯π₯
π£π£2
2.1.3 Differentiation of a function of a function
If π¦π¦ = π’π’(π£π£) dπ¦π¦dπ₯π₯
= dπ’π’dπ£π£
dπ£π£dπ₯π₯
( )f x ( )dfd
xx
c 0
nx 1βnnx
( )sin ax ( )cosa ax
( )cos ax ( )sinβa ax
( )tan ax ( )2seca ax
eax eaxa
( )ln ax 1x
xa lnxa a
loga x 1lnx a
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2.2 Integration
2.2.1 Indefinite integrals
2.2.2 Definite integral
οΏ½ fππ
ππ(π₯π₯) dπ₯π₯ = [F(π₯π₯)]
ππππ = F(ππ) β F(ππ)
2.2.3 Integration by parts
οΏ½π’π’dπ£π£dπ₯π₯
dπ₯π₯ = π’π’π£π£ β οΏ½π£π£dπ’π’dπ₯π₯
dπ₯π₯
f(π₯π₯) οΏ½ f(π₯π₯) dπ₯π₯ (+ππ)
a ax
xn
for n β β1 1
1
+
+
nxn
1π₯π₯
xln
eπππ₯π₯ eπππ₯π₯ππ
πππ₯π₯ πππ₯π₯
lnππ
sin(πππ₯π₯) β cos(πππ₯π₯)ππ
cos(πππ₯π₯) sin(πππ₯π₯)ππ
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3. Algebraic formulae
3.1 Solution of quadratic equation πππ₯π₯2 + πππ₯π₯ + ππ = 0, ππ β 0
β π₯π₯ =βππ Β± βππ2 β 4ππππ
2ππ
3.2 Exponentials/Logarithms
π¦π¦ = eπππ₯π₯ β lnπ¦π¦ = πππ₯π₯
4. Measurement Absolute error = indicated value β true value Relative error = absolute error
true value Absolute correction = true value β indicated value Relative correction = absolute correction
true value
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5. Statistics For a sample, size N, π₯π₯1, π₯π₯2, π₯π₯3, β¦ , π₯π₯ππ, sample mean οΏ½Μ οΏ½π₯ = π₯π₯1+π₯π₯2+π₯π₯3+β―+π₯π₯ππ
ππ
standard deviation ββ==
β=β=N
ii
N
ii xx
Nxx
NS
1
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) β 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 (Ο) = axial force
cross sectional area
axial strain (ΞΎ) = change in length original length
shear stress (Ο) = shear force shear area
Youngβs 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
β’ Class one lever
β’ Class two lever
β’ Class three lever
MA = πΉπΉ0
πΉπΉI= ππ
ππ ππππ = ππ0
ππI= ππ
ππ
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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 Newtonβs equation force = mass x acceleration (πΉπΉ = ππππ) Gravitational potential energy (Wp) = mass x gravitational acceleration x height (mgh) Kinetic energy (Wk) = Β½ mass x velocity2 (1
2πππ£π£2)
Work done = force x distance (Fs) Instantaneous power = force x velocity (Fv) Average power = work done / time (ππ
π‘π‘)
Friction Force β€ coefficient of friction x normal contact force (πΉπΉ β€ ππππ) Momentum of a body = mass x velocity (mv) Pressure = force / area ( πΉπΉ
π΄π΄ )
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6.4 Kinematics Constant acceleration formulae
a β acceleration
s β distance
t β time
u β initial velocity
v β final velocity
6.5 Fluid mechanics Pressure due to a column of liquid
= height of column Γ gravitational acceleration Γ density of liquid (hgΟ) Up-thrust force on a submerged body
= volume of submerged body Γ gravitational acceleration Γ density of liquid (VgΟ) 6.5.1 Energy equations Non-flow energy equation U1 + Q = U2 + W so Q = (U2 β 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 β 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.
π£π£2 = π’π’2 + 2ππππ
ππ = π’π’π’π’ +12πππ’π’2
π£π£ = π’π’ + πππ’π’
ππ =12
(π’π’ + π£π£)π’π’
ππ = π£π£π’π’ β12πππ’π’2
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7. Thermal Physics p β pressure V β volume C β constant T β absolute temperature n β number of moles of a gas R β the gas constant Boyleβs law ππππ = πΆπΆ ππ1ππ1 = ππ2ππ2
Charlesβ law ππππ
= πΆπΆ ππ1ππ1
= ππ2ππ2
Pressure law
ππππ
= πΆπΆ ππ1ππ1
= ππ2ππ2
Combined gas law
ππ1ππ1ππ1
= ππ2ππ2ππ2
Ideal gas law ππππ = ππππππ Characteristic gas law ππππ = ππππππ where m = mass of specific gas and
R = specific gas constant
Efficiency ππ = work outputwork 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 = mcβT
where Q = Energy, m = mass, c = specific heat capacity of substance, βT is change in temperature
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8. Electrical equations Q = charge V = voltage I = current R = resistance Ο = resistivity P = power E = electric field strength (capacitors) C = capacitance L = inductance t = time l = length Ο = time constant W = energy A = cross sectional area Ξ¦ = magnetic flux
N = number of turns Ζ = angle (in radians) f = Frequency (in cycles per second) Ο = 2Οf XL, XC = inductive reactance, capacitive reactance Z = impedance Γ = 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 Ι³ = 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 +β¦
Parallel resistance: 1π π
= 1π π 1
+ 1π π 2
+ 1π π 3
+β¦
Ohms law: R = V/I V = IR I = V/R Resistivity Ο = RA/l Electric field and capacitance E = V/d
C = Q/V W = Β½QV
Inductance and self-inductance L =Ξ¦N/ I WL= Β½LI 2
RC circuits Ο = RC v = v0e-t/RC
AC waveforms v = V sinΖ i = I sinΖ v = V sinΟt i = I sinΟt
AC circuits β resistance and reactance R = V/I XL = V/I and XL = 2ΟfL
XC = V/I and XC = 1
2ΟfC
Series RL and RC circuits
Z = β(R2 + XL2) and cosΓΈ = R/Z
Z = β(R2 + XC2) and cosΓΈ = R/Z
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Series RLC circuits
When XL > XC Z = β[R2 + (XL β XC)2] and cosΓΈ = R/Z When XC>XL Z = β[R2 + (XC β XL)2] and cosΓΈ = R/Z When XL= XC Z = R
DC motor V = E + IaRa DC generator V = E β IaRa DC Series wound self-excited generator
V = E β IaRt Where Rt = Ra + Rf
DC Shunt wound self-excited generator
V = E β IaRa Where Ia = If + Il If = V/Rf Il = P/V
DC Series wound motor
V = E + IaRt Where Rt = Ra + Rf E β Ξ¦n
DC Shunt wound motor - No-load conditions:
V = E1 + IaRa Where Ia = Il β If If = V/Rf
DC Shunt wound motor - Full load conditions:
V = E2 + IaRa Where Ia = Il β If E1/E2 = n1/n2
T1/T2 = (Ξ¦1Ia1)/(Ξ¦2Ia2) Speed control of DC motors - Shunt motor V = E + IaRa
n = (V β IaRa)/(kΞ¦) DC Machine efficiency
Ι³ = output/input Ι³ = 1 β (losses/input)
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