Oxford Cambridge and RSA Level 3 Cambridge Technical in Engineering 05822/05823/05824/05825/05873 C Formula Booklet 3 0 6 - 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. © 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 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 = ππππ 2 Circumference of a circle = 2ππππ Co-ordinate equation of a circle: radius r, centre (a, b) (π₯π₯ − ππ)2 + (π¦π¦ − ππ)2 = ππ 2 Arc and sector: radius r, angle θ for θ expressed in radians Arc length = ππππ, 1 2 Area of sector = ππ 2 ππ, for θ expressed in radians Arc length = ππ ππππ, 180 Area of sector = for θ expressed in degrees ππ ππππ 2 , 360 for θ expressed in degrees Converting between radians and degrees x radians = 180π₯π₯ ππ ππππ degrees x degrees = 180 radians 2 1.1.2 Triangles 1 2 Area = ππβ or 1 ππππ sin π΄π΄ 2 1.2 Volume and Surface area of 3D shapes Cuboid Surface area = 2ππππ + 2π€π€β + 2βππ = 2(ππππ + π€π€β + βππ) Volume = ππππβ Sphere Surface area = 4ππππ 2 4 3 Volume = ππππ 3 Cone Surface area = ππππ 2 + ππππππ 1 3 Volume = ππππ 2 β 3 Cylinder Surface area = 2ππππ 2 + 2ππππβ Volume = ππππ 2 β Rectangular Pyramid Volume = ππππβ 3 Prism Volume = area of shaded cross-section × l Density Density = mass volume 4 1.3 Centroids of planar shapes 1.4 Trigonometry c a θ b ππ sin ππ = ππ cos ππ = tan ππ = ππ ππ ππ ππ Pythagoras’ rule: ππ 2 = ππ2 + ππ 2 C b a A Sine rule: ππ sin π΄π΄ Cosine rule: B c = ππ sin π΅π΅ = ππ sin πΆπΆ ππ2 = ππ 2 + ππ 2 − 2ππππ cos π΄π΄ 5 1.4.1 Trigonometric identities Basic trigonometric values sin 60° = √3 2 cos 60° = 1 2 tan 60° = √3 sin 45° = cos 45° = tan 45° = 1 sin 30° = cos 30° = tan 30° = Trigonometric identities 1 √2 1 2 √3 2 1 √3 sin π΄π΄ = cos(90° − π΄π΄) for angle A in degrees cos π΄π΄ = sin(90° − π΄π΄) for angle A in degrees ππ sin π΄π΄ = cos οΏ½π΄π΄ − οΏ½ 2 ππ cos π΄π΄ = −sin οΏ½π΄π΄ − οΏ½ 2 tan π΄π΄ = sin π΄π΄ cos π΄π΄ sin2 π΄π΄ + cos 2 π΄π΄ = 1 sin(−π΄π΄) = −sin π΄π΄ cos(−π΄π΄) = cos π΄π΄ sin(A ± B) = sinAcosB ± cos AsinB cos(A ± B) = cos Acos B β sinAsinB sin 2π΄π΄ = 2 sin π΄π΄ cos π΄π΄ cos 2π΄π΄ = cos2 π΄π΄ − sin2 π΄π΄ 6 2. Calculus 2.1 Differentiation f ( x) df ( x ) dx c 0 xn nx n −1 sin ( ax ) a cos ( ax ) cos ( ax ) −a sin ( ax ) tan ( ax ) a sec 2 ( ax ) e ax ae ax ln ( ax ) 1 x ax a x ln a log a x 1 x ln a 2.1.1 Differentiation of the product of two functions If π¦π¦ = π’π’ × 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 π’π’ If π¦π¦ = If π¦π¦ = π’π’(π£π£) π£π£ dπ¦π¦ dπ₯π₯ = dπ’π’ dπ£π£ π£π£ dπ₯π₯ − π’π’dπ₯π₯ π£π£ 2 2.1.3 Differentiation of a function of a function dπ¦π¦ dπ₯π₯ = dπ’π’ dπ£π£ dπ£π£ dπ₯π₯ 7 2.2 Integration 2.2.1 Indefinite integrals f(π₯π₯) οΏ½ f(π₯π₯) dπ₯π₯ (+ππ) xn x n +1 n +1 a ax for n ≠ −1 1 π₯π₯ ln x eππππ ππππ e ππ ππ π₯π₯ ln ππ ππ π₯π₯ − cos(ππππ) ππ sin(ππππ) sin(ππππ) ππ cos(ππππ) 2.2.2 Definite integral ππ ππ οΏ½ f (π₯π₯) dπ₯π₯ = [F(π₯π₯)]ππ = F(ππ) − F(ππ) ππ 2.2.3 Integration by parts οΏ½ π’π’ dπ£π£ dπ’π’ dπ₯π₯ = π’π’π’π’ − οΏ½ π£π£ dπ₯π₯ dπ₯π₯ dπ₯π₯ 8 3. Algebraic formulae 3.1 Solution of quadratic equation πππ₯π₯ 2 + ππππ + ππ = 0, ππ ≠ 0 ⇒ π₯π₯ = −ππ ± √ππ 2 − 4ππππ 2ππ 3.2 Exponentials/Logarithms π¦π¦ = eππππ ⇒ ln π¦π¦ = ππππ 4. Measurement Absolute error = indicated value – true value Relative error = absolute error true value Absolute correction = true value – indicated value Relative correction = absolute correction true value 9 5. Statistics For a sample, size N, π₯π₯1 , π₯π₯2 , π₯π₯3 , … , π₯π₯ππ , sample mean π₯π₯Μ = standard deviation π₯π₯1 +π₯π₯2 +π₯π₯3 +β―+π₯π₯ππ ππ S= 1 N N ∑ (x − x) i i =1 2 = 1 N N ∑x i =1 2 i − ( x )2 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) – 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) 10 6. Mechanical equations 6.1 Stress and strain equations axial stress (σ) = axial force cross sectional area axial strain (ξ) = change in length original length shear stress (τ) = shear force shear area Young’s modulus (E) = stress strain Working or allowable stress = ultimate stress Factor of Safety (FOS) 6.2 Mechanisms Mechanical advantage (MA) = Velocity ratio (VR) = velocity of input to a mechanism • Class one lever • Class two lever • Class three lever πΉπΉ0 πΉπΉI = input force (or torque) velocity of output from a mechanism 6.2.1 Levers MA = output force (or torque) ππ ππ ππππ = ππ0 ππI 11 = ππ ππ 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 Newton’s equation force = mass x acceleration (πΉπΉ = ππππ) Gravitational potential energy (Wp) = mass x gravitational acceleration x height (mgh) 1 2 Kinetic energy (Wk) = ½ mass x velocity2 ( πππ£π£ 2 ) Work done = force x distance (Fs) Instantaneous power = force x velocity (Fv) ππ π‘π‘ Average power = work done / time ( ) Friction Force ≤ coefficient of friction x normal contact force (πΉπΉ ≤ ππππ) Momentum of a body = mass x velocity (mv) Pressure = force / area ( πΉπΉ π΄π΄ ) 12 6.4 Kinematics Constant acceleration formulae a – acceleration π£π£ 2 = π’π’2 + 2ππππ s – distance 1 π π = π’π’π’π’ + πππ‘π‘ 2 2 t – time u – initial velocity π£π£ = π’π’ + ππππ v – final velocity 1 π π = (π’π’ + π£π£)π‘π‘ 2 1 π π = π£π£π£π£ − πππ‘π‘ 2 2 6.5 Fluid mechanics Pressure due to a column of liquid = height of column × gravitational acceleration × density of liquid (hgρ) Up-thrust force on a submerged body = volume of submerged body × gravitational acceleration × density of liquid (Vgρ) 6.5.1 Energy equations Non-flow energy equation U1 + Q = U2 + W so Q = (U2 – 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 – 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. 13 7. Thermal Physics p – pressure V – volume C – constant T – absolute temperature n – number of moles of a gas R – the gas constant Boyle’s law Charles’ law ππ Pressure law ππ Combined gas law ππ1 ππ1 Ideal gas law ππ1 ππ1 = ππ2 ππ2 ππππ = πΆπΆ ππ ππ ππ1 = πΆπΆ = πΆπΆ ππ1 = ππ1 ππ1 ππ2 ππ2 ππ1 = = ππ2 ππ2 ππ2 ππ2 ππ2 ππππ = ππππππ Characteristic gas law ππππ = ππππππ where m = mass of specific gas and R = specific gas constant Efficiency ππ = 7.1 Heat formulae work output work input 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 14 8. Electrical equations Q = charge V = voltage I = current R = resistance ρ = resistivity P = power E = electric field strength (capacitors) C = capacitance L = inductance t = time l = length τ = time constant W = energy A = cross sectional area Φ = magnetic flux N = number of turns Ζ = angle (in radians) f = Frequency (in cycles per second) ω = 2πf XL, XC = inductive reactance, capacitive reactance Z = impedance Ø = 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 Drift velocity (current) Power Resistance and Ohms law Q = It V = W/Q W = Pt I = nAve P= VI P = I 2R P = V 2/R Series resistance: R = R1 + R2 + R3 +… Parallel resistance: Resistivity Electric field and capacitance Inductance and self-inductance RC circuits AC waveforms AC circuits – resistance and reactance Series RL and RC circuits Ohms law: R = V/I ρ = RA/l E = V/d C = Q/V W = ½QV L =ΦN/ I WL= ½LI 2 τ = RC v = v e-t/RC 0 1 π π = 1 π π 1 V = IR + 1 π π 2 1 π π 3 I = V/R v = V sinΖ i = I sinΖ v = V sinωt i = I sinωt R = V/I XL = V/I and XL = 2πfL 1 XC = V/I and XC = 2πfC 2 2 Z = √(R + XL ) and cosø = R/Z Z = √(R2 + XC2) and cosø = R/Z 15 + +… Series RLC circuits DC motor DC generator DC Series wound self-excited generator DC Shunt wound self-excited generator DC Series wound motor DC Shunt wound motor - No-load conditions: DC Shunt wound motor - Full load conditions: Speed control of DC motors - Shunt motor DC Machine efficiency When XL > XC Z = √[R2 + (XL – XC)2] and cosø = R/Z When XC>XL Z = √[R2 + (XC – XL)2] and cosø = R/Z When XL= XC Z=R V = E + IaRa V = E − IaRa V = E − IaRt Where Rt = Ra + Rf V = E − IaRa Where Ia = If + Il If = V/Rf Il = P/V V = E + IaRt Where Rt = Ra + Rf E ∝ Φn V = E1 + IaRa Where Ia = Il − If If = V/Rf V = E2 + IaRa Where Ia = Il − If E1/E2 = n1/n2 T1/T2 = (Φ1Ia1)/(Φ2Ia2) V = E + IaRa n = (V − IaRa)/(kΦ) Ι³ = output/input Ι³ = 1 – (losses/input) Copyright Information: OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. 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