Level 3 Cambridge Technical in Engineering

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Level 3 Cmbridge Technicl in Engineering 05822/05823/05824/05825/05873 Formul Booklet Unit 1 Mthemtics for engineering Unit 2 Science for engineering Unit 3 Principles of mechnicl engineering Unit 4

1 Level 3 Cmbridge Technicl in Engineering 05822/05823/05824/05825/05873 Formul Booklet Unit 1 Mthemtics for engineering Unit 2 Science for engineering Unit 3 Principles of mechnicl engineering Unit 4 Principles of electricl nd electronic engineering Unit 23 Applied mthemtics for engineering This booklet contins formule which lerners studying the bove units nd tking ssocited exmintion ppers my need to ccess. Other relevnt formule my be provided in some questions within exmintion ppers. However, in most cses suitble formule will need to be selected nd pplied by the lerner. Clen copies of this booklet will be supplied longside exmintion ppers to be used for reference during exmintions. Formule hve been orgnised by topic rther thn by unit s some my be suitble for use in more thn one unit or context. Note for techers This booklet does not replce the tught content in the unit specifictions or contin n exhustive list of required formule. You should ensure ll unit content is tught before lerners tke ssocited exmintions. OCR 2017 [L/506/7266, R/506/7267, Y/506/7268, D/506/7269, R/506/7270] OCR is n exempt Chrity Version 5 Lst Updted: Oct 2017

2 1. Trigonometry nd Geometry 1.1 Geometry of 2D nd 3D shpes Circles nd rcs Circle: rdius r Are of circle = πr 2 Circumference of circle = 2πr Co-ordinte eqution of circle: rdius r, centre (, b) (x ) 2 + (y b) 2 = r 2 Arc nd sector: rdius r, ngle θ Arc length = θr, for θ expressed in rdins Are of sector = 1 2 r2 θ, for θ expressed in rdins Arc length = θ 180 Are of sector = πr, for θ expressed in degrees θ 360 πr2, for θ expressed in degrees Converting between rdins nd degrees x rdins = 180x π degrees x degrees = πx 180 rdins 2

3 1.1.2 Tringles Are = 1 bh or 1 bc sin A Volume nd Surfce re of 3D shpes Cuboid Surfce re = 2lw + 2wh + 2hl Volume = lwh = 2(lw + wh + hl) Sphere Surfce re = 4πr 2 Volume = 4 3 πr3 Cone Surfce re = πr 2 + πrl Volume = 1 3 πr2 h 3

4 Cylinder Surfce re = 2πr 2 + 2πrl Volume = πr 2 h Rectngulr Pyrmid Volume = lwh 3 Prism Volume = re of shded cross-section l Density Density = mss volume 4

5 1.3 Centroids of plnr shpes 1.4 Trigonometry c θ b sin θ = c cos θ = b c tn θ = b Pythgors rule: c 2 = 2 + b 2 C b A c B Sine rule: = b = c sin A sin B sin C Cosine rule: 2 = b 2 + c 2 2bc cos A 5

6 1.4.1 Trigonometric identities Bsic trigonometric vlues sin 60 = 3 2 cos 60 = 1 2 tn 60 = 3 sin 45 = cos 45 = 1 2 tn 45 = 1 sin 30 = 1 2 cos 30 = 3 2 tn 30 = 1 3 Trigonometric identities sin A = cos(90 A) for ngle A in degrees cos A = sin(90 A) for ngle A in degrees sin A = cos (A π 2 ) cos A = sin (A π 2 ) tn A = sin A cos A sin 2 A + cos 2 A = 1 sin( A) = sin A cos( A) = cos A sin(a ± B) = sinacosb ± cos AsinB cos(a ± B) = cos Acos B sinasinb sin 2A = 2 sin A cos A cos 2A = cos 2 A sin 2 A 6

7 2. Clculus 2.1 Differentition f x df x dx c 0 n n1 x nx sin x cos x cos x sin x tn x 2 sec x e x ln x x e x x 1 x ln log x 1 xln Differentition of the product of two functions If y = u v dy dx dv du = u + v dx dx Differentition of the quotient of two functions If y = u v dy = v dx du dx udv dx v Differentition of function of function If y = u(v) dy = du dx dv dv dx 7

8 2.2 Integrtion Indefinite integrls f(x) f(x) dx (+c) x x n 1 x n for n 1 n 1 1 x e x ln x e x x x ln sin(x) cos(x) cos(x) sin(x) Definite integrl b b f (x) dx = [F(x)] = F(b) F() Integrtion by prts u dv du dx = uv v dx dx dx 8

9 3. Algebric formule 3.1 Solution of qudrtic eqution x 2 + bx + c = 0, Exponentils/Logrithms x = b ± b2 4c 2 y = e x ln y = x 4. Mesurement Absolute error = indicted vlue true vlue Reltive error = bsolute error true vlue Absolute correction = true vlue indicted vlue Reltive correction = bsolute correction true vlue 9

10 5. Sttistics For smple, size N, x 1, x 2, x 3,, x N, smple men x = x 1+x 2 +x 3 + +x N N stndrd devition S 1 N N i1 2 ( x x) i 1 N N i1 x 2 i ( x) Probbility For events A nd B, with probbilities of occurrence P(A) nd P(B), P(A or B) = P(A) + P(B) P(A nd B) If A nd B re mutully exclusive events, P(A nd B) = 0 P(A or B) = P(A) + P(B) If A nd B re independent events, P(A nd B) = P(A) x P(B) 10

11 6. Mechnicl equtions 6.1 Stress nd strin equtions xil stress (σ) = xil force cross sectionl re 6.2 Mechnisms xil strin (ξ) = chnge in length originl length sher stress (τ) = sher force sher re Young s modulus (E) = stress strin Working or llowble stress = ultimte stress Fctor of Sfety (FOS) Mechnicl dvntge (MA) = output force (or torque) input force (or torque) velocity of output from mechnism Velocity rtio (VR) = velocity of input to mechnism Levers Clss one lever Clss two lever Clss three lever MA = F 0 F I = b VR = V 0 V I = b 11

12 6.2.2 Ger systems MA = Number of teeth on output ger Number of teeth on input ger VR = Number of teeth on input ger Number of teeth on output ger Belt nd pulley systems MA = Dimeter of output pulley Dimeter of input pulley VR = Dimeter of input pulley Dimeter of output pulley 6.3 Dynmics Newton s eqution force = mss x ccelertion (F = m) Grvittionl potentil energy (W p ) = mss x grvittionl ccelertion x height (mgh) Kinetic energy (W k ) = ½ mss x velocity 2 ( 1 2 mv2 ) Work done = force x distnce (Fs) Instntneous power = force x velocity (Fv) Averge power = work done / time ( W ) t Friction Force coefficient of friction x norml contct force (F μn) Momentum of body = mss x velocity (mv) Pressure = force / re ( F ) A 12

13 6.4 Kinemtics Constnt ccelertion formule ccelertion s distnce t time u initil velocity v finl velocity v 2 = u 2 + 2s s = ut t2 v = u + t s = 1 (u + v)t 2 s = vt 1 2 t2 6.5 Fluid mechnics Pressure due to column of liquid = height of column grvittionl ccelertion density of liquid (hgρ) Up-thrust force on submerged body = volume of submerged body grvittionl ccelertion density of liquid (Vgρ) Energy equtions Non-flow energy eqution U 1 + Q = U 2 + W so Q = (U 2 U 1 ) + W where Q = energy entering the system W = energy leving the system U 1 = initil energy in the system U 2 = finl energy in the system. Stedy flow energy eqution Q = (W 2 W 1 ) + W where Q = het energy supplied to the system W 1 = energy entering the system W 2 = energy leving the system W = work done by the system. 13

14 7. Therml Physics p pressure V volume C constnt T bsolute temperture n number of moles of gs R the gs constnt Boyle s lw pv = C p 1 V 1 = p 2 V 2 Chrles lw Pressure lw Combined gs lw Idel gs lw Chrcteristic gs lw Efficiency V T = C V 1 T 1 = V 2 T 2 p T = C p 1 T 1 = p 2 T 2 p 1 V 1 T 1 = p 2V 2 T 2 pv = nrt pv = mrt where m = mss of specific gs nd R = specific gs constnt η = work output work input 7.1 Het formule Ltent het formul Het bsorbed or emitted during chnge of stte, Q = ml where Q = Energy, L = ltent het of trnsformtion, m = mss Sensible het formul Het energy, Q = mc T where Q = Energy, m = mss, c = specific het cpcity of substnce, T is chnge in temperture 14

15 8. Electricl equtions Q = chrge V = voltge I = current R = resistnce ρ = resistivity P = power E = electric field strength (cpcitors) C = cpcitnce L = inductnce t = time l = length τ = time constnt W = energy A = cross sectionl re = mgnetic flux N = number of turns Ɵ = ngle (in rdins) f = Frequency (in cycles per second) ω = 2πf X L, X C = inductive rectnce, cpcitive rectnce Z = impednce Ø = phse ngle E = emf (motors) I = rmture current I f = field current I l = lod current R = rmture resistnce R f = field resistnce n = speed (motors) T = torque ɳ = efficiency Chrge nd potentil energy Drift velocity (current) Power Resistnce nd Ohms lw Resistivity Electric field nd cpcitnce Inductnce nd self-inductnce RC circuits AC wveforms AC circuits resistnce nd rectnce Series RL nd RC circuits Q = It V = W/Q W = Pt I = nave P = V I P = I 2 R P = V 2 /R Series resistnce: R = R 1 + R 2 + R 3 + Prllel resistnce: 1 R = 1 R R R 3 + Ohms lw: R = V/I V = IR I = V/R ρ = RA/l E = V/d C = Q/V W = ½QV L =N/ I W L = ½LI 2 τ = RC v = v 0 e -t/rc v = V sinɵ i = I sinɵ v = V sinωt i = I sinωt R = V/I X L = V/I nd X L = 2πfL 1 X C = V/I nd X C = 2πfC Z = (R 2 + X 2 L ) nd cosø = R/Z Z = (R 2 + X 2 C ) nd cosø = R/Z 15

16 Series RLC circuits DC motor DC genertor DC Series wound self-excited genertor DC Shunt wound self-excited genertor DC Series wound motor DC Shunt wound motor - No-lod conditions: DC Shunt wound motor - Full lod conditions: Speed control of DC motors - Shunt motor DC Mchine efficiency When X L > X C Z = [R 2 + (X L X C ) 2 ] nd cosø = R/Z When X C >X L Z = [R 2 + (X C X L ) 2 ] nd cosø = R/Z When X L = X C Z = R V = E + I R V = E I R V = E I R t Where R t = R + R f V = E I R Where I = I f + I l I f = V/R f I l = P/V V = E + I R t Where R t = R + R f E n V = E 1 + I R Where I = I l I f I f = V/R f V = E 2 + I R Where I = I l I f E 1 /E 2 = n 1 /n 2 T 1 /T 2 = ( 1 I 1 )/( 2 I 2 ) V = E + I R n = (V I R )/(k) ɳ = output/input ɳ = 1 (losses/input) Copyright Informtion: OCR is committed to seeking permission to reproduce ll third-prty content tht it uses in its ssessment mterils. OCR hs ttempted to identify nd contct ll copyright holders whose work is used in this pper. To void the issue of disclosure of nswer-relted informtion to cndidtes, ll copyright cknowledgements re reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for ech series of exmintions nd is freely vilble to downlod from our public website ( fter the live exmintion series. If OCR hs unwittingly filed to correctly cknowledge or cler ny third-prty content in this ssessment mteril OCR will be hppy to correct its mistke t the erliest possible opportunity. For queries or further informtion plese contct the Copyright Tem, First Floor, 9 Hills Rod, Cmbridge CB2 1GE. OCR is prt of the Cmbridge Assessment Group. Cmbridge Assessment is the brnd nme of University of Cmbridge Locl Exmintions Syndicte (UCLES), which is itself deprtment of the University of Cmbridge. OCR

6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS

6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS 6. CONCEPTS FOR ADVANCED MATHEMATICS, C (475) AS Objectives To introduce students to number of topics which re fundmentl to the dvnced study of mthemtics. Assessment Emintion (7 mrks) 1 hour 30 minutes.