Virtual Calculator. It is an interactive PDF file just click on the content and you will be directed to the required page

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1 Virtual Calculator Excellent use of Virtual calculator for GATE-2016 It is an interactive PDF file just click on the content and you will be directed to the required page For all Branch of Engineering General Instructions Some functions 1. Exp 2. ln 3. log 4. logy x 5. e x x 7. x y y 8. x 9. x /x 12.sin cos tan sinh cosh tanh 13. sin -1 cos -1 tan -1 sinh -1 cosh - 1 tanh Factorial n (n!) 15. Linear Interpolation 16. Linear regression For Mechanical Engineering Production Engineering Theory of Metal Cutting Shear angle Shear strain Velocity relations Merchant Circle Force Relations Turning Specific Energy Linear Interpolation Tool life equation Linear regression Economics Metrology Rolling Forging Extrusion Wire Drawing Sheet Metal Operation Casting Welding Machine Tools ECM Calculation Strength of Materials Elongation Thermal Stress Principal stresses Deflection of Beams Bending stresses Torsion Spring Theories of column Theories of Failure Theory of Machines Frequency Transmissibility ratio Thermodynamics SFEE Entropy Change Available Energy Heat and Mass Transfer Conduction Unsteady Conduction Heat Exchanger Radiation Industrial Engineering Forecasting Regression Analysis Optimum run size

2 2 P a g e How to use Virtual Calculator General Instructions Operation procedures and sequence of operations are totally different in Virtual calculator. Hence all students are requested to practice the following procedures. It is very weak calculator, can t handle large equation at a time, we have to calculate part by part. Use more and more bracket for calculations BODMAS rule should be followed B Bracket O Order (Power and roots) D Division M Multiplication A Addition S Subtraction For answer must click on = [= means you have to click on this = button] In the starting of any calculation you must click on C [ C means you have to click on this C button] For writing sin30 first write 30 and then click on sin (same procedure should be follow for all trigonometric calculations) [ sin means you have to click on this sin button] Here mod button is simply a showpiece never press mod button. It is indicating calculator is in deg mode or in rad mode. For changing degree mode to radian mode you have to press radio button. Some functions 1. Exp It is actually power of Exp 2 = 100

3 3 P a g e How to use Virtual Calculator 200 GPa 200 Exp 9 = 2e+11 means 2 x Note: Instead of Exp we will use 10 X button often. 2. ln ln2 2 ln = Note: you have to first type value then ln button. 2ln2 2 * 2 ln = ln5 3 * 5 ln =

4 4 P a g e How to use Virtual Calculator 3. log log log = 2 Note: you have to first type value then log button. 5 log50 5 * 50 log = log y x log log y x 10 = 2 Note: you have to first type value of x then logy x button then value of y. Logically value of x should be given first then value of y.

5 5 P a g e How to use Virtual Calculator log logy x 5 = log * ( 50 logy x 5 ) = Note: In this case ( ) is must. if you press 7 * 50 logy x it becomes 350 logx Base y and give wrong answer. But see in case of 5 log50 we simply use 5 * 50 log = and no need of ( ). 5. e X e 2 2 e X = Note: you have to first type value of x then e X button. 5 e 2 5 * 2 e X = e (5 x 3.4 1) 4 * ( 5 x ) e X = e X X = 100 Note: you have to first type value of x then 10 X button.

6 6 P a g e How to use Virtual Calculator 5 x * 2 10 X = /3 (5/3) 10 X = ((1.4 1) ) 1.4 ((1.4 1)/1.4) 10 X = Or you may simplify ( ) (0.4/1.4)10 X = X y x y 3 = 8 Note: you have to first type value of x then x y value of x should be given first then value of y. button then value of y. Logically

7 7 P a g e How to use Virtual Calculator P 2 P 1 γ γ 1 P 2 P 1 γ γ (5/3) x y 1.4/(1.4 1) = y 8. x 5 32 y 32 x 5 = 2 Note: you have to first type value of x then x value of x should be given first then value of y. y button then value of y. Logically We may use x y 5 function also 32 = 32 1/5 = 32 x y (1/5) = 2 But in this case (1/5) is must you can t use 32 x y 1/5 wrong 9. x 5 5 +/- = x = 5

8 8 P a g e How to use Virtual Calculator = Note: you have to first type value then button = = ( 3 x x 2 ) = 5 But ς e = 1 2 ς 1 ς ς 2 ς ς 3 ς 1 2 ς e = Using bracket also we can t calculate it directly, we have to use M +

9 9 P a g e How to use Virtual Calculator x 2 = M + then press C button x 2 = M + then press C button x 2 = M + then press C button Now Press MR button [ It is total value which is under root] Now press button [ it is = ] Now divide it with / 2 = Therefore, ς e = = After the calculation you must press MC button /x This is generally used at middle of calculation. 0.45cos sin12 We first calculate sin12 then use 1/x button * 12 sin =

10 10 P a g e How to use Virtual Calculator Then press 1/x button Then multiply by 0.45 * 12 cos = sin cos tan Calculator must be in degree mode. Always value should be given first then the function.

11 11 P a g e How to use Virtual Calculator sin30 30 sin = 0.5 cos45 45 cos = tan30 30 tan = 0.577

12 12 P a g e How to use Virtual Calculator sin 2 30 (30 sin ) x 2 = 0.25 cos 2 45 (45 cos ) x 2 = 0.5 tan 2 30 (30 tan ) x 2 = sin (A B ) = sin ( ) ( ) sin = cos ( φ + β - α ) = cos ( ) ( ) cos = tan (Φ - α ) = tan ( ) ( ) tan = = = sin 2 θ sin 2 2.0/(20 sin ) x 20 2 = same procedure for sinh cosh tanh 13. sin -1 cos -1 tan -1 Calculator must be in degree mode. If needed in radians calculate by multiplying /180. We may use in rad mode but i will not recommend it because students forget to change the mode to degree and further calculations may go wrong. sin sin -1 = 30 degree

13 13 P a g e How to use Virtual Calculator cos cos -1 = 60 degree tan tan -1 = degree same procedure for sinh -1 cosh -1 tanh Factorial n (n!) You have to first input the value the n! button. 3! 3 n! = 6 5! 5 n! = ! 25 n! = e+25 = x 10 25

14 14 P a g e How to use Virtual Calculator 15. Linear Interpolation formula You have to first calculate upto last form y y 1 y 2 y 1 = x x 1 x 2 x x 10 = x 10 = x = (60 10) * ( ) / ( ) =

15 15 P a g e How to use Virtual Calculator 16. Linear regression analysis Let us assume the equation which best fit the given data y = A + Bx First take summation of both sides y = An + B x.. (i) Next step multiply both side of original equation by x xy = Ax + Bx 2 Again take summation of both sides xy = A x + B x 2.. (ii) Just solve this two equations and find A and B Example: Data x y xy x x x x x = 6 y = 6 xy = 14 x 2 = 14 For x = 6 For y = 6 For xy 1 * * * 3 = 14 For x 2 Use M+ button x 2 M+ then press C button x 2 M+ then press C button x 2 M+ then press C button Then press MR button, Therefore x 2 = 14 Now y = An + B x.. (i) or 6 = 3 A + 6B.. (i)

16 16 P a g e How to use Virtual Calculator and xy = A x + B x 2.. (ii) or 14 = 6A + 14 B.. (ii) Solving (i) and (ii) we get A = 0 and B = 1 y = x is the solution.

17 17 P a g e Production Engineering Theory of Metal Cutting Shear angle (Φ) tan = rcosα 1 rsinα = rcosα 1 rsinα [We have to use one extra bracket in the denominator] tan = 0.45cos sin 12 First find the value of tan 0.45 * 12 cos / ( * 12 sin ) = Then find Just press button tan Shear strain (γ) γ = cot + tan ( α) γ = cot tan ( ) γ = 1 + tan ( ) tan 17.3 It is a long calculation; we have to use M+ 1 tan 17.3 = 1 / 17.3 tan = M+ then press C button tan ( ) = ( ) tan = M+ Then find γ Just press button MR Terefore ( γ) = cot tan ( ) = 3.34

18 18 P a g e Velocity relations V s V = cosα cos α V s 2.5 = cos10 cos V s = 2.5 cos10 cos * 10 cos / (( ) cos ) = Merchant Circle (i) F s = τ s bt = 285 sin sin [we have to use extra bracket for denominator] 285 * 3 * 0.51 / (20.15 sin ) = (ii) F s = Rcos + β α Or R = F s cos + β α = cos [We have to use extra bracket for denominator] / (( ) cos ) = Force Relations F s = F c cos F t sin F s = 900 cos sin * 30 cos * 30 sin =

19 19 P a g e Turning (i) t = fsinλ = 0.32 sin * 75 sin = (ii) F t = F x sinλ = 800 sin 75 [We have to use extra bracket for denominator] 800 / ( 75 sin ) = Specific Energy e = F c = fd [We have to use extra bracket for denominator] 800 / ( 1000 * 0.2 * 2 ) = 2 Linear Interpolation formula You have to first calculate upto last form y y 1 y 2 y 1 = x x 1 x 2 x x 10 = x 10 = x = (60 10) * ( ) / ( ) =

20 20 P a g e Tool life equation (i) V 1 T 1 n = V 2 T 2 n or n = n or 100 = n or 4 3 = 3n or ln 4 3 = nln3 or n = ln 4 3 ln 3 [We have to use extra bracket for denominator] (4/3) ln / ( 3 ln ) = (ii) Find C C = 100 x * 120 x y 0.3 = (iii) V 3 = V 1 T n 1 60 T = * ( 60 / 30 ) x y = (iv) 90 x > 60 x or 90 x = 60 x or 90 x 0.3 = 60 x 0.45 [Make power opposite]

21 21 P a g e or x x 0.3 = or x 0.15 = = 60 xy 0.45 / 90 x y 0.30 = or x = For finding x the just press button x y (1 / 0.15 ) = [Because in the calculator already present] (v) Linear regression analysis Let us assume the equation which best fit the given data y = A + Bx First take summation of both sides y = An + B x.. (i) Next step multiply both side of original equation by x xy = Ax + Bx 2 Again take summation of both sides xy = A x + B x 2.. (ii) Just solve this two equations and find A and B Example: Data X y xy x x x x x = 6 y = 6 xy = 14 x 2 = 14 For x = 6 For y = 6 For xy 1 * * * 3 = 14 For x 2 Use M+ button

22 22 P a g e x 2 M+ then press C button x 2 M+ then press C button x 2 M+ then press C button Then press MR button, Therefore x 2 = 14 Now y = An + B x.. (i) or 6 = 3 A + 6B.. (i) and xy = A x + B x 2.. (ii) or 14 = 6A + 14 B.. (ii) Solving (i) and (ii) we get A = 0 and B = 1 y = x is the solution. Economics in metal cutting T o = T c + C t C m T o = n n To = ( / 0.5 ) (1 0.2 ) / 0.2 = 64 min Now V o T o n = C or V o = 60 or V o = / 64 x y 0.2 = m/min

23 23 P a g e Metrology 3 i = 0.45 D 3 i = D y 0.45 * x 3 = * = Rolling cos α = 1 D = α = 1-5 / 600 = cos -1 = o If you want α in radian after calculating just press * π/180 and you will get α = radian Forging (i) πd = πd d 2 = d = = * ( 50 / 25) = or 100 * 2 = (ii) x s = ln (6 / 2 / 0.25 ) * (1 / 2 / 0.25 ) ln = x s (iii) F sticking = 2 P s + 2K 0 x s x Bdx we have to first integrate without putting values

24 24 P a g e F sticking = 2B P s x + 2K x s x x 2 x s 2 0 F sticking = 2B P s x s + 2K x s 2 x s 2 2 F sticking = 2B P s x s + K x s 2 F Sticking = * 120 * ( * ( 4.04 / 6 ) * x 2 ) = F sticking = N F Sliding = 2 2Ke 2μ x s L L x Bdx F Sliding = 4KB e 2μ x s L L x dx F Sliding = 4KB e2μ L x 2μ L x s F Sliding = 4KB 2μ e 0 e 2μ L x s F Sliding = 2KB μ e 2μ L x s 1 [Note: extra brackets are used] F Sliding = e (2 * 4.04 * 150 * 6 / 0.25) * (((2 * 0.25/6) * ( )) e x - 1) = This is very large calculation; this weak calculator can t handle at once, we have to calculate part by part First calculate (2 * 4.04 * 150 * 6 / 0.25) = Then calculate (((2 * 0.25/6) * ( )) e x - 1) =

25 25 P a g e Now multiply both * = F Sliding = N F Total = F Sticking + F Slding = = N = KN Extrusion F = 2ς o πd o 2 4 ln d o d f F = π 82 4 ln 5 4 It is a long calculation, after some part we press = button then further multiplication is done. 2 * 400 * (π * 8 x 2 / 4) = it gives Now * (5 / 4) ln = N Wire Drawing (i) ς d = ς o 1+B B 1 r f r o 2B ς d = It is a long calculation, First calculate, = 400 * ( ) / = Then calculate, = (1 (5 / 6.25) x y (2 * )) = Now multiply * = MPa [At that time in your calculator is present just multiply it with previous value ]

26 26 P a g e (ii) ς o = ς o 1+B B 1 r fmin r o 2B + r fmin r o 2B ςb 400 = r fmin r fmin Let r fmin = x or 400 = x + x 50 Calculate, = 400 * ( ) / = or 400 = x + x 50 or x = = r fmin or r fmin = or r fmin = 6.25 * 0.4 x y (1 / 2 / ) = mm Sheet Metal Operation (i) C = t τ C = * 1.5 * 294 = mm (ii) F = Ltτ F = 2 a + b tτ = * (100+50) * 5 * 300 = N = 450 KN (iii) D = d 2 + 4d D = [Extra bracket used] ( 25 x * 25 * 15) = mm

27 27 P a g e (iv) t final = t initial e ε 1 e ε 2 = 1.5 e 0.05 e 0.09 [Extra bracket for denominator] 1.5 / ( 0.05 e x * 0.09 e x ) = mm Casting (i) Buoyancy force = πd2 4 ρ liquid ρ core g Buoyancy force = π ( π * 0.12 x 2 / 4 ) * 0.18 * ( ) * 9.81 = N (ii) t s = B V A 2 Find values of V and A separately and then B * (V / A) x 2 = 0 Welding (i) V OCV + I SCC = 1 45 OCV SCC = 1 55 OCV SCC = 1.. (i).. (ii) Now (ii) x 5 - (i) x 4 will give OCV = 5 4 = 1 or OCV = 95 V Now from equation (i)

28 28 P a g e SCC = 1 or 500 SCC = or SCC = / ( 1 45 / 95) = 950 V (ii) H = I 2 Rt = x 2 * 100 * 6 +/- 10 x * = 450 J Machine Tools (i) Turning time ( T ) = L+A+O fn ( L + A + O ) / ( f * N ) = 0 (ii) Drilling time ( T ) = L++A+O fn L = 50 mm = D 2tanα = 15 2 tan59 = 15/ (2 59 tan ) = 4.5 mm A = 2 mm O = 2 mm f = 0.2 mm/rev N = 500 rpm T = ( ) / (0.2 * 500) = min

29 29 P a g e ECM Calculation (i) Find average density of an alloy 1 ρ = x 1 ρ 1 + x 2 ρ 2 + x 3 ρ 3 + x 4 ρ 4 or 1 = ρ First calculate 0.7 / / / / 4.51 = Then just press 1/x button ρ = g/cc (ii) Find equivalent weight of an alloy 1 E = x 1 E 1 + x 2 E 2 + x 3 E 3 + x 4 E 4 or 1 E = x 1v 1 E 1 + x 2v 2 E 2 + x 3v 3 E 3 + x 4v 4 E 4 or 1 = E First calculate 0.7 * 2 / * 2 / * 2 / * 3 / 47.9 = Then just press 1/x button E = Alternate Method 1: First calculate 0.7 * 2 / = Then * 2 / = Then * 2 / = Then * 3 / 47.9 = Then just press 1/x button

30 30 P a g e E = Alternate Method 2: Use M+ button 0.7 * 2 / = press M+ button the press C button 0.2 * 2 / = press M+ button the press C button 0.05 * 2 / = press M+ button the press C button 0.05 * 3 / 47.9 = press M+ button the press MR button Then just press 1/x button E =

31 31 P a g e Strength of Materials (Only for the type of equations which are not yet covered) Elongation (i) δ = PL AE or δ = π mm or δ = π mm [After cancelling common terms from numerator and denominator and one extra bracket in the denominator has to be put] 100 * 4 / ( π * 5 x 2 * 2) = mm Thermal Stress (ii) π First calculate π = π * 0.5 * 4 / (π * 0.01 x 2 * 200 * 6 10 x ) = Then add = Then press button 1/x

32 32 P a g e Then multiply with * 0.5 * 12.5 * 6 +/- 10 x * 20 = Principal stress and principal strain (iii) τ max = σ x σ y τxy 2 τ max = [One bracket for denominator one bracket for square and one for square root] (((80-20) / 2 ) x x 2 ) = 50 MPa For ς 1,2 = ς x +ς y 2 + ς x ς y τxy 2 First calculate ς x +ς y 2 And then calculate ς x ς y τ2 xy Deflection of Beams (iv) δ = wl4 = EI * 3 10 x * 5 x y 4 / (8 * ) = 1 mm

33 33 P a g e Bending stresses (v) ς = My I = = Pa 9.57 * 3 10 x * 12 / (0.2 x y 3 ) = e+7 Pa = MPa Torsion (vi) T J = Gθ L π = π D or D 4 = π First calculate 32 * * 180 = Then calculate π π x 2 * (1 0.7 x y 4) * 80 * 9 10 x = e+11 Now D 4 = = Just press button twice, D = m = mm Spring (vii) δ = 8PD3 n Gd 4

34 34 P a g e *200*310 x 6 +/- 10 x 10 /(80* 9 10 x 8 x y 4 * 12 +/- 10 x ) = m = mm Theories of column (viii) P cr = π2 EI 4L 2 [For one end fixed and other end free] = π π d or = π π d 4 or d 4 = π First calculate * 3 10 x * 4 * 4 x 2 * 64 = e+7 Then calculate π π x 3 * 210 * 9 10 x = e+12 Now d 4 = e e + 12 = Just press button twice, d = m 50 mm Theories of Failure (ix) ς e = 1 2 ς e = 1 2 ς 1 ς ς 2 ς ς 3 ς Using bracket also we can t calculate it directly, we have to use M x 2 = M + then press C button

35 35 P a g e x 2 = M + then press C button x 2 = M + then press C button Now Press MR button [ It is total value which is in under root] Now press button [ it is = ] Now divide it with / 2 = Therefore, ς e = = After the calculation must press MC button.

36 36 P a g e Theory of Machines (Only for the type of equations which are not yet covered) Frequency (i) f n = 1 2π S = 1 M 2π (40 * 10 x 3 / 100 ) / 2 / π = Transmissibility ratio (ii) TR = TR = 1+ 2ξr 2 1 r ξr First calculate 2ξr 2 = (2 * 0.15 * ) x 2 = This data is needed again so PressM+ Next find 1 r 2 2 = ( x 2 ) x 2 = Now find the value of numerator Press MR + 1 = then press Then find denominator Press MR = then press Now Find (TR) Press 1/x and * = TR = (Answer)

37 37 P a g e Thermodynamics (Only for the type of equations which are not yet covered) SFEE (i) 1 + c gz + dq = dm 1 + c gz + dw dm = dw dm M+ M+ M+ M- M- M = Press M+ then press C button 160 x 2 / 2000 = Press M+ then press C button 9.81 * 10 / 1000 = Press M+ then press C button 2600 = Press M- then press C button 100 x 2 / 2000 = Press M- then press C button 9.81 * 6 / 1000 = Press M- Now Press MR and it is answer = dw dm = Entropy Change (ii) S Q S p = c p ln T Q T P Rln P Q P P S Q S p = ln ln M+ M-

38 38 P a g e First calculate ln * (300 / 350 ) ln = Press M+ then press C button Then calculate 0.287ln * (50 /150 ) ln = Press M- Just press MR and it is the answer S = 0.16 KJ/KgK Available Energy (iii) AE = mc p T 2 T 1 T o ln T 2 T 1 AE = ln 1250 First calculate ln ( )-303 * (1250 / 450) ln = Then multiply with * 2000 * 0.5 = KJ = MJ

39 39 P a g e Heat and Mass Transfer (Only for the type of equations which are not covered yet) Conduction (i) Q = 2πL t i t f ln r 2 r1 + ln r3 r2 K A K B 2 π Q = ln ln First calculate denominator ln ln But it is very weak calculator can t calculate two ln in a operation Calculate (0.025 / 0.01) ln / 19 = Press M+ then press C button Then (0.055 / 0.025) ln / 0.2 = Press M+ Then press MR it is denominator Now Press 1/x button Multiply with Numerator 2 π * 2 * π * 600 = W/m 2 π Q = ln ln = W/m

40 40 P a g e Unsteady Conduction (ii) θ θ i = T T a T i T a = e B if o = e 425τ or ln or ln = 425τ = 425τ or τ = ln ((30-300) / ( )) ln = / 425 = / = / 3 +/- 10 x = S Note: Several times use of = is good for this calculator. Heat Exchanger (iii) LMTD = θ i θ o ln θ i θ o = ln (90 / 40) ln = then press 1/x then multiply with numerator * (90 40) = Radiation (iii) Interchange factor f 12 = 1 ε1 +A 1 A2 1 = 1 1 ε First calculate (2 * 3 +/- 10 x / 100) * (1 / ) =

41 41 P a g e Then add 1/ / 0.6 ) = Then press 1/x f12 = Now Q net = f 12 ςa 1 T 1 4 T 2 4 Q net = First calculate * 5.67 * 8 +/- 10 x * 2 * 3 +/- 10 x = e-11 Then multiply with e-11 * (800 x y x y 4) = W Q net = = W

42 42 P a g e Industrial Engineering (Only for the type of equations which are not yet covered) Forecasting (i) u f = αs t + α 1 α S t 1 + α 1 α 2 S t 2 + α 1 α 3 S t 3 u f = M+ M+ M+ M+ 0.4 * 95 = 38 Press M+ then press C button 0.4 * 0.6 * 82 = Press M+ then press C button 0.4 * 0.6 x 2 * 68 = Press M+ then press C button 0.4 * 0.6 x 3 * 70 = Press M+ Then press MR button u f = =73.52 Regression Analysis (ii) Let us assume the equation which best fit the given data y = A + Bx First take summation of both sides y = An + B x.. (i) Next step multiply both side of original equation by x xy = Ax + Bx 2 Again take summation of both sides xy = A x + B x 2.. (ii) Just solve this two equations and find A and B Example:

43 43 P a g e Data x Y Xy x x x x x = 6 y = 6 xy = 14 x 2 = 14 For x = 6 For y = 6 For xy 1 * * * 3 = 14 For x 2 Use M+ button x 2 M+ then press C button x 2 M+ then press C button x 2 M+ then press C button Then press MR button, Therefore x 2 = 14 Now y = An + B x.. (i) or 6 = 3 A + 6B.. (i) and xy = A x + B x 2.. (ii) or 14 = 6A + 14 B.. (ii) Solving (i) and (ii) we get A = 0 and B = 1 y = x is the solution.

44 44 P a g e Optimum run size (iii) Q = 2UR I c I c+i p I p Q = First calculate (2 * *3500 / 2.5) * (( ) / 10) = e+8 Then just press e+8 = END If you got the above points, of the way of calculation then you should be happy enough because we finally succeeded in its usage. Ek Ghatiya Calculator ka Sahi Upyog