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 6. 10 x 7. x y y 8. x 9. x 10. 11.1/x 12.sin cos tan sinh cosh tanh 13. sin -1 cos -1 tan -1 sinh -1 cosh - 1 tanh -1 14. 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 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 10 10 2 1 Exp 2 = 100
3 P a g e How to use Virtual Calculator 200 GPa 200 Exp 9 = 2e+11 means 2 x 10 11 Note: Instead of Exp we will use 10 X button often. 2. ln ln2 2 ln = 0.6931472 Note: you have to first type value then ln button. 2ln2 2 * 2 ln = 1.386294 3ln5 3 * 5 ln = 4.828314
4 P a g e How to use Virtual Calculator 3. log log100 100 log = 2 Note: you have to first type value then log button. 5 log50 5 * 50 log = 8.494850 4. log y x log10 100 100 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 P a g e How to use Virtual Calculator log5 50 50 logy x 5 = 2.430677 7log5 50 7 * ( 50 logy x 5 ) = 17.01474 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 = 8.494850 and no need of ( ). 5. e X e 2 2 e X = 7.389056 Note: you have to first type value of x then e X button. 5 e 2 5 * 2 e X = 36.94528 4 e (5 x 3.4 1) 4 * ( 5 x 3.4 1 ) e X = 3.554444e+7 6. 10 X 10 2 2 10 X = 100 Note: you have to first type value of x then 10 X button.
6 P a g e How to use Virtual Calculator 5 x 10 2 5 * 2 10 X = 500 10 5/3 (5/3) 10 X = 46.41592 10 1.4 1 1.4 10 ((1.4 1) ) 1.4 ((1.4 1)/1.4) 10 X = 1.930698 Or you may simplify 10 1.4 1 1.4 10 (0.4 1.4 ) (0.4/1.4)10 X = 1.930698 7. X y 2 3 2 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 P a g e How to use Virtual Calculator P 2 P 1 γ γ 1 P 2 P 1 γ γ 1 5 3 1.4 1.4 1 (5/3) x y 1.4/(1.4 1) = 5.111263 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 P a g e How to use Virtual Calculator 10. 5 5 = 2.236068 Note: you have to first type value then button. 3 2 + 4 2 = 3 2 + 4 2 = ( 3 x 2 + 4 x 2 ) = 5 But ς e = 1 2 ς 1 ς 2 2 + ς 2 ς 3 2 + ς 3 ς 1 2 ς e = 1 2 97.74 22.96 2 + 22.96 20 2 + 20 97.74 2 Using bracket also we can t calculate it directly, we have to use M +
9 P a g e How to use Virtual Calculator 97.74 22.96 x 2 = 5592.048 M + then press C button 22.96 20 x 2 = 8.7616 M + then press C button 20 97.74 x 2 = 6043.508 M + then press C button Now Press MR button 11644.32 [ It is total value which is under root] Now press button 107.9089 [ it is = 97.74 22.96 2 + 22.96 20 2 + 20 97.74 2 ] Now divide it with 2 107.9089 / 2 = 76.30309 Therefore, ς e = 1 2 97.74 22.96 2 + 22.96 20 2 + 20 97.74 2 = 76.30309 After the calculation you must press MC button. 11. 1/x This is generally used at middle of calculation. 0.45cos12 1 0.45sin12 We first calculate 1 0.45sin12 then use 1/x button. 1 0.45 * 12 sin = 0.9064397
10 P a g e How to use Virtual Calculator Then press 1/x button 1.103217 Then multiply by 0.45 * 12 cos = 0.4855991 12. sin cos tan Calculator must be in degree mode. Always value should be given first then the function.
11 P a g e How to use Virtual Calculator sin30 30 sin = 0.5 cos45 45 cos = 0.707 tan30 30 tan = 0.577
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 = 0.3333333 sin (A B ) = sin (30-10.5) (30 10.5 ) sin = 0.3338 cos ( φ + β - α ) = cos (20.15 + 33-10 ) ( 20.15 + 33-10) cos = 0.729565 tan (Φ - α ) = tan (17.3 10) (17.3 10 ) tan = 0.128103 2.0 = = sin 2 θ sin 2 2.0/(20 sin ) x 20 2 = 17.09726 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 -1 0.5 0.5 sin -1 = 30 degree
13 P a g e How to use Virtual Calculator cos -1 0.5 0.5 cos -1 = 60 degree tan -1 0.5 0.5 tan -1 = 26.565 degree same procedure for sinh -1 cosh -1 tanh -1 14. Factorial n (n!) You have to first input the value the n! button. 3! 3 n! = 6 5! 5 n! = 120 25! 25 n! = 1.551121 e+25 = 1.551121 x 10 25
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 1 1.8 0.8 x 10 = 2.0 0.8 60 10 x 10 = 60 10 x = 10 + 60 10 1.8 0.8 2.0 0.8 1.8 0.8 2.0 0.8 10 + (60 10) * (1.8 0.8) / (2.0 0.8) = 51.66667
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 2 1 1 1 1 x1 1 2 2 2 2 2 x 2 2 2 3 3 3 3 x 3 3 2 x = 6 y = 6 xy = 14 x 2 = 14 For x 1 + 2 + 3 = 6 For y 1 + 2 + 3 = 6 For xy 1 * 1 + 2 * 2 + 3 * 3 = 14 For x 2 Use M+ button 1 2 1 x 2 M+ then press C button 2 2 2 x 2 M+ then press C button 3 2 3 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 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 = 0 + 1. x is the solution.
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 = 0.45cos12 1 0.45sin 12 First find the value of tan 0.45 * 12 cos / ( 1 0.45 * 12 sin ) = 0.4855991 Then find Just press button tan -1 25.901 Shear strain (γ) γ = cot + tan ( α) γ = cot17.3 + tan (17.3 10) γ = 1 + tan (17.3 10) tan 17.3 It is a long calculation; we have to use M+ 1 tan 17.3 = 1 / 17.3 tan = 3.210630 M+ then press C button tan (17.3 10) = (17.3-10) tan = 0.1281029 M+ Then find γ Just press button MR 3.338732 Terefore ( γ) = cot17.3 + tan (17.3 10) = 3.34
18 P a g e Velocity relations V s V = cosα cos α V s 2.5 = cos10 cos 22.94 10 V s = 2.5 cos10 cos 22.94 10 2.5 * 10 cos / ((22.94-10) cos ) = 2.526173 Merchant Circle (i) F s = τ s bt 3 0.51 = 285 sin sin 20.15 [we have to use extra bracket for denominator] 285 * 3 * 0.51 / (20.15 sin ) = 1265.824 (ii) F s = Rcos + β α Or R = F s cos + β α = 1265.8 cos 20.15 + 33 10 [We have to use extra bracket for denominator] 1265.8 / ((20.15 + 33-10) cos ) = 1735.005 Force Relations F s = F c cos F t sin F s = 900 cos30 600 sin30 900 * 30 cos - 600 * 30 sin = 479.4229
19 P a g e Turning (i) t = fsinλ = 0.32 sin75 0.32 * 75 sin = 0.3091 (ii) F t = F x sinλ = 800 sin 75 [We have to use extra bracket for denominator] 800 / ( 75 sin ) = 828.2209 Specific Energy e = F c = 800 1000fd 1000 0.2 2 [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 1 1.8 0.8 x 10 = 2.0 0.8 60 10 x 10 = 60 10 x = 10 + 60 10 1.8 0.8 2.0 0.8 1.8 0.8 2.0 0.8 10 + (60 10) * (1.8 0.8) / (2.0 0.8) = 51.66667
20 P a g e Tool life equation (i) V 1 T 1 n = V 2 T 2 n or 100 10 n = 75 30 n or 100 = 30 75 10 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 ) = 0.2618593 (ii) Find C C = 100 x 120 0.3 100 * 120 x y 0.3 = 420.4887 (iii) V 3 = V 1 T n 1 60 T = 30 3 30 0.204 30 * ( 60 / 30 ) x y 0.204 = 34.55664 (iv) 90 x 1 0.45 > 60 x 1 0.3 or 90 x 1 0.45 = 60 x 1 0.3 or 90 x 0.3 = 60 x 0.45 [Make power opposite]
21 P a g e or x 0.45 600.45 x 0.3 = 90 0.3 or x 0.15 = 600.45 90 0.3 = 60 xy 0.45 / 90 x y 0.30 = 1.636422 or x = 1.636422 1 0.15 For finding x the just press button x y (1 / 0.15 ) = 26.66667 [Because in the calculator 1.636422 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 2 1 1 1 1 x1 1 2 2 2 2 2 x 2 2 2 3 3 3 3 x 3 3 2 x = 6 y = 6 xy = 14 x 2 = 14 For x 1 + 2 + 3 = 6 For y 1 + 2 + 3 = 6 For xy 1 * 1 + 2 * 2 + 3 * 3 = 14 For x 2 Use M+ button
22 P a g e 1 2 1 x 2 M+ then press C button 2 2 2 x 2 M+ then press C button 3 2 3 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 = 0 + 1. x is the solution. Economics in metal cutting T o = T c + C t C m T o = 3 + 6.5 0.5 1 n n 1 0.2 0.2 To = ( 3 + 6.5 / 0.5 ) (1 0.2 ) / 0.2 = 64 min Now V o T o n = C or V o 64 0.2 = 60 or V o = 60 64 0.2 60 / 64 x y 0.2 = 26.11 m/min
23 P a g e Metrology 3 i = 0.45 D 3 i = 0.45 97.98 + 0.001D + 0.001 97.98 y 0.45 * 97.98 x 3 = + 0.001 * 97.98 = 2.172535 Rolling cos α = 1 D = 1 5 600 α = 1-5 / 600 = cos -1 = 7.40198 o If you want α in radian after calculating 7.40198 just press * π/180 and you will get α = 0.129189 radian Forging (i) πd 1 2 4 1 = πd 2 4 2 2 d 2 = d 1 1 2 = 100 50 25 = 100 2 100 * ( 50 / 25) = 141.4214 or 100 * 2 = 141.4214 (ii) x s = 48 6 2 0.25 ln 1 2 0.25 48 (6 / 2 / 0.25 ) * (1 / 2 / 0.25 ) ln = 39.68223 x s (iii) F sticking = 2 P s + 2K 0 x s x Bdx we have to first integrate without putting values
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 = 2 150 16.16 39.68 + 4.04 6 39.68 2 2 * 120 * ( 16.16 * 39.68 + ( 4.04 / 6 ) * 39.68 x 2 ) = 510418.2 F sticking = 510418.2 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 = 2 4.04 150 6 0.25 e 2 0.25 6 48 39.68 1 (2 * 4.04 * 150 * 6 / 0.25) * (((2 * 0.25/6) * (48 39.68)) 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) = 29088 Then calculate (((2 * 0.25/6) * (48 39.68)) e x - 1) = 1.000372
25 P a g e Now multiply both 29088 * 1.000372 = 29098.82 F Sliding = 29098.82 N F Total = F Sticking + F Slding = 510418.2 + 29098.82 = 539517 N = 539.52 KN Extrusion F = 2ς o πd o 2 4 ln d o d f F = 2 400 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 40212.38 Now 40212.38 * (5 / 4) ln = 8973.135 N Wire Drawing (i) ς d = ς o 1+B B 1 r f r o 2B ς d = 400 1 + 1.7145 1.7145 1 5 6.25 2 1.7145 It is a long calculation, First calculate, 400 1+1.7145 1.7145 = 400 * (1 +1.7145) / 1.7145 = 633.3040 Then calculate, 1 5 6.25 2 1.7145 = (1 (5 / 6.25) x y (2 * 1.7145)) = 0.5347402 Now multiply 0.5347402 * 633.3040 = 338.65 MPa [At that time in your calculator 0.5347402 is present just multiply it with previous value 633.3040]
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 = 400 1 + 1.7145 1.7145 1 r fmin 6.25 2 1.7145 + r fmin 6.25 2 1.7145 50 Let r fmin 6.25 2 1.7145 = x or 400 = 400 1+1.7145 1.7145 1 x + x 50 Calculate, 400 1+1.7145 1.7145 = 400 * (1 +1.7145) / 1.7145 = 633.3 or 400 = 633.3 1 x + x 50 or x = 633.3 400 0.4 = r fmin 633.3 50 6.25 2 1.7145 or r fmin = 6.25 0.4 1 2 1.7145 or r fmin = 6.25 * 0.4 x y (1 / 2 / 1.7145) = 4.784413 mm Sheet Metal Operation (i) C = 0.0032t τ C = 0.0032 1.5 294 0.0032 * 1.5 * 294 = 0.08230286 mm (ii) F = Ltτ F = 2 a + b tτ = 2 100 + 50 5 300 2 * (100+50) * 5 * 300 = 450000 N = 450 KN (iii) D = d 2 + 4d D = 25 2 + 4 25 15 [Extra bracket used] ( 25 x 2 + 4 * 25 * 15) = 46.09772 mm
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 ) = 1.304038 mm Casting (i) Buoyancy force = πd2 4 ρ liquid ρ core g Buoyancy force = π 0.1202 4 0.180 11300 1600 9.81 ( π * 0.12 x 2 / 4 ) * 0.18 * (11300-1600) * 9.81 = 193.7161 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 + 500 SCC = 1 55 OCV + 400 SCC = 1.. (i).. (ii) Now (ii) x 5 - (i) x 4 will give 55 5 45 4 OCV = 5 4 = 1 or OCV = 95 V Now from equation (i)
28 P a g e 45 95 + 500 SCC = 1 or 500 SCC = 1 45 95 or SCC = 500 1 45 95 500 / ( 1 45 / 95) = 950 V (ii) H = I 2 Rt = 30000 2 100 10 6 0.005 30000 x 2 * 100 * 6 +/- 10 x * 0.005 = 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 = 50 + 4.5 + 2 + 2 0.2 500 (50 + 4.5 + 2 + 2 ) / (0.2 * 500) = 0.585 min
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 = 0.7 + 0.2 + 0.05 + 0.05 ρ 8.9 7.19 7.86 4.51 First calculate 0.7 / 8.9 +0.2 / 7.19 +0.05 / 7.86 +0.05 / 4.51 = 0.1239159 Then just press 1/x button ρ = 8.069989 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 = 0.7 2 + 0.2 2 + 0.05 2 + 0.05 3 E 58.71 51.99 55.85 47.9 First calculate 0.7 * 2 / 58.71+0.2 * 2 / 51.99+0.05 * 2 / 55.85+0.05 * 3 / 47.9 = 0.03646185 Then just press 1/x button E = 27.42593 Alternate Method 1: First calculate 0.7 * 2 / 58.71 = 0.02384602 Then 0.02384602 + 0.2 * 2 / 51.99 = 0.03153981 Then 0.03153981 + 0.05 * 2 / 55.85 = 0.03333032 Then 0.03333032 + 0.05 * 3 / 47.9 = 0.03646185 Then just press 1/x button
30 P a g e E = 27.42593 Alternate Method 2: Use M+ button 0.7 * 2 / 58.71 = 0.02384602 press M+ button the press C button 0.2 * 2 / 51.99 = 0.007693788 press M+ button the press C button 0.05 * 2 / 55.85 = 0.001790511 press M+ button the press C button 0.05 * 3 / 47.9 = 0.003131524 press M+ button the press MR button Then just press 1/x button E = 27.42593
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 = 10 103 1000 π 5 2 4 200 103 mm or δ = 100 4 π 5 2 2 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) = 2.546480 mm Thermal Stress (ii) 0.5 12.5 10 6 20 1+ 50 0.5 π 0.01 2 4 200 10 6 First calculate 50 0.5 π 0.01 2 200 10 4 6 = 50 0.5 4 π 0.01 2 200 10 6 50 * 0.5 * 4 / (π * 0.01 x 2 * 200 * 6 10 x ) = 0.001591550 Then add 1 0.001591550 + 1 = 1.001592 Then press button 1/x
32 P a g e 0.9984105 Then multiply with 0.5 12.5 10 6 20 0.9984105 * 0.5 * 12.5 * 6 +/- 10 x * 20 = 0.0001248013 Principal stress and principal strain (iii) τ max = σ x σ y 2 2 + τxy 2 τ max = 80 20 2 2 + 40 2 [One bracket for denominator one bracket for square and one for square root] (((80-20) / 2 ) x 2 + 40 x 2 ) = 50 MPa For ς 1,2 = ς x +ς y 2 + ς x ς y 2 2 + τxy 2 First calculate ς x +ς y 2 And then calculate ς x ς y 2 2 + τ2 xy Deflection of Beams (iv) δ = wl4 = 10 103 5 4 8EI 8 781250 10 * 3 10 x * 5 x y 4 / (8 * 781250 ) = 1 mm
33 P a g e Bending stresses (v) ς = My I = 9.57 103 0.1 0.1 0.2 3 12 = 9.57 103 12 0.2 3 Pa 9.57 * 3 10 x * 12 / (0.2 x y 3 ) = 1.435500e+7 Pa = 14.355 MPa Torsion (vi) T J = Gθ L 409.256 π = 80 109 π 32 1 0.74 D 4 1 180 or D 4 = 32 409.256 180 π 2 1 0.7 4 80 10 9 First calculate 32 * 409.256 * 180 = 2357315 Then calculate π 2 1 0.7 4 80 10 9 π x 2 * (1 0.7 x y 4) * 80 * 9 10 x = 5.999930e+11 Now D 4 = 2357315 5.999930 1011 = 0.000003928904 Just press button twice, D = 0.04452130 m = 44.52 mm Spring (vii) δ = 8PD3 n Gd 4
34 P a g e 8 200 10 3 10 6 10 80 10 9 8 4 10 12 8*200*310 x 6 +/- 10 x 10 /(80* 9 10 x 8 x y 4 * 12 +/- 10 x ) = 0.04882813 m = 48.83 mm Theories of column (viii) P cr = π2 EI 4L 2 [For one end fixed and other end free] 10 10 3 = π 2 210 10 9 π d4 64 4 4 2 or 10 10 3 4 4 2 64 = π 2 210 10 9 π d 4 or d 4 = 10 103 4 4 2 64 π 3 210 10 9 First calculate 10 10 3 4 4 2 64 10 * 3 10 x * 4 * 4 x 2 * 64 = 4.096000e+7 Then calculate π 3 210 10 9 π x 3 * 210 * 9 10 x = 6.511319e+12 Now d 4 = 4.096000e + 7 6.511319e + 12 = 0.000006290584 Just press button twice, d = 0.05008097 m 50 mm Theories of Failure (ix) ς e = 1 2 ς e = 1 2 ς 1 ς 2 2 + ς 2 ς 3 2 + ς 3 ς 1 2 97.74 22.96 2 + 22.96 20 2 + 20 97.74 2 Using bracket also we can t calculate it directly, we have to use M + 97.74 22.96 x 2 = 5592.048 M + then press C button
35 P a g e 22.96 20 x 2 = 8.7616 M + then press C button 20 97.74 x 2 = 6043.508 M + then press C button Now Press MR button 11644.32 [ It is total value which is in under root] Now press button 107.9089 [ it is = 97.74 22.96 2 + 22.96 20 2 + 20 97.74 2 ] Now divide it with 2 107.9089 / 2 = 76.30309 Therefore, ς e = 1 2 97.74 22.96 2 + 22.96 20 2 + 20 97.74 2 = 76.30309 After the calculation must press MC button.
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 3 100 (40 * 10 x 3 / 100 ) / 2 / π = 3.183099 Transmissibility ratio (ii) TR = TR = 1+ 2ξr 2 1 r 2 2 + 2ξr 2 1 + 2 0.15 18.85 2 1 18.85 2 2 + 2 0.15 18.85 2 First calculate 2ξr 2 = 2 0.15 18.85 2 (2 * 0.15 * 18.85 ) x 2 = 31.97903 This data is needed again so PressM+ Next find 1 r 2 2 = 1 18.85 2 2 (1 18.85 x 2 ) x 2 = 125544.4 Now find the value of numerator Press MR + 1 = then press 5.742737 Then find denominator Press MR + 125544.4 = then press 354.3676 Now Find (TR) Press 1/x and * 5.742737 = 0.01620559 TR = 0.01620559 (Answer)
37 P a g e Thermodynamics (Only for the type of equations which are not yet covered) SFEE (i) 1 + c 1 2 + gz + dq = 2000 1000 dm 1 + c 2 1 + gz + dw 2000 1000 dm 3200 + 1602 9.81 10 + + 0 = 2600 + 1002 2000 1000 2000 + 9.81 6 1000 + dw dm M+ M+ M+ M- M- M- 3200 = 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 = 607.8392400000004 dw dm 1602 9.81 10 = 3200 + + 2600 1002 2000 1000 2000 9.81 6 1000 Entropy Change (ii) S Q S p = c p ln T Q T P Rln P Q P P S Q S p = 1.005 ln 300 350 0.287ln 50 150 M+ M-
38 P a g e First calculate 1.005 ln 300 350 1.005 * (300 / 350 ) ln = -0.1549214 Press M+ then press C button Then calculate 0.287ln 50 150 0.287 * (50 /150 ) ln = -0.3153016 Press M- Just press MR and it is the answer 0.16038020000000003 S = 0.16 KJ/KgK Available Energy (iii) AE = mc p T 2 T 1 T o ln T 2 T 1 AE = 2000 0.5 1250 450 303ln 1250 First calculate 1250 450 303ln 1250 450 450 (1250-450)-303 * (1250 / 450) ln = 490.4397 Then multiply with 2000 0.5 490.4397 * 2000 * 0.5 = 490439.7 KJ = 490.44 MJ
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 1 1200 600 Q = ln 0.025 0.055 ln 0.01 + 0.025 19 0.2 First calculate denominator ln 0.025 0.01 19 + ln 0.055 0.025 0.2 But it is very weak calculator can t calculate two ln in a operation Calculate (0.025 / 0.01) ln / 19 = 0.04822583 Press M+ then press C button Then (0.055 / 0.025) ln / 0.2 = 3.942287 Press M+ Then press MR it is denominator 3.9905128299999996 Now Press 1/x button 0.2505944 Multiply with Numerator 2 π 1 1200 600 0.2505944 * 2 * π * 600 = 944.7186 W/m 2 π 1 1200 600 Q = ln 0.025 0.055 ln 0.01 + 0.025 19 0.2 = 944.72 W/m
40 P a g e Unsteady Conduction (ii) θ θ i = T T a T i T a = e B if o 298 300 30 300 = e 425τ 2.3533 10 3 or ln 298 300 30 300 or ln 30 300 298 300 = 425τ 2.3533 10 3 = 425τ 2.3533 10 3 or τ = 30 300 ln 298 300 425 2.3533 10 3 ((30-300) / (298-300)) ln = / 425 = / 2.3533 = / 3 +/- 10 x = 4.904526 S Note: Several times use of = is good for this calculator. Heat Exchanger (iii) LMTD = θ i θ o ln θ i θ o = 90 40 ln 90 40 (90 / 40) ln = then press 1/x then multiply with numerator * (90 40) = 61.65760 Radiation (iii) Interchange factor f 12 = 1 ε1 +A 1 A2 1 = 1 1 ε2 1 1 0.6 +2 10 3 100 1 0.3 1 First calculate 2 10 3 100 1 0.3 1 (2 * 3 +/- 10 x / 100) * (1 / 0.3 1 ) = 0.00004666666
41 P a g e Then add 1/0.6 0.00004666666 + 1 / 0.6 ) = 1.666714 Then press 1/x 0.5999830 f12 =0.5999830 0.6 Now Q net = f 12 ςa 1 T 1 4 T 2 4 Q net = 0.6 5.67 10 8 2 10 3 800 4 300 4 First calculate 0.6 5.67 10 8 2 10 3 0.6 * 5.67 * 8 +/- 10 x * 2 * 3 +/- 10 x = 6.804000e-11 Then multiply with 800 4 300 4 6.804000e-11 * (800 x y 4-300 x y 4) = 27.31806 W Q net = 0.6 5.67 10 8 2 10 3 800 4 300 4 = 27.32 W
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 = 0.4 95 + 0.4 0.6 82 + 0.4 0.6 2 68 + 0.4 0.6 3 70 M+ M+ M+ M+ 0.4 * 95 = 38 Press M+ then press C button 0.4 * 0.6 * 82 = 19.68 Press M+ then press C button 0.4 * 0.6 x 2 * 68 = 19.68 Press M+ then press C button 0.4 * 0.6 x 3 * 70 = 6.048 Press M+ Then press MR button 73.52 u f = 0.4 95 + 0.4 0.6 82 + 0.4 0.6 2 68 + 0.4 0.6 3 70 =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 P a g e Data x Y Xy x 2 1 1 1 1 x1 1 2 2 2 2 2 x 2 2 2 3 3 3 3 x 3 3 2 x = 6 y = 6 xy = 14 x 2 = 14 For x 1 + 2 + 3 = 6 For y 1 + 2 + 3 = 6 For xy 1 * 1 + 2 * 2 + 3 * 3 = 14 For x 2 Use M+ button 1 2 1 x 2 M+ then press C button 2 2 2 x 2 M+ then press C button 3 2 3 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 = 0 + 1. x is the solution.
44 P a g e Optimum run size (iii) Q = 2UR I c I c+i p I p Q = 2 30000 3500 2.5 2.5 + 10 10 First calculate 2 30000 3500 2.5 2.5+10 10 (2 * 30000 *3500 / 2.5) * ((2.5 + 10) / 10) = 1.050000e+8 Then just press 1.050000e+8 = 10246.95 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