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

For Mechanical Engineering

General Instructions

Production Engineering

Some functions

Theory of Metal Cutting

1. Exp

Shear angle

2. ln 3. log 4. logyx 5. ex 6.

10x

7.

xy 𝒚

Shear strain Velocity relations Merchant Circle Force Relations Turning Specific Energy Linear Interpolation

8. √𝒙

Tool life equation

10. √

Economics

11.1/x

Metrology

12.sin cos tan sinh cosh tanh

Rolling

9. |𝒙|

13. sin-1 cos-1 tan-1 sinh-1 cosh1

tanh-1

14. Factorial n (n!) 15. Linear Interpolation 16. Linear regression

Linear regression

Forging Extrusion Wire Drawing Sheet Metal Operation Casting Welding Machine Tools

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

ECM Calculation

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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 10 102

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1 Exp 2 =

100

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200 GPa

200 Exp 9 =

2e+11 means 2 x 1011

Note: Instead of Exp we will use 10X 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

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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. logyx log10100

100

logyx

10 =

2

Note: you have to first type value of x then logyx button then value of y. Logically value of x should be given first then value of y.

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log550

50 logyx 5

7log550

7 * ( 50 logyx 5 ) =

=

2.430677 17.01474

Note: In this case ( ) is must. if you press 7 * 50 logyx 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. eX e2

2 eX

=

7.389056

Note: you have to first type value of x then eX button. 5 e2

5 * 2 eX

=

36.94528

4 e(5 x 3.4 – 1)

4 * ( 5 x 3.4 – 1 ) eX =

3.554444e+7

6. 10X 102

2 10X

=

100

Note: you have to first type value of x then 10X button.

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5 x 102

5 * 2 10X

=

105/3

(5/3) 10X

=

10

1.4−1 1.4

(1.4−1) ) 1.4

10(

Or you may simplify

10

1.4−1 1.4

0.4

10(1.4)

500

46.41592

((1.4 − 1)/1.4) 10X = (0.4/1.4)10X =

1.930698

1.930698

7. Xy 23

2 xy

3 =

8

Note: you have to first type value of x then xy button then value of y. Logically value of x should be given first then value of y.

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𝛾

1.4

𝛾

𝑃2 𝛾−1 𝑃2 (𝛾−1) 5 (1.4−1) ⟹ � � ⟹ � � � � 𝑃1 𝑃1 3 (5/3) xy 1.4/(1.4 – 1) =

5.111263

𝑦

8. √𝑥 5

√32

32 √𝑥 5 =

2

𝑦

Note: you have to first type value of x then √𝑥 button then value of y. Logically value of x should be given first then value of y. 𝑦

We may use xy function also √32 = 321/5 = 32 xy (1/5) =

2

5

But in this case (1/5) is must you can’t use 32 xy 1/5 → wrong

9. |𝑥 |

|−5|

Made Easy

5 +/- = |𝑥| =

5

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10. √ √5

5 √

=

2.236068

Note: you have to first type value then √ button. √32 + 42 =�(32 + 42 ) = ( 3 x2 + 4 x2 ) √

=

5

But

𝜎𝑒 =

𝜎𝑒 =

1

√2 1

√2

�[(𝜎1 − 𝜎2 )2 + (𝜎2 − 𝜎3 )2 + (𝜎3 − 𝜎1 )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+

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(97.74 − 22.96) x2 = (22.96 − 20)

(20 − 97.74)

5592.048

M+ then press C button

x2 =

8.7616

M+ then press C button

x2 =

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 √ Therefore, 𝜎𝑒 =

= 1

√2

76.30309

�[(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

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=

0.9064397

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Then press 1/x button

Then multiply by 0.45 * 12 cos =

12. sin  

cos

1.103217

0.4855991

tan

Calculator must be in degree mode. Always value should be given first then the function.

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

sin30

30 sin

=

0.5



cos45

45 cos



tan30

30 tan =

Made Easy

=

0.707

0.577

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

sin230

(30 sin ) x2

=

0.25

cos245

(45 cos ) x2 =

0.5

tan230

(30 tan ) x2 =

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





ℎ

𝑠𝑖𝑛 2 𝜃

2.0

= 𝑠𝑖𝑛 2 20 = 2.0/(20 sin ) x2

same procedure for

13. sin-1

cos-1

=

17.09726

sinh cosh tanh

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-10.5

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0.5 sin-1

=

30

degree

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

cos-10.5

0.5 cos-1

=

60

degree



tan-10.5

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!

Made Easy

=

1.551121 e+25 = 1.551121 x 1025

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How to use Virtual Calculator

15. Linear Interpolation formula You have to first calculate upto last form 𝑦 − 𝑦1 𝑥 − 𝑥1 = 𝑦2 − 𝑦1 𝑥2 − 𝑥1

𝑥 − 10 1.8 − 0.8 = 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) =

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51.66667

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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 + Bx2 Again take summation of both sides

∑𝑥𝑦 = 𝐴∑𝑥 + 𝐵∑𝑥 2

Just solve this two equations and find A and B

… … … … . . (𝑖𝑖)

Example: Data 1 2 3

x 1 2 3 ∑𝑥 = 6 For ∑𝑥 1 + 2 + 3 = 6

y 1 2 3 ∑𝑦 = 6

xy 1 x1 2x2 3x3 ∑𝑥𝑦 = 14

x2 12 22 32 ∑𝑥 2 = 14

For ∑𝑦 1 + 2 + 3 = 6

For ∑𝑥𝑦 1 * 1 + 2 * 2 + 3 * 3 = 14 For ∑𝑥 2 Use M+ button 12

1 x2 M+

then press C button

22

2 x2 M+

then press C button

32

3 x2 M+

then press C button

Then press MR button, Therefore ∑𝑥 2 = 14 Now ∑𝑦 = 𝐴𝑛 + 𝐵∑𝑥 or

6 = 3 𝐴 + 6𝐵

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… … … … . . (𝑖)

… … … … . . (𝑖)

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and ∑𝑥𝑦 = 𝐴∑𝑥 + 𝐵∑𝑥 2 or

14 = 6A + 14 B

… … … … . . (𝑖𝑖)

… … … … . . (𝑖𝑖)

Solving (i) and (ii) we get A = 0 and B = 1 y = 0 + 1. x is the solution.

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Production Engineering Theory of Metal Cutting Shear angle (Φ) 𝑟𝑐𝑜𝑠𝛼

𝑟𝑐𝑜𝑠𝛼

𝑡𝑎𝑛∅ = 1−𝑟𝑠𝑖𝑛𝛼 = (1−𝑟𝑠𝑖𝑛𝛼 )

[We have to use one extra bracket in the denominator]

0.45𝑐𝑜𝑠 12

𝑡𝑎𝑛∅ = (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) 1

𝛾 = 𝑡𝑎𝑛 17.3 + tan⁡ (17.3 − 10)

It is a long calculation; we have to use M+ 1

𝑡𝑎𝑛 17.3

= 1 / 17.3 tan

=

tan⁡ (17.3 − 10) = (17.3 - 10) tan Then find 𝛾

Just press button MR

3.210630 =

0.1281029 M+

3.338732

𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 ( 𝛾) = 𝑐𝑜𝑡17.3 + tan⁡ (17.3 − 10) = 3.34

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M+ then press C button

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Velocity relations 𝑉𝑠 𝑐𝑜𝑠𝛼 = 𝑉 𝑐𝑜𝑠(∅ − 𝛼)

𝑉𝑠 𝑐𝑜𝑠10 = 2.5 𝑐𝑜𝑠(22.94 − 10) 𝑉𝑠 = 2.5 ×

𝑐𝑜𝑠10 𝑐𝑜𝑠(22.94 − 10)

2.5 * 10 cos / ((22.94 - 10) cos )

=

2.526173

Merchant Circle (i)

𝑏𝑡

3×0.51

𝐹𝑠 = 𝜏𝑠 × 𝑠𝑖𝑛 ∅ = 285 × (𝑠𝑖𝑛 20.15) [we have to use extra bracket for denominator]

285 * 3 * 0.51 / (20.15 sin )

(ii)

=

1265.824

𝐹𝑠 = 𝑅𝑐𝑜𝑠(∅ + 𝛽 − 𝛼)

𝑂𝑟 𝑅 =

1265.8 𝐹𝑠 = 𝑐𝑜𝑠(∅ + 𝛽 − 𝛼) �𝑐𝑜𝑠(20.15 + 33 − 10)�

[We have to use extra bracket for denominator]

1265.8 / ((20.15 + 33 - 10) cos )

=

1735.005

Force Relations 𝐹𝑠 = 𝐹𝑐 𝑐𝑜𝑠∅ − 𝐹𝑡 𝑠𝑖𝑛∅

𝐹𝑠 = 900 𝑐𝑜𝑠30 − 600 𝑠𝑖𝑛30

900 * 30 cos - 600 * 30 sin

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=

479.4229

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Turning (i)

𝑡 = 𝑓𝑠𝑖𝑛𝜆 = 0.32 𝑠𝑖𝑛75

0.32 * 75 sin (ii)

𝐹

=

800

𝑥 𝐹𝑡 = 𝑠𝑖𝑛𝜆 = (𝑠𝑖𝑛 75)

800 / ( 75 sin )

=

0.3091 [We have to use extra bracket for denominator] 828.2209

Specific Energy 𝐹

800

𝑒 = 1000𝑐 𝑓𝑑 = (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 − 𝑦1 𝑥 − 𝑥1 = 𝑦2 − 𝑦1 𝑥2 − 𝑥1

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) =

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51.66667

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Tool life equation (i)

𝑉1 𝑇1𝑛 = 𝑉2 𝑇2𝑛

or 100 × 10𝑛 = 75 × 30𝑛 or

100

or

4

75

3

30 𝑛

= �10 �

= 3𝑛 4

or 𝑙𝑛 �3� = 𝑛𝑙𝑛3 or 𝑛 =

4 3

𝑙𝑛 � �

[We have to use extra bracket for denominator]

(𝑙𝑛 3)

(4/3) ln / ( 3 ln )

(ii)

=

0.2618593

Find C C = 100 x 1200.3 100 * 120 xy 0.3 =

(iii)

𝑛

𝑇

420.4887

60 0.204

𝑉3 = 𝑉1 × �𝑇1 � = 30 × �30 � 3

30 * ( 60 / 30 ) xy 0.204 =

(iv)

1

90 0.45 � � 𝑥

1

>

90 0.45 or � � 𝑥

or

90 0.3

� � 𝑥

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1

60 0.3 � � 𝑥

=

34.55664

1

60 0.3 � � 𝑥

60 0.45

=� � 𝑥

[Make power opposite]

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or

𝑥 0.45 𝑥 0.3

or 𝑥

=

0.15

60 0.45

=

900.3

60 0.45 900.3

= 60 xy 0.45

/ 90 xy 0.30 = 1.636422

1

or 𝑥 = (1.636422)0.15

For finding x the just press button xy (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 + Bx2 Again take summation of both sides

∑𝑥𝑦 = 𝐴∑𝑥 + 𝐵∑𝑥 2

Just solve this two equations and find A and B

… … … … . . (𝑖𝑖)

Example: Data 1 2 3

X 1 2 3 ∑𝑥 = 6 For ∑𝑥 1 + 2 + 3 = 6

y 1 2 3 ∑𝑦 = 6

xy 1 x1 2x2 3x3 ∑𝑥𝑦 = 14

x2 12 22 32 ∑𝑥 2 = 14

For ∑𝑦 1 + 2 + 3 = 6

For ∑𝑥𝑦 1 * 1 + 2 * 2 + 3 * 3 = 14 For ∑𝑥 2 Use M+ button

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12

1 x2 M+

then press C button

22

2 x2 M+

then press C button

32

3 x2 M+

then press C button

Then press MR button, Therefore ∑𝑥 2 = 14 Now ∑𝑦 = 𝐴𝑛 + 𝐵∑𝑥 or

6 = 3 𝐴 + 6𝐵

… … … … . . (𝑖)

and ∑𝑥𝑦 = 𝐴∑𝑥 + 𝐵∑𝑥 2

or

… … … … . . (𝑖)

14 = 6A + 14 B

… … … … . . (𝑖𝑖)

… … … … . . (𝑖𝑖)

Solving (i) and (ii) we get A = 0 and B = 1 y = 0 + 1. x is the solution.

Economics in metal cutting 𝑇𝑜 = �𝑇𝑐 + 𝑇𝑜 = �3 +

𝐶𝑡 1 − 𝑛 �� � 𝐶𝑚 𝑛

6.5 1 − 0.2 �� � 0.5 0.2

To = ( 3 + 6.5 / 0.5 ) (1 – 0.2 ) / 0.2 =

64 min

Now 𝑉𝑜 𝑇𝑜𝑛 = 𝐶

or 𝑉𝑜 (64)0.2 = 60 60

or 𝑉𝑜 = 64 0.2

60 / 64 xy 0.2 =

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26.11 m/min

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Metrology 3

𝑖 = 0.45 √𝐷 + 0.001𝐷 3

𝑖 = 0.45 √97.98 + 0.001 × 97.98 𝒚

0.45 * 97.98 √𝒙 3

=

+ 0.001 * 97.98

=

2.172535

Rolling cos 𝛼 = 1 −

∆ℎ 5 = 1− 𝐷 600

𝜶 = 1 - 5 / 600

=

cos-1

=

7.40198o

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 12 4

× ℎ1 =

𝑑2 = 𝑑1 × �

𝜋𝑑 22 4

× ℎ2

ℎ1 50 = 100 × � = 100 × √2 ℎ2 25

100 * ( 50 / 25) √

or 100 * 2 √

(ii)

=

=

141.4214

6

𝑥𝑠 = 48 − �

141.4214

2×0.25

� 𝑙𝑛 �

1

2×0.25

�

48 – (6 / 2 / 0.25 ) * (1 / 2 / 0.25 ) ln

(iii)

𝑥

𝐹𝑠𝑡𝑖𝑐𝑘𝑖𝑛𝑔 = 2 ∫0 𝑠 �𝑃𝑠 +

2𝐾 ℎ

=

39.68223

(𝑥𝑠 − 𝑥)� 𝐵𝑑𝑥

we have to first integrate without putting values

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𝐹𝑠𝑡𝑖𝑐𝑘𝑖𝑛𝑔 = 2𝐵 �𝑃𝑠 𝑥 +

2𝐾 ℎ

�𝑥𝑠 𝑥 −

2𝐾

𝐹𝑠𝑡𝑖𝑐𝑘𝑖𝑛𝑔 = 2𝐵 �𝑃𝑠 𝑥𝑠 +

ℎ

𝐾

�𝑥𝑠2 −

𝐹𝑠𝑡𝑖𝑐𝑘𝑖𝑛𝑔 = 2𝐵 �𝑃𝑠 𝑥𝑠 + 𝑥𝑠2 � ℎ

𝑥2 2

𝑥 𝑠2 2

��

��

𝑥𝑠

0

4.04

𝐹𝑆𝑡𝑖𝑐𝑘𝑖𝑛𝑔 = 2 × 150 × �16.16 × 39.68 + �

6

� × 39.682 �

2 * 120 * ( 16.16 * 39.68 + ( 4.04 / 6 ) * 39.68 x2 ) =

510418.2

𝐹𝑠𝑡𝑖𝑐𝑘𝑖𝑛𝑔 = 510418.2 𝑁 𝐿

2𝜇

𝐹𝑆𝑙𝑖𝑑𝑖𝑛𝑔 = 2 � 2𝐾𝑒 ℎ 𝑥𝑠

𝐿

(𝐿−𝑥)

2𝜇

𝐹𝑆𝑙𝑖𝑑𝑖𝑛𝑔 = 4𝐾𝐵 � 𝑒 ℎ 𝑥𝑠

𝐹𝑆𝑙𝑖𝑑𝑖𝑛𝑔 = 4𝐾𝐵 � 𝐹𝑆𝑙𝑖𝑑𝑖𝑛𝑔 =

2𝜇

𝑒ℎ

−

𝐵𝑑𝑥

(𝐿−𝑥)

(𝐿−𝑥)

2𝜇 ℎ

�

𝑑𝑥

𝐿 𝑥𝑠

2𝜇 4𝐾𝐵 (𝐿−𝑥 𝑠 ) �𝑒 0 − 𝑒 ℎ � 2𝜇 �− � ℎ 2𝐾𝐵ℎ

𝐹𝑆𝑙𝑖𝑑𝑖𝑛𝑔 = �

𝐹𝑆𝑙𝑖𝑑𝑖𝑛𝑔 = �

𝜇

� �𝑒

��

2𝜇 �(𝐿−𝑥 𝑠 )� ℎ

− 1�

[Note: extra brackets are used]

2×0.25 2 × 4.04 × 150 × 6 �� �(48−39.68 )� 6 � �𝑒 − 1� 0.25

(2 * 4.04 * 150 * 6 / 0.25) * (((2 * 0.25/6) * (48 – 39.68)) ex - 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)) ex - 1) =

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Now multiply both 29088 * 1.000372 =

29098.82

𝐹𝑆𝑙𝑖𝑑𝑖𝑛𝑔 = 29098.82 𝑁

𝐹𝑇𝑜𝑡𝑎𝑙 = 𝐹𝑆𝑡𝑖𝑐𝑘𝑖𝑛𝑔 + 𝐹𝑆𝑙𝑑𝑖𝑛𝑔 = 510418.2 + 29098.82 = 539517 𝑁 = 539.52 𝐾𝑁

Extrusion 𝐹 = 2𝜎𝑜 ×

𝜋𝑑𝑜2 𝑑𝑜 × 𝑙𝑛 � � 4 𝑑𝑓

5 𝜋 × 82 � 𝑙𝑛 � � 𝐹 = 2 × 400 × � 4 4

It is a long calculation, after some part we press multiplication is done . 2 * 400 * (𝝅 * 8 x2 / 4) = Now 40212.38 * (5 / 4) ln

=

button then further

it gives 40212.38 =

8973.135 N

Wire Drawing (i)

𝜎𝑑 = 𝜎𝑜

(1+𝐵) 𝐵

𝑟

2𝐵

�1 − �𝑟𝑓 � � 𝑜

(1 + 1.7145) 5 2×1.7145 𝜎𝑑 = 400 × �1 − � � � 1.7145 6.25

It is a long calculation, First calculate, 400 ×

(1+1.7145 )

Then calculate, 5

2×1.7145

�1 − �6.25 �

1.7145

= 400 * (1 +1.7145) / 1.7145

� = (1 –(5 / 6.25) xy (2 * 1.7145)) =

Now multiply 0.5347402 * 633.3040

=

=

633.3040

0.5347402

338.65 MPa

[At that time in your calculator 0.5347402 is present just multiply it with previous value 633.3040]

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(ii)

𝜎𝑜 = 𝜎𝑜

(1+𝐵) 𝐵

400 = 400 × 𝑟 𝑓𝑚𝑖𝑛

Let �

or

6.25

�

𝑟 𝑓𝑚𝑖𝑛

�1 − �

𝑟 𝑓𝑚𝑖𝑛 𝑟𝑜

2×1.7145

=𝑥

400 = 400 ×

(1+1.7145)

1.7145

× 𝜎𝑏

(1+1.7145 ) 1.7145

[1 − 𝑥] + 𝑥 × 50

= 400 * (1 +1.7145) / 1.7145

400 = 633.3[1 − 𝑥] + 𝑥 × 50

or 𝑥 =

2𝐵

�

(1 + 1.7145) 𝑟𝑓𝑚𝑖𝑛 2×1.7145 𝑟𝑓𝑚𝑖𝑛 2×1.7145 �1 − � � �+� � × 50 6.25 6.25 1.7145

Calculate, 400 ×

or

𝑟𝑜

2𝐵

� �+�

(633.3−400) (633.3−50)

𝑟 𝑓𝑚𝑖𝑛

≈ 0.4 = � 1

6.25

or 𝑟𝑓𝑚𝑖𝑛 = 6.25 × (0.4)2×1.7145

�

=

633.3

2×1.7145

or 𝒓𝒇𝒎𝒊𝒏 = 6.25 * 0.4 xy (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 √

(ii)

=

0.08230286 mm

𝐹 = 𝐿𝑡𝜏

𝐹 = 2(𝑎 + 𝑏)𝑡𝜏 = 2(100 + 50) × 5 × 300 2 * (100+50) * 5 * 300

(iii)

=

𝐷 = √𝑑2 + 4𝑑ℎ

𝐷 = �(252 + 4 × 25 × 15)

( 25 x2 + 4 * 25 * 15) √

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450000 N = 450 KN

[Extra bracket used] =

46.09772 mm

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(iv)

𝑡𝑓𝑖𝑛𝑎𝑙 =

𝑡 𝑖𝑛𝑖𝑡𝑖𝑎𝑙

𝑒 𝜀 1 ×𝑒 𝜀 2

1.5

= (𝑒 0.05

×𝑒 0.09 )

1.5 / ( 0.05 ex * 0.09 ex ) =

[Extra bracket for denominator] 1.304038 mm

Casting (i)

𝐵𝑢𝑜𝑦𝑎𝑛𝑐𝑦 𝑓𝑜𝑟𝑐𝑒 =

𝜋𝑑 2 4

× ℎ�𝜌𝑙𝑖𝑞𝑢𝑖𝑑 − 𝜌𝑐𝑜𝑟𝑒 � × 𝑔

𝜋 × 0.1202 � × 0.180 × (11300 − 1600) × 9.81 𝐵𝑢𝑜𝑦𝑎𝑛𝑐𝑦 𝑓𝑜𝑟𝑐𝑒 = � 4 ( 𝝅 * 0.12 x2 / 4 ) * 0.18 * (11300 - 1600) * 9.81 =

(ii)

193.7161 N

𝑉 2

𝑡𝑠 = 𝐵 �𝐴 �

Find values of V and A separately and then B * (V / A) x2

=

0

Welding (i)

𝑉

𝑂𝐶𝑉

𝐼

+ 𝑆𝐶𝐶 = 1

45 500 + =1 𝑂𝐶𝑉 𝑆𝐶𝐶

55 400 + =1 𝑂𝐶𝑉 𝑆𝐶𝐶

… … . . (𝑖)

… … . . (𝑖𝑖)

Now (ii) x 5 - (i) x 4 will give

(55 × 5 − 45 × 4) = (5 − 4) = 1 𝑂𝐶𝑉 or OCV = 95 V

Now from equation (i)

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45 500 + =1 95 𝑆𝐶𝐶 500

45

or 𝑆𝐶𝐶 = �1 − 95� or 𝑆𝐶𝐶 =

500

45 95

�1− �

500 / ( 1 – 45 / 95)

(ii)

=

950 V

𝐻 = 𝐼 2 𝑅𝑡 = 300002 × 100 × 10−6 × 0.005 30000 x2 * 100 * 6 +/- 10x * 0.005

=

450 J

Machine Tools (i)

Turning time ( T ) =

(𝐿+𝐴+𝑂) (𝑓𝑁 )

(L+A+O) / (f *N)

(ii)

Drilling time ( T ) =

0

(𝐿+ℎ+𝐴+𝑂)

L = 50 mm ℎ=

=

(𝑓𝑁 )

𝐷 15 = = 15/ (2 ∗59 tan ) = 4.5 𝑚𝑚 2𝑡𝑎𝑛𝛼 (2 × 𝑡𝑎𝑛59)

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) =

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ECM Calculation (i)

Find average density of an alloy 1 𝑥1 𝑥2 𝑥3 𝑥4 = + + + 𝜌 𝜌1 𝜌2 𝜌3 𝜌4 or

1

𝜌

=

0.7

8.9

+

0.2

7.19

First calculate

+

0.05 7.86

+

0.05 4.51

0.7 / 8.9 +0.2 / 7.19 +0.05 / 7.86 +0.05 / 4.51 =

0.1239159

Then just press 1/x button

(ii)

𝜌 = 8.069989 𝑔/𝑐𝑐

Find equivalent weight of an alloy 1 𝑥1 𝑥2 𝑥3 𝑥4 = + + + 𝐸 𝐸1 𝐸2 𝐸3 𝐸4 or

or

1

𝐸 1

𝐸

= =

𝑥 1 𝑣1 𝐸1

+

0.7×2 58.71

𝑥 2 𝑣2

+

𝐸2

+

0.2×2 51.99

First calculate

𝑥 3 𝑣3 𝐸3

+

+

𝑥 4 𝑣4

0.05×2 55.85

𝐸4

+

0.05×3 47.9

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

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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

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Strength of Materials (Only for the type of equations which are not yet covered)

Elongation (i)

𝛿=

𝑃𝐿

𝐴𝐸

10×10 3 ×1000

or 𝛿 =

𝜋 ×5 2 ×200×10 3 4

or 𝛿 =

(𝜋 ×52 ×2)

100×4

𝑚𝑚

𝑚𝑚

[After cancelling common terms from numerator and denominator and one extra bracket in the denominator has to be put] 100 * 4 / ( 𝝅 * 5 x2 * 2)

=

2.546480 mm

Thermal Stress (ii)

0.5×12.5×10 −6 ×20

50×0.5 � 𝜋 ×0.01 2 6 � ×200 ×10 � 4

�1+

First calculate

�

50×0.5

𝜋 ×0.01 2 4

×200×10 6 �

50×0.5×4

=(𝜋×0.012

×200×10 6 )

50 * 0.5 * 4 / (𝝅 * 0.01 x2 * 200 * 6 10x ) =

0.001591550

Then add 1 0.001591550 + 1

=

1.001592

Then press button 1/x

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0.9984105 Then multiply with 0.5 × 12.5 × 10−6 × 20

0.9984105 * 0.5 * 12.5 * 6 +/- 10x * 20 =

0.0001248013

Principal stress and principal strain (iii)

𝝈𝒙 −𝝈𝒚 𝟐

𝝉𝒎𝒂𝒙 = �� 𝜏𝑚𝑎𝑥

𝟐

� + 𝝉𝟐𝒙𝒚 2

80 − 20 = ���� �� + 402 � 2

[One bracket for denominator one bracket for square and one for square root] (((80-20) / 2 ) x2 + 40 x2 ) √ For

𝜎1,2 =

𝜎𝑥 +𝜎𝑦

First calculate

2

50 MPa

𝜎𝑥 −𝜎𝑦 2

+ ��

2 � + 𝜏𝑥𝑦

2

�𝜎𝑥 +𝜎𝑦 �

And then calculate

=

2

𝜎𝑥 −𝜎𝑦

����

2

2

2 � �� + 𝜏𝑥𝑦

Deflection of Beams (iv)

𝛿=

𝑤𝐿4 8𝐸𝐼

10×10 3 ×54

= (8×781250 )

10 * 3 10x * 5 xy 4 / (8 * 781250 ) =

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Bending stresses (v)

𝜎

9.57×10 = 𝑀𝑦 = 𝐼

3

×0.1

3 0.1×0.2 � 12

�

=

Pa

9.57 × 103 × 12 0.23

9.57 * 3 10x * 12 / (0.2 xy 3 ) = 1.435500e+7 Pa = 14.355 MPa

Torsion (vi)

𝑇 𝐽

=

𝐺𝜃 𝐿

409.256

𝜋 (1−0.74 )𝐷 4 32

or

𝐷4 =

=

80×10 9 ×𝜋 1×180

32×409.256×180

𝜋 2 ×(1−0.74 )×80×10 9

First calculate 32 * 409.256 * 180 = 2357315 Then calculate 𝜋 2 × (1 − 0.74 ) × 80 × 109

𝝅 x2 * (1 – 0.7 xy 4) * 80 * 9 10x Now 𝐷 4

=

2357315

5.999930×10 11

=

5.999930e+11

= 0.000003928904

Just press √ button twice , D = 0.04452130 m = 44.52 mm

Spring (vii)

𝛿=

8𝑃𝐷 3 𝑛

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8×200×10 3 ×10 −6 ×10 (80×10 9 ×84 ×10 −12 )

8*200*310x 6 +/- 10x 10 /(80* 9 10x 8 xy 4 * 12 +/- 10x ) =

0.04882813 m

= 48.83 mm

Theories of column (viii) 𝑃𝑐𝑟

=𝜋

2 𝐸𝐼

[For one end fixed and other end free]

2

4𝐿

3

10 × 10 = 3

𝜋 ×𝑑 4 64

𝜋 2 ×210×10 9 × 4×4 2

9

2

4

or 10 × 10 × 4 × 4 × 64 = 𝜋2 × 210 × 10 × 𝜋 × 𝑑

or 𝑑 4 =

10×10 3 ×4×42 ×64

𝜋 3 ×210×10 9

First calculate 10 × 103 × 4 × 42 × 64 10 * 3 10x * 4 * 4 x2 * 64

Then calculate 𝜋 3 × 210 × 109 𝝅 x3 * 210 * 9 10x

𝑁𝑜𝑤 𝑑4 =

= 4.096000e+7

= 6.511319e+12

4.096000e + 7 = 0.000006290584 6.511319𝑒 + 12

Just press √ button twice, d = 0.05008097 m ≈ 50 mm

Theories of Failure (ix)

𝜎𝑒 = 𝜎𝑒 =

1

√2 1

√2

�[(𝜎1 − 𝜎2 )2 + (𝜎2 − 𝜎3 )2 + (𝜎3 − 𝜎1 )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) x2 =

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5592.048

M+ then press C button

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(22.96 − 20)

(20 − 97.74)

x2 =

8.7616

M+ then press C button

x2 =

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 √ Therefore, 𝜎𝑒 =

= 1

√2

76.30309

�[(97.74 − 22.96)2 + (22.96 − 20)2 + (20 − 97.74)2 ] = 76.30309

After the calculation must press MC button.

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Theory of Machines (Only for the type of equations which are not yet covered)

Frequency (i)

𝑓𝑛 =

1

2𝜋

𝑆

� = 𝑀

1

2𝜋

40×10 3

��

100

�

(40 * 10 x3 / 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 − 18.852 )2 + (2 × 0.15 × 18.85)2

First calculate (2𝜉𝑟)2 = (2 × 0.15 × 18.85)2 (2 * 0.15 PressM+

* 18.85 ) x2

31.97903 This data is needed again so

=

Next find (1 − 𝑟 2 )2 =(1 − 18.852 )2 (1 – 18.85 x2 ) x2

=

125544.4

Now find the value of numerator Press MR + 1 = Then find denominator

then press √

Press MR + 125544.4 = Now Find (TR)

5.742737

then press √

Press 1/x and * 5.742737 =

354.3676

0.01620559

TR = 0.01620559 (Answer)

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Thermodynamics (Only for the type of equations which are not yet covered)

SFEE (i)

ℎ1 +

𝑐12

2000

+

𝑔𝑍

1000

+

𝑑𝑄

𝑑𝑚

= ℎ1 +

𝑐12

2000

+

𝑔𝑍

1000

+

𝑑𝑊 𝑑𝑚

1602 9.81 × 10 1002 9.81 × 6 𝑑𝑊 3200 + + + 0 = 2600 + + + 1000 1000 𝑑𝑚 2000 2000 M+

3200

M+

=

M+

Press M+

M-

M-

M-

then press C button

160 x2 / 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 x2 / 2000 = Press M-

then press C button

9.81 * 6 / 1000 = Press MNow Press MR and it is answer = 607.8392400000004

1602 9.81 × 10 1002 9.81 × 6 𝑑𝑊 = 3200 + + − 2600 − − 1000 1000 𝑑𝑚 2000 2000

Entropy Change (ii)

𝑇

𝑃

𝑆𝑄 − 𝑆𝑝 = 𝑐𝑝 𝑙𝑛 � 𝑄 � − 𝑅𝑙𝑛 � 𝑄 � 𝑇 𝑃 𝑃

𝑆𝑄 − 𝑆𝑝 = 1.005 𝑙𝑛 � M+

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𝑃

50 300 � − 0.287𝑙𝑛 � � 150 350 M-

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First calculate

300

1.005 𝑙𝑛 �

350

�

-0.1549214 Press M+ then press C button

1.005 * (300 / 350 ) ln = 50

Then calculate 0.287𝑙𝑛 �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)

𝑇

𝐴𝐸 = 𝑚𝑐𝑝 �(𝑇2 − 𝑇1 ) − 𝑇𝑜 𝑙𝑛 � 2 �� 𝑇 1

1250

𝐴𝐸 = 2000 × 0.5 �(1250 − 450) − 303𝑙𝑛 � 450 �� 1250

First calculate �(1250 − 450) − 303𝑙𝑛 � 450 �� (1250-450)-303 * (1250 / 450) ln

Then multiply with 2000 × 0.5 490.4397 * 2000 * 0.5 =

Made Easy

=

490.4397

490439.7 KJ = 490.44 MJ

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Heat and Mass Transfer (Only for the type of equations which are not covered yet)

Conduction (i)

𝑄= 𝑄=

2𝜋𝐿 �𝑡 𝑖 −𝑡 𝑓 �

𝑟 𝑟 𝑙𝑛 �𝑟 2 � 𝑙𝑛 �𝑟 3 � 1 + 2 𝐾𝐴 𝐾𝐵

2 × 𝜋 × 1 × (1200 − 600) 0.055 0.025 𝑙𝑛 � 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 =

∴ 𝑄=

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944.7186

W/m

2 × 𝜋 × 1 × (1200 − 600) = 944.72 𝑊/𝑚 0.055 0.025 𝑙𝑛 𝑙𝑛 � 0.01 � + �0.025� 19 0.2

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Unsteady Conduction (ii)

𝜃

𝜃𝑖

𝑇−𝑇

= 𝑇 −𝑇𝑎 = 𝑒 −𝐵𝑖 𝐹𝑜 𝑖

𝑎

298 − 300 −3 = 𝑒 −425𝜏×2.3533 ×10 30 − 300 or 𝑙𝑛 �

or 𝑙𝑛 � or 𝜏

298−300 30−300

30−300

298−300

=

� = −425𝜏 × 2.3533 × 10−3 � = 425𝜏 × 2.3533 × 10−3

(30−300 )

𝑙𝑛 �(298 −300 )�

425×2.3533 ×10 −3

((30-300) / (298-300)) ln = Note: Several times use of

/ 425 = =

/ 2.3533 =

/ 3 +/- 10x =

4.904526 S

is good for this calculator.

Heat Exchanger (iii)

𝐿𝑀𝑇𝐷 =

𝜃 𝑖 −𝜃𝑜

𝜃 𝑙𝑛 � 𝑖 � 𝜃𝑜

=

90−40 90 40

𝑙𝑛 � �

(90 / 40) ln = then press 1/x then multiply with numerator * (90 – 40) = 61.65760

Radiation (iii)

Interchange factor

𝑓12 =

1

1 𝐴1 1 + � −1� 𝜀1 𝐴2 𝜀2

First calculate �

2×10 −3 100

=

1

1 2×10 −3 1 � −1� + 0.6 0.3 100

1

� �0.3 − 1�

(2 * 3 +/- 10x / 100) * (1 / 0.3 – 1 ) =

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0.00004666666

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Then add 1/0.6 0.00004666666 + 1 / 0.6 ) =

1.666714

Then press 1/x 0.5999830 f12 =0.5999830 ≈0.6 Now 𝑄𝑛𝑒𝑡

= 𝑓12 𝜎𝐴1 (𝑇14 − 𝑇24 )

𝑄𝑛𝑒𝑡 = 0.6 × 5.67 × 10−8 × 2 × 10−3 (8004 − 3004 )

First calculate 0.6 × 5.67 × 10−8 × 2 × 10−3

0.6 * 5.67 * 8 +/- 10x * 2 * 3 +/- 10x =

6.804000e-11

Then multiply with (8004 − 3004 )

6.804000e-11 * (800 xy 4 - 300 xy 4) = 27.31806 W

𝑄𝑛𝑒𝑡 = 0.6 × 5.67 × 10−8 × 2 × 10−3 (8004 − 3004 ) = 27.32 𝑊

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Industrial Engineering (Only for the type of equations which are not yet covered)

Forecasting (i)

𝑢𝑓 = 𝛼𝑆𝑡 + 𝛼(1 − 𝛼)𝑆𝑡−1 + 𝛼(1 − 𝛼)2 𝑆𝑡−2 + 𝛼(1 − 𝛼)3 𝑆𝑡−3 𝑢𝑓 = 0.4 × 95 + 0.4 × 0.6 × 82 + 0.4 × 0.62 × 68 + 0.4 × 0.63 × 70 M+

0.4 * 95

M+

=

0.4 * 0.6 * 82

38

M+

M+

Press M+ then press C button

= 19.68 Press M+ then press C button

0.4 * 0.6 x2 * 68

=

19.68 Press M+ then press C button

0.4 * 0.6 x3 * 70

=

6.048 Press M+

Then press MR button 73.52 𝑢𝑓 = 0.4 × 95 + 0.4 × 0.6 × 82 + 0.4 × 0.62 × 68 + 0.4 × 0.63 × 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 + Bx2 Again take summation of both sides

∑𝑥𝑦 = 𝐴∑𝑥 + 𝐵∑𝑥 2

Just solve this two equations and find A and B

… … … … . . (𝑖𝑖)

Example:

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Data 1 2 3

x 1 2 3 ∑𝑥 = 6 For ∑𝑥 1 + 2 + 3 = 6

Y 1 2 3 ∑𝑦 = 6

Xy 1 x1 2x2 3x3 ∑𝑥𝑦 = 14

x2 12 22 32 2 ∑𝑥 = 14

For ∑𝑦 1 + 2 + 3 = 6

For ∑𝑥𝑦 1 * 1 + 2 * 2 + 3 * 3 = 14 For ∑𝑥 2 Use M+ button 12

1 x2 M+

then press C button

22

2 x2 M+

then press C button

32

3 x2 M+

then press C button

Then press MR button, Therefore ∑𝑥 2 = 14 Now ∑𝑦 = 𝐴𝑛 + 𝐵∑𝑥 or

6 = 3 𝐴 + 6𝐵

… … … … . . (𝑖)

and ∑𝑥𝑦 = 𝐴∑𝑥 + 𝐵∑𝑥 2

or

14 = 6A + 14 B

… … … … . . (𝑖)

… … … … . . (𝑖𝑖)

… … … … . . (𝑖𝑖)

Solving (i) and (ii) we get A = 0 and B = 1 y = 0 + 1. x is the solution.

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Optimum run size (iii)

𝑄=�

2𝑈𝑅 𝐼𝑐

�

𝐼𝑐 +𝐼𝑝 𝐼𝑝

�

2 × 30000 × 3500 2.5 + 10 𝑄=� ×� � 2.5 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”

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