virtual calculator

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

Some functions

1. Exp

2. ln

3. log

4. logyx

5. ex

6. 10x

7. xy

8. √𝒙𝒚

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

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

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2 | P a g e How to use Virtual Calculator

Made Easy By: S K Mondal

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 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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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 = 2.430677

7log550 7 * ( 50 logyx 5 ) = 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 = 500

105/3 (5/3) 10X = 46.41592

101.4−1

1.4 10((1.4−1)1.4 ) ((1.4 − 1)/1.4) 10X = 1.930698

Or you may simplify

101.4−1

1.4 10(0.41.4) (0.4/1.4)10X = 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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�𝑃2𝑃1�

𝛾𝛾−1

⟹ �𝑃2𝑃1�

𝛾(𝛾−1)

⟹ �53�

1.4(1.4−1)

(5/3) xy 1.4/(1.4 – 1) = 5.111263

8. √𝑥𝑦

√325 32 √𝑥𝑦 5 = 2

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.

We may use xy function also √325 = 321/5 = 32 xy (1/5) = 2

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

9. |𝑥| |−5| 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 = 1√2

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

𝜎𝑒 = 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+

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(97.74 − 22.96) x2 = 5592.048 M+ then press C button

(22.96 − 20) x2 = 8.7616 M+ then press C button

(20 − 97.74) 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 √ = 76.30309

Therefore, 𝜎𝑒 = 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.

0.45𝑐𝑜𝑠121 − 0.45𝑠𝑖𝑛12

We first calculate 1 – 0.45sin12 then use 1/x button.

1 – 0.45 * 12 sin = 0.9064397

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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.

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sin30 30 sin = 0.5

cos45 45 cos = 0.707

tan30 30 tan = 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 2𝜃 = 2.0𝑠𝑖𝑛 220 = 2.0/(20 sin ) x2 = 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-10.5 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! = 1.551121 e+25 = 1.551121 x 1025

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15. Linear Interpolation formula You have to first calculate upto last form

𝑦 − 𝑦1𝑦2 − 𝑦1

= 𝑥 − 𝑥1𝑥2 − 𝑥1

1.8 − 0.82.0 − 0.8 = 𝑥 − 10

60 − 10

𝑥 − 10 = (60 − 10) × 1.8 − 0.82.0 − 0.8

𝑥 = 10 + (60 − 10) × 1.8 − 0.82.0 − 0.8

10 + (60 – 10) * (1.8 – 0.8) / (2.0 – 0.8) = 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∑𝑥2 ………… . . (𝑖𝑖)

Just solve this two equations and find A and B

Example:

Data x y xy x2

1 1 1 1 x1 12

2 2 2 2 x 2 22

3 3 3 3 x 3 32

∑𝑥 = 6 ∑𝑦 = 6 ∑𝑥𝑦 = 14 ∑𝑥2 = 14 For ∑𝑥 1 + 2 + 3 = 6

For ∑𝑦 1 + 2 + 3 = 6

For ∑𝑥𝑦 1 * 1 + 2 * 2 + 3 * 3 = 14

For ∑𝑥2 Use M+ button

12 1 x2 M+ then press C button



Then press MR button, Therefore ∑𝑥2 = 14

Now ∑𝑦 = 𝐴𝑛 + 𝐵∑𝑥 ………… . . (𝑖)

or 6 = 3 𝐴 + 6𝐵 ………… . . (𝑖)

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and ∑𝑥𝑦 = 𝐴∑𝑥 + 𝐵∑𝑥2 ………… . . (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.

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17 | P a g e How to use Virtual Calculator in Mechanical Engineering


Production Engineering

Theory of Metal Cutting

Shear angle (Φ)

𝑡𝑎𝑛∅ = 𝑟𝑐𝑜𝑠𝛼1−𝑟𝑠𝑖𝑛𝛼 = 𝑟𝑐𝑜𝑠𝛼

(1−𝑟𝑠𝑖𝑛𝛼 ) [We have to use one extra bracket in the denominator]

𝑡𝑎𝑛∅ = 0.45𝑐𝑜𝑠12(1−0.45𝑠𝑖𝑛12)

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 = 1𝑡𝑎𝑛 17.3 + tan(17.3 − 10)

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

1𝑡𝑎𝑛 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

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

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Velocity relations

𝑉𝑠𝑉 = 𝑐𝑜𝑠𝛼

𝑐𝑜𝑠(∅ − 𝛼)

𝑉𝑠2.5 = 𝑐𝑜𝑠10

𝑐𝑜𝑠(22.94 − 10)

𝑉𝑠 = 2.5 × 𝑐𝑜𝑠10𝑐𝑜𝑠(22.94 − 10)

2.5 * 10 cos / ((22.94 - 10) cos ) = 2.526173

Merchant Circle

(i) 𝐹𝑠 = 𝜏𝑠 × 𝑏𝑡𝑠𝑖𝑛∅ = 285 × 3×0.51

(𝑠𝑖𝑛20.15) [we have to use extra bracket for denominator]

285 * 3 * 0.51 / (20.15 sin ) = 1265.824

(ii) 𝐹𝑠 = 𝑅𝑐𝑜𝑠(∅ + 𝛽 − 𝛼)

𝑂𝑟 𝑅 = 𝐹𝑠𝑐𝑜𝑠(∅ + 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

𝐹𝑠 = 𝐹𝑐𝑐𝑜𝑠∅ − 𝐹𝑡𝑠𝑖𝑛∅

𝐹𝑠 = 900 𝑐𝑜𝑠30 − 600 𝑠𝑖𝑛30

900 * 30 cos - 600 * 30 sin = 479.4229

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Turning

(i) 𝑡 = 𝑓𝑠𝑖𝑛𝜆 = 0.32 𝑠𝑖𝑛75

0.32 * 75 sin = 0.3091

(ii) 𝐹𝑡 = 𝐹𝑥𝑠𝑖𝑛𝜆 = 800

(𝑠𝑖𝑛75) [We have to use extra bracket for denominator]

800 / ( 75 sin ) = 828.2209

Specific Energy

𝑒 = 𝐹𝑐1000𝑓𝑑 = 800

(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 − 𝑦1𝑦2 − 𝑦1

= 𝑥 − 𝑥1𝑥2 − 𝑥1

1.8 − 0.82.0 − 0.8 = 𝑥 − 10

60 − 10

𝑥 − 10 = (60 − 10) × 1.8 − 0.82.0 − 0.8

𝑥 = 10 + (60 − 10) × 1.8 − 0.82.0 − 0.8

10 + (60 – 10) * (1.8 – 0.8) / (2.0 – 0.8) = 51.66667

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Tool life equation (i) 𝑉1𝑇1

𝑛 = 𝑉2𝑇2𝑛

or 100 × 10𝑛 = 75 × 30𝑛

or 10075 = �30

10�𝑛

or 43 = 3𝑛

or 𝑙𝑛 �43� = 𝑛𝑙𝑛3

or 𝑛 = 𝑙𝑛�43�

(𝑙𝑛3) [We have to use extra bracket for denominator]

(4/3) ln / ( 3 ln ) = 0.2618593

(ii) Find C

C = 100 x 1200.3

100 * 120 xy 0.3 = 420.4887

(iii) 𝑉3 = 𝑉1 × �𝑇1𝑇3�𝑛

= 30 × �6030�

0.204

30 * ( 60 / 30 ) xy 0.204 = 34.55664

(iv) �90𝑥 �

10.45 > �60

𝑥 �1

0.3

or �90𝑥 �

10.45 = �60

𝑥 �1

0.3

or �90𝑥 �

0.3= �60

𝑥 �0.45

[Make power opposite]

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or 𝑥0.45

𝑥0.3 = 600.45

900.3

or 𝑥0.15 = 600.45

900.3 = 60 xy 0.45 / 90 xy 0.30 = 1.636422

or 𝑥 = (1.636422)1

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



xy = Ax + Bx2



Example:

Data X y xy x2

1 1 1 1 x1 12

2 2 2 2 x 2 22

3 3 3 3 x 3 32

∑𝑥 = 6 ∑𝑦 = 6 ∑𝑥𝑦 = 14 ∑𝑥2 = 14 For ∑𝑥 1 + 2 + 3 = 6

For ∑𝑦 1 + 2 + 3 = 6

For ∑𝑥𝑦 1 * 1 + 2 * 2 + 3 * 3 = 14


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Now ∑𝑦 = 𝐴𝑛 + 𝐵∑𝑥 ………… . . (𝑖)

or 6 = 3 𝐴 + 6𝐵 ………… . . (𝑖)

and ∑𝑥𝑦 = 𝐴∑𝑥 + 𝐵∑𝑥2 ………… . . (𝑖𝑖)

or 14 = 6A + 14 B ………… . . (𝑖𝑖)



Economics in metal cutting

𝑇𝑜 = �𝑇𝑐 + 𝐶𝑡𝐶𝑚

� �1 − 𝑛𝑛 �

𝑇𝑜 = �3 + 6.50.5� �

1 − 0.20.2 �

To = ( 3 + 6.5 / 0.5 ) (1 – 0.2 ) / 0.2 = 64 min

Now 𝑉𝑜𝑇𝑜𝑛 = 𝐶

or 𝑉𝑜(64)0.2 = 60

or 𝑉𝑜 = 60640.2

60 / 64 xy 0.2 = 26.11 m/min

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Metrology 𝑖 = 0.45√𝐷3 + 0.001𝐷

𝑖 = 0.45√97.983 + 0.001 × 97.98

0.45 * 97.98 √𝒙𝒚 3 = + 0.001 * 97.98 = 2.172535

Rolling

cos𝛼 = 1 − ∆ℎ𝐷 = 1 − 5

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) 𝜋𝑑1

2

4 × ℎ1 = 𝜋𝑑22

4 × ℎ2

𝑑2 = 𝑑1 × �ℎ1ℎ2

= 100 × �5025 = 100 × √2

100 * ( 50 / 25) √ = 141.4214

or 100 * 2 √ = 141.4214

(ii) 𝑥𝑠 = 48 − � 62×0.25� 𝑙𝑛 �

12×0.25�

48 – (6 / 2 / 0.25 ) * (1 / 2 / 0.25 ) ln = 39.68223

(iii) 𝐹𝑠𝑡𝑖𝑐𝑘𝑖𝑛𝑔 = 2∫ �𝑃𝑠 + 2𝐾ℎ (𝑥𝑠 − 𝑥)�𝐵𝑑𝑥𝑥𝑠

0

we have to first integrate without putting values

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𝐹𝑠𝑡𝑖𝑐𝑘𝑖𝑛𝑔 = 2𝐵 �𝑃𝑠𝑥 + 2𝐾ℎ �𝑥𝑠𝑥 −

𝑥2

2 ��0

𝑥𝑠

𝐹𝑠𝑡𝑖𝑐𝑘𝑖𝑛𝑔 = 2𝐵 �𝑃𝑠𝑥𝑠 + 2𝐾ℎ �𝑥𝑠2 −

𝑥𝑠22 ��

𝐹𝑠𝑡𝑖𝑐𝑘𝑖𝑛𝑔 = 2𝐵 �𝑃𝑠𝑥𝑠 + 𝐾ℎ 𝑥𝑠

2�

𝐹𝑆𝑡𝑖𝑐𝑘𝑖𝑛𝑔 = 2 × 150 × �16.16 × 39.68 + �4.046 � × 39.682�

2 * 120 * ( 16.16 * 39.68 + ( 4.04 / 6 ) * 39.68 x2 ) = 510418.2

𝐹𝑠𝑡𝑖𝑐𝑘𝑖𝑛𝑔 = 510418.2 𝑁

𝐹𝑆𝑙𝑖𝑑𝑖𝑛𝑔 = 2 � 2𝐾𝑒2𝜇ℎ (𝐿−𝑥)𝐵𝑑𝑥

𝐿

𝑥𝑠

𝐹𝑆𝑙𝑖𝑑𝑖𝑛𝑔 = 4𝐾𝐵 �𝑒2𝜇ℎ (𝐿−𝑥)𝑑𝑥

𝐿

𝑥𝑠

𝐹𝑆𝑙𝑖𝑑𝑖𝑛𝑔 = 4𝐾𝐵 �𝑒2𝜇ℎ (𝐿−𝑥)

− 2𝜇ℎ

�𝑥𝑠

𝐿

𝐹𝑆𝑙𝑖𝑑𝑖𝑛𝑔 = 4𝐾𝐵�− 2𝜇

ℎ ��𝑒0 − 𝑒

2𝜇ℎ (𝐿−𝑥𝑠)�

𝐹𝑆𝑙𝑖𝑑𝑖𝑛𝑔 = �2𝐾𝐵ℎ𝜇 � �𝑒��

2𝜇ℎ �(𝐿−𝑥𝑠)� − 1� [Note: extra brackets are used]

𝐹𝑆𝑙𝑖𝑑𝑖𝑛𝑔 = �2 × 4.04 × 150 × 60.25 � �𝑒��

2×0.256 �(48−39.68)� − 1�

(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) = 1.000372

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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 × 𝜋𝑑𝑜24 × 𝑙𝑛 �𝑑𝑜𝑑𝑓

�

𝐹 = 2 × 400 × �𝜋 × 82

4 � 𝑙𝑛 �54�

It is a long calculation, after some part we press = button then further multiplication is done .

2 * 400 * (𝝅 * 8 x2 / 4) = it gives 40212.38

Now 40212.38 * (5 / 4) ln = 8973.135 N

Wire Drawing

(i) 𝜎𝑑 = 𝜎𝑜(1+𝐵)𝐵 �1 − �𝑟𝑓𝑟𝑜 �

2𝐵�

𝜎𝑑 = 400 ×(1 + 1.7145)

1.7145 �1 − � 56.25�

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 − � 56.25�

2×1.7145� = (1 –(5 / 6.25) xy (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]

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(ii) 𝜎𝑜 = 𝜎𝑜(1+𝐵)𝐵 �1 − �𝑟𝑓𝑚𝑖𝑛𝑟𝑜

�2𝐵� + �𝑟𝑓𝑚𝑖𝑛𝑟𝑜

�2𝐵

× 𝜎𝑏

400 = 400 ×(1 + 1.7145)

1.7145 �1 − �𝑟𝑓𝑚𝑖𝑛6.25�

2×1.7145� + �

𝑟𝑓𝑚𝑖𝑛6.25�

2×1.7145× 50

Let �𝑟𝑓𝑚𝑖𝑛6.25 �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)(633.3−50) ≈ 0.4 = �𝑟𝑓𝑚𝑖𝑛6.25 �

2×1.7145

or 𝑟𝑓𝑚𝑖𝑛 = 6.25 × (0.4)1

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 √ = 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 = √𝑑2 + 4𝑑ℎ

𝐷 = �(252 + 4 × 25 × 15) [Extra bracket used]

( 25 x2 + 4 * 25 * 15) √ = 46.09772 mm

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(iv) 𝑡𝑓𝑖𝑛𝑎𝑙 = 𝑡𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑒𝜀1 ×𝑒𝜀2 = 1.5

(𝑒0.05 ×𝑒0.09) [Extra bracket for denominator]

1.5 / ( 0.05 ex * 0.09 ex ) = 1.304038 mm

Casting

(i) 𝐵𝑢𝑜𝑦𝑎𝑛𝑐𝑦 𝑓𝑜𝑟𝑐𝑒 = 𝜋𝑑2

4 × ℎ�𝜌𝑙𝑖𝑞𝑢𝑖𝑑 − 𝜌𝑐𝑜𝑟𝑒 � × 𝑔

𝐵𝑢𝑜𝑦𝑎𝑛𝑐𝑦 𝑓𝑜𝑟𝑐𝑒 = �𝜋 × 0.1202

4 � × 0.180 × (11300 − 1600) × 9.81

( 𝝅 * 0.12 x2 / 4 ) * 0.18 * (11300 - 1600) * 9.81 = 193.7161 N

(ii) 𝑡𝑠 = 𝐵 �𝑉𝐴�2

Find values of V and A separately and then

B * (V / A) x2 = 0

Welding

(i) 𝑉𝑂𝐶𝑉 + 𝐼

𝑆𝐶𝐶 = 1

45𝑂𝐶𝑉 + 500

𝑆𝐶𝐶 = 1 …… . . (𝑖)

55𝑂𝐶𝑉 + 400

𝑆𝐶𝐶 = 1 …… . . (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)

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4595 + 500

𝑆𝐶𝐶 = 1

or 500𝑆𝐶𝐶 = �1 − 45

95�

or 𝑆𝐶𝐶 = 500�1−45

95�

500 / ( 1 – 45 / 95) = 950 V

(ii) 𝐻 = 𝐼2𝑅𝑡 = 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 ) = 0

(ii) Drilling time ( T ) = (𝐿+ℎ+𝐴+𝑂)(𝑓𝑁)

L = 50 mm

ℎ = 𝐷2𝑡𝑎𝑛𝛼 = 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

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ECM Calculation (i) Find average density of an alloy

1𝜌 = 𝑥1

𝜌1+ 𝑥2𝜌2

+ 𝑥3𝜌3

+ 𝑥4𝜌4

or 1𝜌 = 0.7

8.9 + 0.27.19 + 0.05

7.86 + 0.054.51

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

1𝐸 = 𝑥1

𝐸1+ 𝑥2𝐸2

+ 𝑥3𝐸3

+ 𝑥4𝐸4

or 1𝐸 = 𝑥1𝑣1

𝐸1+ 𝑥2𝑣2

𝐸2+ 𝑥3𝑣3

𝐸3+ 𝑥4𝑣4

𝐸4

or 1𝐸 = 0.7×2

58.71 + 0.2×251.99 + 0.05×2

55.85 + 0.05×347.9

First calculate

0.7 * 2 / 58.71+0.2 * 2 / 51.99+0.05 * 2 / 55.85+0.05 * 3 / 47.9 = 0.03646185


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


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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



0.05 * 3 / 47.9 = 0.003131524 press M+ button the press MR 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 = 𝑃𝐿𝐴𝐸

or 𝛿 = 10×103×1000𝜋×52

4 ×200×103 𝑚𝑚

or 𝛿 = 100×4(𝜋×52×2) 𝑚𝑚

[After cancelling common terms from numerator and denominator and one extra bracket in the denominator has to be put]

100 * 4 / ( 𝝅 * 5 x2 * 2) = 2.546480 mm

Thermal Stress

(ii) 0.5×12.5×10−6×20

�1+ 50×0.5

�𝜋×0.0124 ×200×106�

�

First calculate 50×0.5

�𝜋×0.0124 ×200×106�

= 50×0.5×4(𝜋×0.012×200×106)

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) 𝝉𝒎𝒂𝒙 = ��𝝈𝒙−𝝈𝒚𝟐 �𝟐

+ 𝝉𝒙𝒚𝟐

𝜏𝑚𝑎𝑥 = ��80 − 202 ��

2+ 402�

[One bracket for denominator one bracket for square and one for square root]

(((80-20) / 2 ) x2 + 40 x2 ) √ = 50 MPa

For 𝜎1,2 = 𝜎𝑥+𝜎𝑦2 + ��𝜎𝑥−𝜎𝑦2 �

2+ 𝜏𝑥𝑦2

First calculate �𝜎𝑥+𝜎𝑦�

2

And then calculate ��𝜎𝑥−𝜎𝑦2 ��2

+ 𝜏𝑥𝑦2 �

Deflection of Beams

(iv) 𝛿 = 𝑤𝐿4

8𝐸𝐼 = 10×103×54

(8×781250 )

10 * 3 10x * 5 xy 4 / (8 * 781250 ) = 1 mm

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Bending stresses

(v) 𝜎 = 𝑀𝑦𝐼 = 9.57×103×0.1

�0.1×0.2312 �

Pa

= 9.57 × 103 × 120.23

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

Torsion

(vi) 𝑇𝐽 = 𝐺𝜃

𝐿

409.256𝜋

32(1−0.74)𝐷4 = 80×109×𝜋1×180

or 𝐷4 = 32×409.256×180𝜋2×(1−0.74)×80×109

First calculate 32 * 409.256 * 180 = 2357315

Then calculate 𝜋2 × (1 − 0.74) × 80 × 109

𝝅 x2 * (1 – 0.7 xy 4) * 80 * 9 10x = 5.999930e+11

Now 𝐷4 = 23573155.999930×1011 = 0.000003928904

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

Spring

(vii) 𝛿 = 8𝑃𝐷3𝑛𝐺𝑑4

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8×200×103×10−6×10(80×109×84×10−12)

8*200*310x6 +/- 10x10 /(80* 9 10x 8 xy 4 * 12 +/- 10x ) = 0.04882813 m

= 48.83 mm

Theories of column

(viii) 𝑃𝑐𝑟 = 𝜋2𝐸𝐼4𝐿2 [For one end fixed and other end free]

10 × 103 = 𝜋2×210×109×𝜋×𝑑464

4×42 or 10 × 103 × 4 × 42 × 64 = 𝜋2 × 210 × 109 × 𝜋 × 𝑑4

or 𝑑4 = 10×103×4×42×64𝜋3×210×109

First calculate 10 × 103 × 4 × 42 × 64

10 * 3 10x * 4 * 4 x2 * 64 = 4.096000e+7

Then calculate 𝜋3 × 210 × 109

𝝅 x3 * 210 * 9 10x = 6.511319e+12

𝑁𝑜𝑤 𝑑4 = 4.096000e + 76.511319𝑒 + 12 = 0.000006290584

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

Theories of Failure

(ix) 𝜎𝑒 = 1√2�[(𝜎1 − 𝜎2)2 + (𝜎2 − 𝜎3)2 + (𝜎3 − 𝜎1)2]

𝜎𝑒 = 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) x2 = 5592.048 M+ then press C button

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(22.96 − 20) x2 = 8.7616 M+ then press C button

(20 − 97.74) 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 √ = 76.30309

Therefore, 𝜎𝑒 = 1√2 �[(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


Frequency

(i) 𝑓𝑛 = 12𝜋 �

𝑆𝑀 = 1

2𝜋��40×103

100 �

(40 * 10 x3 / 100 ) √ / 2 / 𝝅 = 3.183099

Transmissibility ratio

(ii) 𝑇𝑅 = �1+(2𝜉𝑟 )2

�(1−𝑟2)2+(2𝜉𝑟 )2

𝑇𝑅 = �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 * 18.85 ) x2 = 31.97903 This data is needed again so PressM+

Next find (1 − 𝑟2)2 =(1 − 18.852)2

(1 – 18.85 x2 ) x2 = 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)

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Thermodynamics


SFEE

(i) ℎ1 + 𝑐12

2000 + 𝑔𝑍1000 + 𝑑𝑄

𝑑𝑚 = ℎ1 + 𝑐12

2000 + 𝑔𝑍1000 + 𝑑𝑊

𝑑𝑚

3200 + 1602

2000 + 9.81 × 101000 + 0 = 2600 + 1002

2000 + 9.81 × 61000 + 𝑑𝑊

𝑑𝑚

M+ M+ M+ M- M- M-

3200 = Press 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 M-

Now Press MR and it is answer = 607.8392400000004

𝑑𝑊𝑑𝑚 = 3200 + 1602

2000 + 9.81 × 101000 − 2600 − 1002

2000 − 9.81 × 61000

Entropy Change

(ii) 𝑆𝑄 − 𝑆𝑝 = 𝑐𝑝𝑙𝑛 �𝑇𝑄𝑇𝑃� − 𝑅𝑙𝑛 �𝑃𝑄𝑃𝑃�

𝑆𝑄 − 𝑆𝑝 = 1.005 𝑙𝑛 �300350� − 0.287𝑙𝑛 � 50

150�

M+ M-

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First calculate 1.005 𝑙𝑛 �300350�

1.005 * (300 / 350 ) ln = -0.1549214 Press M+ then press C button

Then calculate 0.287𝑙𝑛 � 50150�

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 = 𝑚𝑐𝑝 �(𝑇2 − 𝑇1)− 𝑇𝑜𝑙𝑛 �𝑇2𝑇1��

𝐴𝐸 = 2000 × 0.5 �(1250 − 450) − 303𝑙𝑛 �1250450 ��

First calculate �(1250 − 450) − 303𝑙𝑛 �1250450 ��

(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

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Heat and Mass Transfer

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

Conduction

(i) 𝑄 = 2𝜋𝐿�𝑡𝑖−𝑡𝑓�𝑙𝑛 �𝑟2

𝑟1�𝐾𝐴

+𝑙𝑛 �𝑟3

𝑟2�𝐾𝐵

𝑄 = 2 × 𝜋 × 1 × (1200− 600)

𝑙𝑛 �0.0250.01 �19 +

𝑙𝑛 �0.0550.025�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

∴ 𝑄 = 2 × 𝜋 × 1 × (1200− 600)

𝑙𝑛 �0.0250.01 �19 +

𝑙𝑛 �0.0550.025�0.2

= 944.72 𝑊/𝑚

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Unsteady Conduction

(ii) 𝜃𝜃𝑖

= 𝑇−𝑇𝑎𝑇𝑖−𝑇𝑎

= 𝑒−𝐵𝑖𝐹𝑜

298 − 30030 − 300 = 𝑒−425𝜏×2.3533×10−3

or 𝑙𝑛 �298−30030−300 � = −425𝜏 × 2.3533 × 10−3

or 𝑙𝑛 � 30−300298−300� = 425𝜏 × 2.3533 × 10−3

or 𝜏 =𝑙𝑛� (30−300)

(298−300)�425×2.3533×10−3

((30-300) / (298-300)) ln = / 425 = / 2.3533 = / 3 +/- 10x = 4.904526 S

Note: Several times use of = is good for this calculator.

Heat Exchanger

(iii) 𝐿𝑀𝑇𝐷 = 𝜃𝑖−𝜃𝑜𝑙𝑛�𝜃𝑖𝜃𝑜�

= 90−40𝑙𝑛�90

40�

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

Radiation

(iii) Interchange factor

𝑓12 = 11𝜀1

+𝐴1𝐴2� 1𝜀2−1�

= 11

0.6+2×10−3100 � 1

0.3−1�

First calculate �2×10−3

100 � � 10.3 − 1�

(2 * 3 +/- 10x / 100) * (1 / 0.3 – 1 ) = 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 = 𝑓12𝜎𝐴1(𝑇14 − 𝑇2

4)

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


Forecasting

(i) 𝑢𝑓 = 𝛼𝑆𝑡 + 𝛼(1 − 𝛼)𝑆𝑡−1 + 𝛼(1 − 𝛼)2𝑆𝑡−2 + 𝛼(1 − 𝛼)3𝑆𝑡−3

𝑢𝑓 = 0.4 × 95 + 0.4 × 0.6 × 82 + 0.4 × 0.62 × 68 + 0.4 × 0.63 × 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 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



xy = Ax + Bx2



Example:

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Data x Y Xy x2

1 1 1 1 x1 12

2 2 2 2 x 2 22

3 3 3 3 x 3 32

∑𝑥 = 6 ∑𝑦 = 6 ∑𝑥𝑦 = 14 ∑𝑥2 = 14 For ∑𝑥 1 + 2 + 3 = 6

For ∑𝑦 1 + 2 + 3 = 6

For ∑𝑥𝑦 1 * 1 + 2 * 2 + 3 * 3 = 14






Now ∑𝑦 = 𝐴𝑛 + 𝐵∑𝑥 ………… . . (𝑖)

or 6 = 3 𝐴 + 6𝐵 ………… . . (𝑖)

and ∑𝑥𝑦 = 𝐴∑𝑥 + 𝐵∑𝑥2 ………… . . (𝑖𝑖)

or 14 = 6A + 14 B ………… . . (𝑖𝑖)



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Optimum run size

(iii) 𝑄 = �2𝑈𝑅𝐼𝑐

× �𝐼𝑐+𝐼𝑝𝐼𝑝

�

𝑄 = �2 × 30000 × 35002.5 × �2.5 + 10

10 �

First calculate �2×30000 ×35002.5 � × �(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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