Post on 28-Jan-2016
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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. βππ
9. |π| 10. β
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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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, canβt 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 β Bracket O β Order (Power and roots) D β Division M β Multiplication A β Addition S β 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 β 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 β 1) 4 * ( 5 x 3.4 β 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
101.4β1
1.4 10((1.4β1)1.4 ) ((1.4 β 1)/1.4) 10X = 1.930698
Or you may simplify
101.4β1
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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οΏ½π2π1οΏ½
πΎπΎβ1
βΉ οΏ½π2π1οΏ½
πΎ(πΎβ1)
βΉ οΏ½53οΏ½
1.4(1.4β1)
(5/3) xy 1.4/(1.4 β 1) = 5.111263
8. βπ₯π¦
β325 32 βπ₯π¦ 5 = 2
Note: you have to first type value of x then βπ₯π¦ button then value of y. Logically value of x should be given first then value of y.
We may use xy function also β325 = 321/5 = 32 xy (1/5) = 2
But in this case (1/5) is must you canβt use 32 xy 1/5 β wrong
9. |π₯| |β5| 5 +/- = |π₯| = 5
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10. β
β5 5 β = 2.236068
Note: you have to first type value then β button.
β32 + 42 =οΏ½(32 + 42) = ( 3 x2 + 4 x2 ) β = 5
But
ππ = 1β2
οΏ½[(π1 β π2)2 + (π2 β π3)2 + (π3 β π1)2]
ππ = 1β2οΏ½[(97.74 β 22.96)2 + (22.96 β 20)2 + (20 β 97.74)2]
Using bracket also we canβt calculate it directly, we have to use M+
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(97.74 β 22.96) x2 = 5592.048 M+ then press C button
(22.96 β 20) x2 = 8.7616 M+ then press C button
(20 β 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 β button 107.9089
[ it is = οΏ½[(97.74 β 22.96)2 + (22.96 β 20)2 + (20 β 97.74)2] ]
Now divide it with β2
107.9089 / 2 β = 76.30309
Therefore, ππ = 1β2 οΏ½[(97.74 β 22.96)2 + (22.96 β 20)2 + (20 β 97.74)2] = 76.30309
After the calculation you must press MC button.
11. 1/x
This is generally used at middle of calculation.
0.45πππ 121 β 0.45π ππ12
We first calculate 1 β 0.45sin12 then use 1/x button.
1 β 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 β B ) = sin (30-10.5)
(30 β 10.5 ) sin = 0.3338
cos ( Ο + Ξ² - Ξ± ) = cos (20.15 + 33 -10 ) ( 20.15 + 33 - 10) cos = 0.729565
tan (Ξ¦ - Ξ± ) = tan (17.3 β 10) (17.3 β 10 ) tan = 0.128103
β
π ππ 2π = 2.0π ππ 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
π¦ β π¦1π¦2 β π¦1
= π₯ β π₯1π₯2 β π₯1
1.8 β 0.82.0 β 0.8 = π₯ β 10
60 β 10
π₯ β 10 = (60 β 10) Γ 1.8 β 0.82.0 β 0.8
π₯ = 10 + (60 β 10) Γ 1.8 β 0.82.0 β 0.8
10 + (60 β 10) * (1.8 β 0.8) / (2.0 β 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 βπ¦ = π΄π + π΅βπ₯ β¦β¦β¦β¦ . . (π)
Next step multiply both side of original equation by x
xy = Ax + Bx2
Again take summation of both sides βπ₯π¦ = π΄βπ₯ + π΅βπ₯2 β¦β¦β¦β¦ . . (ππ)
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
βπ₯ = 6 βπ¦ = 6 βπ₯π¦ = 14 βπ₯2 = 14 For βπ₯ 1 + 2 + 3 = 6
For βπ¦ 1 + 2 + 3 = 6
For βπ₯π¦ 1 * 1 + 2 * 2 + 3 * 3 = 14
For βπ₯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 βπ₯2 = 14
Now βπ¦ = π΄π + π΅βπ₯ β¦β¦β¦β¦ . . (π)
or 6 = 3 π΄ + 6π΅ β¦β¦β¦β¦ . . (π)
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and βπ₯π¦ = π΄βπ₯ + π΅βπ₯2 β¦β¦β¦β¦ . . (ππ)
or 14 = 6A + 14 B β¦β¦β¦β¦ . . (ππ)
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 (Ξ¦)
π‘ππβ = ππππ πΌ1βππ πππΌ = ππππ πΌ
(1βππ πππΌ ) [We have to use one extra bracket in the denominator]
π‘ππβ = 0.45πππ 12(1β0.45π ππ12)
First find the value of π‘ππβ
0.45 * 12 cos / ( 1 β 0.45 * 12 sin ) = 0.4855991
Then find β
Just press button tan-1 25.901
Shear strain (Ξ³)
πΎ = πππ‘β + tan(β β πΌ)
πΎ = πππ‘17.3 + tan(17.3 β 10)
πΎ = 1π‘ππ 17.3 + tan(17.3 β 10)
It is a long calculation; we have to use M+
1π‘ππ 17.3 = 1 / 17.3 tan = 3.210630 M+ then press C button
tan(17.3 β 10) = (17.3 - 10) tan = 0.1281029 M+
Then find πΎ
Just press button MR 3.338732
πβπππππππ ( πΎ) = πππ‘17.3 + tan(17.3 β 10) = 3.34
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Velocity relations
ππ π = πππ πΌ
πππ (β β πΌ)
ππ 2.5 = πππ 10
πππ (22.94 β 10)
ππ = 2.5 Γ πππ 10πππ (22.94 β 10)
2.5 * 10 cos / ((22.94 - 10) cos ) = 2.526173
Merchant Circle
(i) πΉπ = ππ Γ ππ‘π ππβ = 285 Γ 3Γ0.51
(π ππ20.15) [we have to use extra bracket for denominator]
285 * 3 * 0.51 / (20.15 sin ) = 1265.824
(ii) πΉπ = π πππ (β + π½ β πΌ)
ππ π = πΉπ πππ (β + π½ β πΌ) = 1265.8
οΏ½πππ (20.15 + 33 β 10)οΏ½
[We have to use extra bracket for denominator]
1265.8 / ((20.15 + 33 - 10) cos ) = 1735.005
Force Relations
πΉπ = πΉππππ β β πΉπ‘π ππβ
πΉπ = 900 πππ 30 β 600 π ππ30
900 * 30 cos - 600 * 30 sin = 479.4229
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Turning
(i) π‘ = ππ πππ = 0.32 π ππ75
0.32 * 75 sin = 0.3091
(ii) πΉπ‘ = πΉπ₯π πππ = 800
(π ππ75) [We have to use extra bracket for denominator]
800 / ( 75 sin ) = 828.2209
Specific Energy
π = πΉπ1000ππ = 800
(1000Γ0.2Γ2) [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
π¦ β π¦1π¦2 β π¦1
= π₯ β π₯1π₯2 β π₯1
1.8 β 0.82.0 β 0.8 = π₯ β 10
60 β 10
π₯ β 10 = (60 β 10) Γ 1.8 β 0.82.0 β 0.8
π₯ = 10 + (60 β 10) Γ 1.8 β 0.82.0 β 0.8
10 + (60 β 10) * (1.8 β 0.8) / (2.0 β 0.8) = 51.66667
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Tool life equation (i) π1π1
π = π2π2π
or 100 Γ 10π = 75 Γ 30π
or 10075 = οΏ½30
10οΏ½π
or 43 = 3π
or ππ οΏ½43οΏ½ = πππ3
or π = πποΏ½43οΏ½
(ππ3) [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) π3 = π1 Γ οΏ½π1π3οΏ½π
= 30 Γ οΏ½6030οΏ½
0.204
30 * ( 60 / 30 ) xy 0.204 = 34.55664
(iv) οΏ½90π₯ οΏ½
10.45 > οΏ½60
π₯ οΏ½1
0.3
or οΏ½90π₯ οΏ½
10.45 = οΏ½60
π₯ οΏ½1
0.3
or οΏ½90π₯ οΏ½
0.3= οΏ½60
π₯ οΏ½0.45
[Make power opposite]
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or π₯0.45
π₯0.3 = 600.45
900.3
or π₯0.15 = 600.45
900.3 = 60 xy 0.45 / 90 xy 0.30 = 1.636422
or π₯ = (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
First take summation of both sides βπ¦ = π΄π + π΅βπ₯ β¦β¦β¦β¦ . . (π)
Next step multiply both side of original equation by x
xy = Ax + Bx2
Again take summation of both sides βπ₯π¦ = π΄βπ₯ + π΅βπ₯2 β¦β¦β¦β¦ . . (ππ)
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
βπ₯ = 6 βπ¦ = 6 βπ₯π¦ = 14 βπ₯2 = 14 For βπ₯ 1 + 2 + 3 = 6
For βπ¦ 1 + 2 + 3 = 6
For βπ₯π¦ 1 * 1 + 2 * 2 + 3 * 3 = 14
For βπ₯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 βπ₯2 = 14
Now βπ¦ = π΄π + π΅βπ₯ β¦β¦β¦β¦ . . (π)
or 6 = 3 π΄ + 6π΅ β¦β¦β¦β¦ . . (π)
and βπ₯π¦ = π΄βπ₯ + π΅βπ₯2 β¦β¦β¦β¦ . . (ππ)
or 14 = 6A + 14 B β¦β¦β¦β¦ . . (ππ)
Solving (i) and (ii) we get A = 0 and B = 1
y = 0 + 1. x is the solution.
Economics in metal cutting
ππ = οΏ½ππ + πΆπ‘πΆπ
οΏ½ οΏ½1 β ππ οΏ½
ππ = οΏ½3 + 6.50.5οΏ½ οΏ½
1 β 0.20.2 οΏ½
To = ( 3 + 6.5 / 0.5 ) (1 β 0.2 ) / 0.2 = 64 min
Now πππππ = πΆ
or ππ(64)0.2 = 60
or ππ = 60640.2
60 / 64 xy 0.2 = 26.11 m/min
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Metrology π = 0.45βπ·3 + 0.001π·
π = 0.45β97.983 + 0.001 Γ 97.98
0.45 * 97.98 βππ 3 = + 0.001 * 97.98 = 2.172535
Rolling
cosπΌ = 1 β ββπ· = 1 β 5
600
πΆ = 1 - 5 / 600 = cos-1 = 7.40198o
If you want πΌ in radian after calculating 7.40198 just press * π/180 and you will get πΌ = 0.129189 ππππππ
Forging
(i) ππ1
2
4 Γ β1 = ππ22
4 Γ β2
π2 = π1 Γ οΏ½β1β2
= 100 Γ οΏ½5025 = 100 Γ β2
100 * ( 50 / 25) β = 141.4214
or 100 * 2 β = 141.4214
(ii) π₯π = 48 β οΏ½ 62Γ0.25οΏ½ ππ οΏ½
12Γ0.25οΏ½
48 β (6 / 2 / 0.25 ) * (1 / 2 / 0.25 ) ln = 39.68223
(iii) πΉπ π‘ππππππ = 2β« οΏ½ππ + 2πΎβ (π₯π β π₯)οΏ½π΅ππ₯π₯π
0
we have to first integrate without putting values
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πΉπ π‘ππππππ = 2π΅ οΏ½ππ π₯ + 2πΎβ οΏ½π₯π π₯ β
π₯2
2 οΏ½οΏ½0
π₯π
πΉπ π‘ππππππ = 2π΅ οΏ½ππ π₯π + 2πΎβ οΏ½π₯π 2 β
π₯π 22 οΏ½οΏ½
πΉπ π‘ππππππ = 2π΅ οΏ½ππ π₯π + πΎβ π₯π
2οΏ½
πΉππ‘ππππππ = 2 Γ 150 Γ οΏ½16.16 Γ 39.68 + οΏ½4.046 οΏ½ Γ 39.682οΏ½
2 * 120 * ( 16.16 * 39.68 + ( 4.04 / 6 ) * 39.68 x2 ) = 510418.2
πΉπ π‘ππππππ = 510418.2 π
πΉπππππππ = 2 οΏ½ 2πΎπ2πβ (πΏβπ₯)π΅ππ₯
πΏ
π₯π
πΉπππππππ = 4πΎπ΅ οΏ½π2πβ (πΏβπ₯)ππ₯
πΏ
π₯π
πΉπππππππ = 4πΎπ΅ οΏ½π2πβ (πΏβπ₯)
β 2πβ
οΏ½π₯π
πΏ
πΉπππππππ = 4πΎπ΅οΏ½β 2π
β οΏ½οΏ½π0 β π
2πβ (πΏβπ₯π )οΏ½
πΉπππππππ = οΏ½2πΎπ΅βπ οΏ½ οΏ½ποΏ½οΏ½
2πβ οΏ½(πΏβπ₯π )οΏ½ β 1οΏ½ [Note: extra brackets are used]
πΉπππππππ = οΏ½2 Γ 4.04 Γ 150 Γ 60.25 οΏ½ οΏ½ποΏ½οΏ½
2Γ0.256 οΏ½(48β39.68)οΏ½ β 1οΏ½
(2 * 4.04 * 150 * 6 / 0.25) * (((2 * 0.25/6) * (48 β 39.68)) ex - 1) =
This is very large calculation; this weak calculator canβt 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 β 39.68)) ex - 1) = 1.000372
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Now multiply both 29088 * 1.000372 = 29098.82
πΉπππππππ = 29098.82 π
πΉπππ‘ππ = πΉππ‘ππππππ + πΉππππππ = 510418.2 + 29098.82 = 539517 π = 539.52 πΎπ
Extrusion
πΉ = 2ππ Γ πππ24 Γ ππ οΏ½ππππ
οΏ½
πΉ = 2 Γ 400 Γ οΏ½π Γ 82
4 οΏ½ ππ οΏ½54οΏ½
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) ππ = ππ(1+π΅)π΅ οΏ½1 β οΏ½ππππ οΏ½
2π΅οΏ½
ππ = 400 Γ(1 + 1.7145)
1.7145 οΏ½1 β οΏ½ 56.25οΏ½
2Γ1.7145οΏ½
It is a long calculation,
First calculate, 400 Γ (1+1.7145)1.7145 = 400 * (1 +1.7145) / 1.7145 = 633.3040
Then calculate,
οΏ½1 β οΏ½ 56.25οΏ½
2Γ1.7145οΏ½ = (1 β(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) ππ = ππ(1+π΅)π΅ οΏ½1 β οΏ½πππππππ
οΏ½2π΅οΏ½ + οΏ½πππππππ
οΏ½2π΅
Γ ππ
400 = 400 Γ(1 + 1.7145)
1.7145 οΏ½1 β οΏ½πππππ6.25οΏ½
2Γ1.7145οΏ½ + οΏ½
πππππ6.25οΏ½
2Γ1.7145Γ 50
Let οΏ½πππππ6.25 οΏ½2Γ1.7145
= π₯
or 400 = 400 Γ (1+1.7145)1.7145 [1β π₯] + π₯ Γ 50
Calculate, 400 Γ (1+1.7145)1.7145 = 400 * (1 +1.7145) / 1.7145 = 633.3
or 400 = 633.3[1 β π₯]+ π₯Γ 50
or π₯ = (633.3β400)(633.3β50) β 0.4 = οΏ½πππππ6.25 οΏ½
2Γ1.7145
or πππππ = 6.25 Γ (0.4)1
2Γ1.7145
or πππππ = 6.25 * 0.4 xy (1 / 2 / 1.7145) = 4.784413 mm
Sheet Metal Operation (i) πΆ = 0.0032π‘βπ
πΆ = 0.0032 Γ 1.5 Γ β294
0.0032 * 1.5 * 294 β = 0.08230286 mm
(ii) πΉ = πΏπ‘π
πΉ = 2(π + π)π‘π = 2(100 + 50) Γ 5 Γ 300
2 * (100+50) * 5 * 300 = 450000 N = 450 KN
(iii) π· = βπ2 + 4πβ
π· = οΏ½(252 + 4 Γ 25 Γ 15) [Extra bracket used]
( 25 x2 + 4 * 25 * 15) β = 46.09772 mm
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(iv) π‘πππππ = π‘ππππ‘πππππ1 Γππ2 = 1.5
(π0.05 Γπ0.09) [Extra bracket for denominator]
1.5 / ( 0.05 ex * 0.09 ex ) = 1.304038 mm
Casting
(i) π΅π’ππ¦ππππ¦ πππππ = ππ2
4 Γ βοΏ½πππππ’ππ β πππππ οΏ½ Γ π
π΅π’ππ¦ππππ¦ πππππ = οΏ½π Γ 0.1202
4 οΏ½ Γ 0.180 Γ (11300 β 1600) Γ 9.81
( π * 0.12 x2 / 4 ) * 0.18 * (11300 - 1600) * 9.81 = 193.7161 N
(ii) π‘π = π΅ οΏ½ππ΄οΏ½2
Find values of V and A separately and then
B * (V / A) x2 = 0
Welding
(i) πππΆπ + πΌ
ππΆπΆ = 1
45ππΆπ + 500
ππΆπΆ = 1 β¦β¦ . . (π)
55ππΆπ + 400
ππΆπΆ = 1 β¦β¦ . . (ππ)
Now (ii) x 5 - (i) x 4 will give
(55 Γ 5 β 45 Γ 4)ππΆπ = (5 β 4) = 1
or OCV = 95 V
Now from equation (i)
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4595 + 500
ππΆπΆ = 1
or 500ππΆπΆ = οΏ½1 β 45
95οΏ½
or ππΆπΆ = 500οΏ½1β45
95οΏ½
500 / ( 1 β 45 / 95) = 950 V
(ii) π» = πΌ2π π‘ = 300002 Γ 100 Γ 10β6 Γ 0.005
30000 x2 * 100 * 6 +/- 10x * 0.005 = 450 J
Machine Tools
(i) Turning time ( T ) = (πΏ+π΄+π)(ππ)
( L + A + O ) / ( f * N ) = 0
(ii) Drilling time ( T ) = (πΏ+β+π΄+π)(ππ)
L = 50 mm
β = π·2π‘πππΌ = 15
(2 Γ π‘ππ59) = 15/ (2 β59 tan ) = 4.5 ππ
A = 2 mm
O = 2 mm
f = 0.2 mm/rev
N = 500 rpm
π =(50 + 4.5 + 2 + 2)
(0.2 Γ 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
1π = π₯1
π1+ π₯2π2
+ π₯3π3
+ π₯4π4
or 1π = 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
π = 8.069989 π/ππ
(ii) Find equivalent weight of an alloy
1πΈ = π₯1
πΈ1+ π₯2πΈ2
+ π₯3πΈ3
+ π₯4πΈ4
or 1πΈ = π₯1π£1
πΈ1+ π₯2π£2
πΈ2+ π₯3π£3
πΈ3+ π₯4π£4
πΈ4
or 1πΈ = 0.7Γ2
58.71 + 0.2Γ251.99 + 0.05Γ2
55.85 + 0.05Γ347.9
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
πΈ = 27.42593
Alternate Method β 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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πΈ = 27.42593
Alternate Method β 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
πΈ = 27.42593
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Strength of Materials
(Only for the type of equations which are not yet covered)
Elongation
(i) πΏ = ππΏπ΄πΈ
or πΏ = 10Γ103Γ1000πΓ52
4 Γ200Γ103 ππ
or πΏ = 100Γ4(πΓ52Γ2) ππ
[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) 0.5Γ12.5Γ10β6Γ20
οΏ½1+ 50Γ0.5
οΏ½πΓ0.0124 Γ200Γ106οΏ½
οΏ½
First calculate 50Γ0.5
οΏ½πΓ0.0124 Γ200Γ106οΏ½
= 50Γ0.5Γ4(πΓ0.012Γ200Γ106)
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 Γ 12.5 Γ 10β6 Γ 20
0.9984105 * 0.5 * 12.5 * 6 +/- 10x * 20 = 0.0001248013
Principal stress and principal strain
(iii) ππππ = οΏ½οΏ½ππβπππ οΏ½π
+ ππππ
ππππ₯ = οΏ½οΏ½οΏ½οΏ½80 β 202 οΏ½οΏ½
2+ 402οΏ½
[One bracket for denominator one bracket for square and one for square root]
(((80-20) / 2 ) x2 + 40 x2 ) β = 50 MPa
For π1,2 = ππ₯+ππ¦2 + οΏ½οΏ½ππ₯βππ¦2 οΏ½
2+ ππ₯π¦2
First calculate οΏ½ππ₯+ππ¦οΏ½
2
And then calculate οΏ½οΏ½οΏ½οΏ½ππ₯βππ¦2 οΏ½οΏ½2
+ ππ₯π¦2 οΏ½
Deflection of Beams
(iv) πΏ = π€πΏ4
8πΈπΌ = 10Γ103Γ54
(8Γ781250 )
10 * 3 10x * 5 xy 4 / (8 * 781250 ) = 1 mm
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Bending stresses
(v) π = ππ¦πΌ = 9.57Γ103Γ0.1
οΏ½0.1Γ0.2312 οΏ½
Pa
= 9.57 Γ 103 Γ 120.23
9.57 * 3 10x * 12 / (0.2 xy 3 ) = 1.435500e+7 Pa = 14.355 MPa
Torsion
(vi) ππ½ = πΊπ
πΏ
409.256π
32(1β0.74)π·4 = 80Γ109Γπ1Γ180
or π·4 = 32Γ409.256Γ180π2Γ(1β0.74)Γ80Γ109
First calculate 32 * 409.256 * 180 = 2357315
Then calculate π2 Γ (1 β 0.74) Γ 80 Γ 109
π x2 * (1 β 0.7 xy 4) * 80 * 9 10x = 5.999930e+11
Now π·4 = 23573155.999930Γ1011 = 0.000003928904
Just press β button twice , D = 0.04452130 m = 44.52 mm
Spring
(vii) πΏ = 8ππ·3ππΊπ4
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8Γ200Γ103Γ10β6Γ10(80Γ109Γ84Γ10β12)
8*200*310x6 +/- 10x10 /(80* 9 10x 8 xy 4 * 12 +/- 10x ) = 0.04882813 m
= 48.83 mm
Theories of column
(viii) πππ = π2πΈπΌ4πΏ2 [For one end fixed and other end free]
10 Γ 103 = π2Γ210Γ109ΓπΓπ464
4Γ42 or 10 Γ 103 Γ 4 Γ 42 Γ 64 = π2 Γ 210 Γ 109 Γ π Γ π4
or π4 = 10Γ103Γ4Γ42Γ64π3Γ210Γ109
First calculate 10 Γ 103 Γ 4 Γ 42 Γ 64
10 * 3 10x * 4 * 4 x2 * 64 = 4.096000e+7
Then calculate π3 Γ 210 Γ 109
π x3 * 210 * 9 10x = 6.511319e+12
πππ€ π4 = 4.096000e + 76.511319π + 12 = 0.000006290584
Just press β button twice, d = 0.05008097 m β 50 mm
Theories of Failure
(ix) ππ = 1β2οΏ½[(π1 β π2)2 + (π2 β π3)2 + (π3 β π1)2]
ππ = 1β2οΏ½[(97.74 β 22.96)2 + (22.96 β 20)2 + (20 β 97.74)2]
Using bracket also we canβt calculate it directly, we have to use M+
(97.74 β 22.96) x2 = 5592.048 M+ then press C button
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(22.96 β 20) x2 = 8.7616 M+ then press C button
(20 β 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 β button 107.9089
[ it is = οΏ½[(97.74 β 22.96)2 + (22.96 β 20)2 + (20 β 97.74)2] ]
Now divide it with β2
107.9089 / 2 β = 76.30309
Therefore, ππ = 1β2 οΏ½[(97.74 β 22.96)2 + (22.96 β 20)2 + (20 β 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) ππ = 12π οΏ½
ππ = 1
2ποΏ½οΏ½40Γ103
100 οΏ½
(40 * 10 x3 / 100 ) β / 2 / π = 3.183099
Transmissibility ratio
(ii) ππ = οΏ½1+(2ππ )2
οΏ½(1βπ2)2+(2ππ )2
ππ = οΏ½1 + (2 Γ 0.15 Γ 18.85)2
οΏ½(1 β 18.852)2 + (2 Γ 0.15 Γ 18.85)2
First calculate (2ππ)2 = (2 Γ 0.15 Γ 18.85)2
(2 * 0.15 * 18.85 ) x2 = 31.97903 This data is needed again so PressM+
Next find (1 β π2)2 =(1 β 18.852)2
(1 β 18.85 x2 ) x2 = 125544.4
Now find the value of numerator
Press MR + 1 = then press β 5.742737
Then find denominator
Press MR + 125544.4 = then press β 354.3676
Now Find (TR)
Press 1/x and * 5.742737 = 0.01620559
TR = 0.01620559 (Answer)
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Thermodynamics
(Only for the type of equations which are not yet covered)
SFEE
(i) β1 + π12
2000 + ππ1000 + ππ
ππ = β1 + π12
2000 + ππ1000 + ππ
ππ
3200 + 1602
2000 + 9.81 Γ 101000 + 0 = 2600 + 1002
2000 + 9.81 Γ 61000 + ππ
ππ
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
ππππ = 3200 + 1602
2000 + 9.81 Γ 101000 β 2600 β 1002
2000 β 9.81 Γ 61000
Entropy Change
(ii) ππ β ππ = ππππ οΏ½πππποΏ½ β π ππ οΏ½πππποΏ½
ππ β ππ = 1.005 ππ οΏ½300350οΏ½ β 0.287ππ οΏ½ 50
150οΏ½
M+ M-
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First calculate 1.005 ππ οΏ½300350οΏ½
1.005 * (300 / 350 ) ln = -0.1549214 Press M+ then press C button
Then calculate 0.287ππ οΏ½ 50150οΏ½
0.287 * (50 /150 ) ln = -0.3153016 Press M-
Just press MR and it is the answer 0.16038020000000003
β΄ βπ = 0.16 πΎπ½/πΎππΎ
Available Energy
(iii) π΄πΈ = πππ οΏ½(π2 β π1)β ππππ οΏ½π2π1οΏ½οΏ½
π΄πΈ = 2000 Γ 0.5 οΏ½(1250 β 450) β 303ππ οΏ½1250450 οΏ½οΏ½
First calculate οΏ½(1250 β 450) β 303ππ οΏ½1250450 οΏ½οΏ½
(1250-450)-303 * (1250 / 450) ln = 490.4397
Then multiply with 2000 Γ 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) π = 2ππΏοΏ½π‘πβπ‘ποΏ½ππ οΏ½π2
π1οΏ½πΎπ΄
+ππ οΏ½π3
π2οΏ½πΎπ΅
π = 2 Γ π Γ 1 Γ (1200β 600)
ππ οΏ½0.0250.01 οΏ½19 +
ππ οΏ½0.0550.025οΏ½0.2
First calculate denominator πποΏ½0.025
0.01 οΏ½19 + πποΏ½0.055
0.025οΏ½0.2
But it is very weak calculator canβt 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 Γ π Γ 1 Γ (1200 β 600)
0.2505944 * 2 * π * 600 = 944.7186 W/m
β΄ π = 2 Γ π Γ 1 Γ (1200β 600)
ππ οΏ½0.0250.01 οΏ½19 +
ππ οΏ½0.0550.025οΏ½0.2
= 944.72 π/π
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Unsteady Conduction
(ii) πππ
= πβππππβππ
= πβπ΅ππΉπ
298 β 30030 β 300 = πβ425πΓ2.3533Γ10β3
or ππ οΏ½298β30030β300 οΏ½ = β425π Γ 2.3533 Γ 10β3
or ππ οΏ½ 30β300298β300οΏ½ = 425π Γ 2.3533 Γ 10β3
or π =πποΏ½ (30β300)
(298β300)οΏ½425Γ2.3533Γ10β3
((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) πΏπππ· = ππβπππποΏ½πππποΏ½
= 90β40πποΏ½90
40οΏ½
(90 / 40) ln = then press 1/x then multiply with numerator * (90 β 40) = 61.65760
Radiation
(iii) Interchange factor
π12 = 11π1
+π΄1π΄2οΏ½ 1π2β1οΏ½
= 11
0.6+2Γ10β3100 οΏ½ 1
0.3β1οΏ½
First calculate οΏ½2Γ10β3
100 οΏ½ οΏ½ 10.3 β 1οΏ½
(2 * 3 +/- 10x / 100) * (1 / 0.3 β 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 β0.6
Now ππππ‘ = π12ππ΄1(π14 β π2
4)
ππππ‘ = 0.6 Γ 5.67 Γ 10β8 Γ 2 Γ 10β3(8004 β 3004)
First calculate 0.6 Γ 5.67 Γ 10β8 Γ 2 Γ 10β3
0.6 * 5.67 * 8 +/- 10x * 2 * 3 +/- 10x = 6.804000e-11
Then multiply with (8004 β 3004)
6.804000e-11 * (800 xy 4 - 300 xy 4) = 27.31806 W
ππππ‘ = 0.6 Γ 5.67 Γ 10β8 Γ 2 Γ 10β3(8004 β 3004) = 27.32 π
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Industrial Engineering
(Only for the type of equations which are not yet covered)
Forecasting
(i) π’π = πΌππ‘ + πΌ(1 β πΌ)ππ‘β1 + πΌ(1 β πΌ)2ππ‘β2 + πΌ(1 β πΌ)3ππ‘β3
π’π = 0.4 Γ 95 + 0.4 Γ 0.6 Γ 82 + 0.4 Γ 0.62 Γ 68 + 0.4 Γ 0.63 Γ 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
π’π = 0.4 Γ 95 + 0.4 Γ 0.6 Γ 82 + 0.4 Γ 0.62 Γ 68 + 0.4 Γ 0.63 Γ 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 βπ¦ = π΄π + π΅βπ₯ β¦β¦β¦β¦ . . (π)
Next step multiply both side of original equation by x
xy = Ax + Bx2
Again take summation of both sides βπ₯π¦ = π΄βπ₯ + π΅βπ₯2 β¦β¦β¦β¦ . . (ππ)
Just solve this two equations and find A and B
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
βπ₯ = 6 βπ¦ = 6 βπ₯π¦ = 14 βπ₯2 = 14 For βπ₯ 1 + 2 + 3 = 6
For βπ¦ 1 + 2 + 3 = 6
For βπ₯π¦ 1 * 1 + 2 * 2 + 3 * 3 = 14
For βπ₯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 βπ₯2 = 14
Now βπ¦ = π΄π + π΅βπ₯ β¦β¦β¦β¦ . . (π)
or 6 = 3 π΄ + 6π΅ β¦β¦β¦β¦ . . (π)
and βπ₯π¦ = π΄βπ₯ + π΅βπ₯2 β¦β¦β¦β¦ . . (ππ)
or 14 = 6A + 14 B β¦β¦β¦β¦ . . (ππ)
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) π = οΏ½2ππ πΌπ
Γ οΏ½πΌπ+πΌππΌπ
οΏ½
π = οΏ½2 Γ 30000 Γ 35002.5 Γ οΏ½2.5 + 10
10 οΏ½
First calculate οΏ½2Γ30000 Γ35002.5 οΏ½ Γ οΏ½(2.5+10)
10 οΏ½
(2 * 30000 *3500 / 2.5) * ((2.5 + 10) / 10) = 1.050000e+8
Then just press β
1.050000e+8 β = 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.
βEk Ghatiya Calculator ka Sahi Upyogβ
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