Calculator Techniques for Engineering Mechanics Using Casio Calculators
Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.
Engineering mechanics is an applied section in the field of Mathematics that includes a variety of problems that are realistic. Whenever you see trusses, you might be wondering of the value of the axial forces of each member. When you see a cable hanging, you might think of the tension forces in each cable. It is the branch of physics that supports the subdivided fields like Kinematics and Dynamics. In terms of Mathematics, mechanics is one of the most used subjects in realworld scenarios. It is subdivided into 3 minor sections or branches namely:
 Statics
 Dynamics
 Kinetics.
Engineering Mechanics plays a vital role in board examinations. Most of the mathematical problems require you to use Engineering Mechanics either it is in the static, kinematic or dynamics field. It is also the basic foundation in learning Structural Theory. Structural Theory is a course related to Structural Engineering that covers applied physics and performance of different materials and geometric properties. In fact, Engineering Mechanics is the widely used component of Physics by American Society of Civil Engineers.
What You'll Learn:
 Right Triangle Technique
 Components of a 2D Force
 Concurrent  Coplanar Force System
 Non  Concurrent Coplanar Force System
 Components of a 3D Force
 Parallel  Noncoplanar Force System
 Equilibrium of Coplanar Forces
 Beams
 Centroid or Moment of Inertia
Calculator Techniques in Engineering Mechanics are very easy to learn. It just uses 'STAT' mode and 'EQN' mode of your calculators. Casio calculators are highly recommended in learning the following calculator techniques.
Right Triangle Technique
In the figure below, find the remaining side and angles if A=5 and B=7.
CALCULATOR TECHNIQUE
You might have heard rectangular and polar coordinates in your Analytic Geometry class. These functions are to use in this kind of problem in Mechanics. Calculators today have functions called 'Rec' and 'Pol' Functions. These functions are located in the lower right portion of your calculator where plus and minus operations are situated.
'Rec' function converts polar coordinates (r,θ) into rectangular coordinates (x,y). On the other hand, 'Pol' function converts rectangular coordinates to polar coordinates. To use the 'Rec' function in your calculator, press [SHIFT] then press the [] operation on the lower right corner. To use the other function which is the 'Pol' function, press [SHIFT] then [+].
To start, let x=5 and y=7.
1. Remain your calculator in computation mode [1: COMP]. Then, use the 'Pol' function.
Command: SHIFT]  [+]
2. Input the values x, y in the form (x,y).
Command: [5]  [SHIFT]  [ ) ]  [7]  [ ) ]
In this step, your screen should display: Pol(5,7)
3. Press Enter or equal sign (=). There are two answers displayed in your calculator. The first one is the 'r' and beside it is 'θ'. The first answer 'r' is considered the length of the hypotenuse C while 'θ' corresponds to the angle opposite B which is β. From these values, you may now be able to get the last angle α by subtracting 90° and β from 180°.
Command: [ = ]
Answers:
r = 8.602325267 ≅ C
θ = 54.46232221 ≅ β°
Components of a 2D Force
In the figure shown, find the xcomponent and ycomponent of the force if F = 5000 N and angle θ = 35°.
CALCULATOR TECHNIQUE
Let r = 5000 N and θ = 35°.
1. Apply the right triangle technique but this time around using 'Rec' function.
Command: [SHIFT]  []
2. Input the values r and θ in the form (r,θ).
Command: [5000]  [SHIFT]  [ ) ]  [35]  [ ) ]
It should display: Rec(5000,35)
3. Lastly, press the equal sign. This time, the calculator displays values in rectangular coordinates, x, and y. The 'x' coordinate corresponds to the xcomponent of the force F while the 'y' coordinate corresponds to the ycomponent of the force F.
Command: [ = ]
Answers:
x = 4095. 760221 ≅ Fx
y = 2867.882182 ≅ Fy
Concurrent  CoPlanar Force System
For the forces in the figure shown,
a. Find the magnitude of the horizontal component of the resultant.
b. Find the magnitude of the resultant.
c. Find the angle that the resultant makes with the xaxis.
CALCULATOR TECHNIQUE
1. Complex mode approach is the approach to answering this problem. Start solving by entering the mode setup in the top righthand corner of your calculator buttons. Pick the complex mode.
Command: [MODE]  [2:CMPLX]
2. Input the values magnitude of forces and their angles with respect to the positive xaxis counterclockwise. Remember that θ is always measured from the origin counterclockwise. To familiarize, use this guide:
1st Quadrant: as is
2nd Quadrant: 180  θ
3rd Quadrant: 180 + θ
4th Quadrant: θ or 360  θ
Now, input the values in your calculator in the form 'r<θ' where 'r' is the force and θ is the corresponding angle of the force from the origin or positive xaxis. Just like the simple summation of forces, get the summation of 'r<θ'.
Command: [86.50]  [SHIFT]  [ (  ) ]  [30]  [+]  [70.5]  [SHIFT]  [ (  ) ]  [18045]  [68.50]  [SHIFT]  [ (  ) ]  [270]
In this step, your calculator must display 86.50<(30)+70.5<(18045)+68.50<(270).
The answer displayed should be:
25.06016935+24.60102807i
As you can observe, the answer is in the form a+bi which is in rectangular form. The first term in the answer corresponds to your horizontal component Fx of the resultant force while the other term paired with 'i' corresponds to the vertical component Fy of the resultant force.
3. The last step is to solve the magnitude of the resultant and the angle made by the resultant force.
Command: [SHIFT]  [2]  [3: r<θ]  [ = ]
The answer displayed should be:
35.11727026<44.47028849
Answers:
a. Fx = 25.06016935 N
b. F = 35. 11727026 N
c. θ = 44.47028849°
NonConcurrent  CoPlanar Force System
Consider the coplanar force system shown.
a. Find the magnitude of the resultant.
b. Compute the location of the resultant from the yaxis.
c. Compute the location of the resultant from the xaxis.
CALCULATOR TECHNIQUE
1. Complex mode approach is the approach to answering this problem. Start solving by entering the mode setup in the top righthand corner of your calculator buttons. Pick the complex mode.
Command: [MODE]  [2:CMPLX]
2. Get the resultant force of the 2 linear forces using the [SHIFT]  [ (  ) ] command again. The answer should be 5.997 + 12.9963i.
Command: [10]  [SHIFT]  [ (  ) ]  [90]  [ + ]  [6.7]  [SHIFT]  [ (  ) ] 
[tan^{1}(1/2)]
3. Solve for the magnitude of the resultant force using the 'r<θ' function. This part should display an answer of 14.31142<65.24538. Now we have the final magnitude of the resultant force from the 2 linear forces 10 Newtons and 6.7 Newtons.
Command: [SHIFT]  [ 2 ]  [ 3:r<θ ]  [ = ]
4. You may now start calculating the value of the moment force using statistics mode or 'STAT' mode. This is a different approach from what we have done in the previous example problems. But before continue solving, you must compute the xcomponent and ycomponent of the 6.7 N force. If to use trigonometric identities, the final xcomponent of the force is 5.993 N while the final ycomponent of the force is 2.996 N. These values are needed in the following computations.
Command: [MODE]  [3:STAT]  [2:A+BX]
5. After entering, the calculator displays an empty table for the values of x and y. Turn on the frequency table.
Note: [ v ] means down button.
Command: [SHIFT]  [MODE]  [ v ]  [4:STAT]  [1:ON]
6. Input the values in the table. X is the force in the coplanar force system. Y is the moment arm and frequency are 1 for clockwise rotations and 1 for counterclockwise rotations. The calculator must display:
X  Y  FREQ 

10  5  1 
2.996  8  1 
5.993  2  1 
30  1  1 
7. Get the summation of xy which is the equal moment at 0.
Command: [AC]  [SHIFT]  [1:STAT]  [3:SUM]  [5]  [ = ]
8. Solve the location of the resultant force using the answer in step number 7.
(12.9963)(I_{x}) = 31.982
(5.9927)(I_{y}) = 31.982
Answers:
31.982 Nm
I_{x} = 2.461 m
I_{y} = 5.34 m
Components of a 3D Force
In the figure shown, find the x, y, and z component of the force F if F = 200 N and passes from the point A(0,0,0) to the point B(3,4,5). Its direction cosines and the angle that it makes with the coordinate axes.
CALCULATOR TECHNIQUE
1. This type of problem is solved using the 'Vector' mode of your calculator. Since the force diagram is in 3 dimensional, then set 3D in the settings of your calculator. Input the values (3,4,5)
Command: [MODE]  [8:VECTOR]  [1:VctA]  [1:3]  (3,4,5)  [AC]
2. Next step is to get the direction cosines or unit vector.
Command: [SHIFT]  [5:VECTOR]  [3:VctA]  [ / ]  [SHIFT]  [hyp]  [SHIFT]  [5:VECTOR]  [3:VctA]  [ = ]
The calculator should display VctA / Abs(VctA). The answer is [0.4242640687,0.5656854249,0.7071067812]. Simply multiply them to 100 to get the magnitude of the forces. To get the angles it makes with the coordinate axes, take the inverse cosine of the direction cosines.
Answers:
Fx = 42.4260687 N
Fy = 56.56854249 N
Fz = 70.71067812 N
Parallel  Non Coplanar Force System
Find the resultant of the four forces acting on the plane shown. Also, find its position.
CALCULATOR TECHNIQUE
This is another problem that uses the stat mode function of the calculator.
1. Start by pressing the mode setup button. Then press number 3 which is the statistic button and the second option which is a+bx. Again, add a frequency column by following the steps below. After setting all this up, input all the values and don't forget to consider the signs depending on what axis the line is going. The table is shown below.
Command: [MODE]  [3:STAT]  [2:A+BX]  [SHIFT]  [MODE]  [ v ]  [4:STAT]  [1:ON]
X is the moment arm for the moment about the xaxis and Y is the moment arm for the moment about the yaxis. Frequency is the magnitude of the force.
X  Y  FREQ 

2  6  70 
6  6  65 
6  0  60 
0  0  80 
3. M_{x} is the summation of all the xcomponents. Now to get M_{x} (summation of all xcomponents), follow the following command. The answer should be the same as the one written below which is 956 lbft.
Command: [SHIFT]  [1:STAT]  [3:SUM]  [2:Summation of X]  [ = ]
Mx= 956 lbft
4. On the other hand, M_{y} is the summation of all the ycomponents. Now, to get M_{y} (summation of all ycomponents), do the same procedure with step number three but this time it is for the 'Y'. The answer is 804 lbft.
Command: [SHIFT]  [1:STAT]  [3:SUM]  [4:Summation of Y]  [ = ]
My= 804 lbft
5. After getting the M_{x} and M_{y} values, compute the summation of the forces. Given the following forces 80, 60, 70 and 65 pounds, the resultant force is 275 pounds.
R = 80 + 60 + 70 +65
R = 275 lb.
6. Now present your answer in the form ayi +bxj.
Therefore, M_{o }= 275yi  275xj = 956i  804j
Equilibrium of Coplanar Forces (Application)
Determine the tensile forces in each chord which supports 600 lb weight as shown in the illustration below.
Note that forces going upward and rightward have positive signs. On the other hand, all forces going to the left and downward are negative.
CALCULATOR TECHNIQUE
1. Equation mode approach is the approach to answering this problem. Start solving by entering the mode setup in the top righthand corner of your calculator buttons.Pick the equation mode.
Command: [MODE]  [5:EQN]
2. Pick the equation with the form ax+by = c. In casio calculators(or other calculators), it looks like this.
Command: [1 : anx + bny =cn]
BC  BA  APPLIED LOADS  

cos30  cos60  0  xcomponent 
sin30  sin60  600  ycomponent 
3. Now, just enter the equal button twice and you'll get the following answers wherein x corresponds to BC while y corresponds to BA.
X = BC = 300 lb.
Y = BA = 519.62 lb.
Beams (Application)
Find the magnitude of the reaction in the simply supported beam as shown in the figure below.
CALCULATOR TECHNIQUE
1. Equation mode approach is the approach to answering this problem. Start solving by entering the mode setup in the top righthand corner of your calculator buttons.Pick the equation mode.
Command: [MODE]  [5:EQN]
2. Pick the equation with the form ax+by = c. In casio calculators(or other calculators), it looks like this.
Command: [1 : anx + bny =cn]
3. Take note that there are certain things to consider. Consider the reaction at A, R_{a} be equal to x and consider the reaction at B, R_{b} equal to y.
4. Let M_{a} or M_{b} be equal to 0 and Fy equal to 0.
a  b  c 

8  0  397.5 
1  1  132.5 
5. From the computation in the table, you'll get the following values. Reaction at A is 30.3125 kN and reaction at B is 73.125 kN.
R_{a} = x = 30.3125 kN
R_{b} = y = 73.125 kN
Centroid and Moment of Inertia (Application)
Get the centroid and moment of inertia of the figure below.
CALCULATOR TECHNIQUE IN GETTING THE CENTROID
This is another problem that uses the stat mode function of the calculator.
1. Start by pressing the mode setup button. Then press number 3 which is the statistic button and the second option which is a+bx. Again, add a frequency column by following the steps below. After setting all this up, input all the values and don't forget to consider the signs depending on what axis the line is going. The table is shown below.
Command: [MODE]  [3:STAT]  [2:A+BX]  [SHIFT]  [MODE]  [ v ]  [4:STAT]  [1:ON]
2. Input the following values. Remember not to put anything for the xvalues since the centroid of the figure with respect to the yaxis is the centroid of the two areas from the figure. The centroids coincide with each other.
Stand by  C.G. OF EACH AREA FROM XAXIS  AREA  

x  y  frequency  
 30  60 x 200  AREA 1 
 60+300/2  300 x 25  AREA 2 
3. To get the centroid, perform the following procedure. Enter shift and look for the 'sum' option. Then, pick the summation of y. The main command is dividing the value of the summation of y with the variable n. The value of the centroid is 99.23 mm from the xaxis or from the foot of the figure.
Command: [AC]  [SHIFT]  [1]  [3:SUM]  [4:Summation of Y]  [divide sign] [SHIFT]  [1]  [4:Var]  [1:n]  [=]
Centroid = 99.2307 mm
CALCULATOR TECHNIQUE IN GETTING THE MOMENT OF INERTIA
Now in getting the moment of inertia of the figure, store the value of centroid from xaxis at 'A'.
1. Start by pressing the mode setup button. Then press number 3 which is the statistic button and the second option which is a+bx. Again, add a frequency column by following the steps below. After setting all this up, input all the values and don't forget to consider the signs depending on what axis the line is going. The table is shown below.
Command: [MODE]  [3:STAT]  [2:A+BX]  [SHIFT]  [MODE]  [ v ]  [4:STAT]  [1:ON]
2. Next step is to enter the shift button. Enter RCL button and Display the value of A.
Command: [SHIFT[  [RCL]  [DISPLAY: 'A']
3. Now, edit the data from getting the centroid of the figure. This time, put something on the x column. The values are shown in the table below.
Command: [SHIFT]  [1]  [2:DATA]
Y MINUS A  C.G. OF EACH AREA FROM XAXIS  AREA  

X  Y  FREQUENCY 

30  A  30  60 X 200  AREA 1 
60 + 300/2  A  60 + 300/2  300 X 25  AREA 2 
4. Finally, to get the centroidal moment of inertia (I_{gg}). Enter the Shift button. Next, pick the 'STAT' button and choose the 'SUM' option. Lastly, choose the summation of x squared. Then there you have the value of the centroidal moment of inertia.
Command: [SHIFT]  [1]  [3]  [1]
Learning Assessment
© 2018 Ray
Comments
Umesh Chandra Bhatt from Kharghar, Navi Mumbai, India on February 19, 2019:
Excellent and well explained.