The problems provided in the questionnaire are solved and the solutions provided as below:
Observations:
Solution#1
A) While calculating the total moment of two system of forces, the moment of each force about the point O calculated and later the moments are added to find the total moment due to the two forces. For calculating the moments vector product of force and the distances are used and totaled.
While calculating the resultant of the system of forces, each of the forces present there in the system is resolved along the X and Y direction and finally the resultant of all the three forces is calculated.
Resultant force magnitude and the angle it makes with the X-axis in anticlock wise direction(Negative ) calculated and indicated there in the solution.
Solution#2
(i) Matrix addition, multiplication, inverse calculations are directly performed.
(ii) Linear equations of two unknowns and three unknowns are also calculated directly from the problem conditions.
(iii) Also the solutions found are verified from the traditional analytical procedures.
TASK
A) F1 = (3i +2J) A = (2i - j) F2=(-3i +J) B = (-3i -J)
The given point for moment calculation = (I + 2J)
Moment = OA X F1 + OB X F2
= (i-3J) X(3i +2J) + (-4i -3J) X (-3i + J)
= 11K + 5K
=16K
B) Determination of the resultant of the Forces:
From F.B.D shown, the
Resolving all the forces along X and Y directions,
Rx = 4 cos30 + 3 Cos10 - 5 Sin20
Ry = 4 Sin30 - 3 Sin10 - 5 Cos20
Rx = 4.71N
Ry = -3.22
Hence the magnitude of R = |R| = sqrt(Rx^2 + Ry^2) = 5.71
Tan(theta) = Ry/Rx
Theta = -34.350
Angle is negative implies in anti-clock wise direction.
TASK
A) (1) Matrix Addition
(2 4) + (-2 7) = (0 11)
(2) Matrix Mulitplication
(3 5) (2 -5) = (6+25 -15+10) = (31 -5)
(2 4) (5 2) (4+20 -10 + 8) (24 - 2)
(3) Determinant of the matrix
|5 9| = 5*4 - 9 *4 = -16
|4 4|
(4) Inverse of (3 2) = [ 1/(3*8 -2 *((-2)) ] * (8 -2)= (1/28)* = (8 -2)
(-2 -8) (2 3) (2 3)
(2/7 -1/14)
= (1/14 3/28)
(B) Given Equations
3x -5y =-17.6
-2x +7y =22.0
Matrix form:
(3 -5) (x) (-17 .6)
(-2 7) * (y) = (22 .0)
Hence by Rearranging the terms,
A*X = B
X = Inv(A) * B
Inv (A) = [1/(21-10)]* (7 5) = (1/11 5/11)
(2 3) (2/11 3/11)
X = Inv (A) *B
X = 7/11*(-17.6) + 5/11* 22.0 = -1.2
Y = 2/11*(-17.6) + 3/11* 22.0 = 2.8
(C) Given Equations
3x + 2y -3z =-7 -------(R1)
X +4y +z = 25 ----------(R2)
2x +6y +3z =49 ---------(R3)
Making the following set of operations,
R1--------R1/3
R2--------R2 - R1/3
R3 ---------R3-2R1
R2----------3/10 R2
R3--------------R3 -14/3 R2
Hence, it yields,
Which further on simplification will yield,
Z= 7, y = 4 and x =2
Hence the solution matrix
(2)
X = (4)
(7)
(D )There do exist alternative procedures for solving the equations set given in the set B and set C, the following procedures indicate how to solve the equations set,
3x -5y = -17.6
-2x +7y =22.0
Multiply the first equation by 2 and second equation by 3 and adding the two will provide an equation,
-10y +21y = -17.6 x2 +22 x3 = 30.8
Or alternatively +11y =30.8
Y =2.8
Alternatively on substitution in equation 1 it will provide 3x = -17.6 + 5*2.8 =-3.6
Which indicate that x = -1.2, hence the result is correct as per matrix methods.
3x + 2y -3z =-7
X + 4y +z =25
Multipling the second equation by 3 and subtracting the first will give
-10y + 6Z =82-----------------(1)
Also
X + 4y +z =25
2x + 6Y +3Z =49
Multiplying the second by 1 and from it subtracting the twice of first equation will provide
-2y + z = -1 -------------------- (2)
From 1 and 2 equations,
X = 2 and y =4 and z=7, which indicates that the matrix formulation is correct and the procedures employed before are correct.
Analysis of the procedures employed:
Advantages:
The matrix based linear equation solving is much simplified and the solution is less prone to go erroroneous. Similar procedure using Gaussian principles of matrix solutions can be done with any number of equations.
Disadvantages:
More the number of equations, more will be steps involved to solve the matrix of equations using Gaussian reduction methodologies.