Respuesta :
Answer:
M_AD = 1359.17 lb-in
Explanation:
Given:
- T_ef = 46 lb
Find:
- Moment of that force T_ef about the line joining points A and D.
Solution:
- Find the position of point E:
mag(BC) = sqrt ( 48^2 + 36^2) = 60 in
BE / BC = 45 / 60 = 0.75
Hence, E = < 0.75*48 , 96 , 36*0.75> = < 36 , 96 , 27 > in
- Find unit vector EF:
mag(EF) = sqrt ( (21-36)^2 + (96+14)^2 + (57-27)^2 ) = 115 in
vec(EF) = < -15 , -110 , 30 >
unit(EF) = (1/115) * < -15 , -110 , 30 >
- Tension T_EF = (46/115) * < -15 , -110 , 30 > = < -6 , -44 , 12 > lb
- Find unit vector AD:
mag(AD) = sqrt ( (48)^2 + (-12)^2 + (36)^2 ) = 12*sqrt(26) in
vec(AD) = < 48 , -12 , 36 >
unit(AD) = (1/12*sqrt(26)) * < 48 , -12 , 36 >
unit (AD) = <0.7845 , -0.19612 , 0.58835 >
Next:
M_AD = unit(AD) . ( E x T_EF)
[tex]M_d = \left[\begin{array}{ccc}0.7845&-0.19612&0.58835\\36&96&27\\-6&-44&12\end{array}\right][/tex]
M_AD = 1835.73 + 116.49528 - 593.0568
M_AD = 1359.17 lb-in
The moment of the force about the line joining points A and D is; 617.949 lb.in
What is the moment of the force?
We are given;
Force exerted by cable EF at E; T_EF = 46 lb.
From the diagram of the guy wire, we can draw a triangle and we will have the following coordinates;
A(0, 0, 0)
D(48, -12, 36)
E(E_x, 96, E_z)
Also, we can get that;
BC² = 48² + 36²
BC = √(48² + 36²)
BC = 60 in
Also, from similar triangles, we will have the coordinate of E as;
E(36, 96, 27)
Position of Vector of EF is;
EF = {(21 - 36)i + (-14 - 96)j + (57 - 27)k} in
EF = {-15i - 110j + 30k} in
Magnitude of EF from online calculation = 115 in
Force along cable EF is;
F_EF = 46{(-15i - 110j + 30k)/115}
F_EF = {-6i - 44j + 12k} lb
Position vector of AE is {36i + 96j + 27k} in
Position vector of AD is {48i - 12j + 36k} in
Magnitude of AD = 61.188 N
Unit vector of AD; λ_ad = {48i - 12j + 36k}/61.188
λ_ad = 0.7845i - 0.1961j + 0.5883k
M_ad = λ_ad × r_ea × T_EF
M_ad = [tex]\left[\begin{array}{ccc}0.7845&-0.1961&0.5883\\36&96&27\\6&-44&12\end{array}\right][/tex]
Solving this gives;
M_ad = 617.949 lb.in
Read more about moment of a force at; https://brainly.com/question/25329636
