Total left direction moments of a coplanar system of forces are given by G, 2G and 4G about the points (0,0) , (1,2) and (2,5) respectively. Find the magnitude and the equation of the line of action of the resultant. (G doesn't equal 0)
I know that the line of equation of the resultant is given by
G = G0 - Yx + Xy
Where G= moment of forcea about any point(let A),
G0 = moment of forces about point 0(origin) ,
Y,X = y an x components of the resultant force ,
x,y(x,y coordinates of point A)
Since there are no hints given on the magnitude of forces how do we solve this one?
To find the magnitude and equation of the line of action of the resultant force, we need to take into account the left direction moments of the forces and their locations.
Given that the total left direction moments about points (0,0), (1,2), and (2,5) are G, 2G, and 4G respectively, we can set up the following equations:
For point (0,0):
G = 0 - Y(0) + X(0) = 0
For point (1,2):
2G = 0 - Y(1) + X(2)
For point (2,5):
4G = 0 - Y(2) + X(5)
To solve these equations, we can use matrix methods. Rewrite the equations in matrix form:
| 0 0 -1 | | X | | 0 |
| -1 2 -1 | | Y | = | 0 |
| -2 5 -1 | | G | | 0 |
Here, the first column represents the coefficients of X, Y, and G in the first equation, and so on. The second column represents the unknowns X, Y, and G, and the third column represents the constants on the right-hand side of the equations.
To find X, Y, and G, we need to solve the system of equations AX = B, where A and B are the matrices above. Thus, we have:
X = A^(-1) * B
Calculate the inverse of matrix A and multiply it by matrix B to find X, Y, and G.
Once you have the values of X, Y, and G, you can substitute them into the equation G = G0 - Yx + Xy to determine the equation of the line of action of the resultant force.