Forces P, Q, R act along the line x=0, y=0 and  x.cos A +y.sin A = p. Find the magnitude of the resultant and the equation of line of action.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Notice that the force P acts along x axis, the force Q acts along y axis and the force R acts on a support that expresses the hypotenuse of right triangle that has as legs the supports of P and Q forces. This support line intercepts x axis at A and y axis at B.

You need to project the origin O to hypotenuse of right triangle. This orthogonal projection falls in the point M.

The line MO makes the angle `alpha`  to x axis, thus `ltAOM= 90^o - alpha`  and the angle `ltBAO = 90^o + alpha`

You need to write the x axis equilibrium equation of forces such that:

`X = Q*cos 0^o + P*cos 90^o+ R*cos(90^o + alpha)`

You need to write the y axis equilibrium equation of forces such that:

`Y= Q*sin 0^o + P*sin 90^o + R*sin(90^o + alpha)`

Thus, evaluating the resultant of forces acting as problem suggests yields:

`resultant = sqrt(X^2+Y^2)=gt`  resultant = `sqrt(P^2 + Q^2 + R^2 - 2R(2Qsin alpha+2Pcos alpha))`

Hence, evaluating the resultant of forces under given conditions yields resultant =`sqrt(P^2 + Q^2 + R^2 - 2R(2Qsin alpha+2Pcos alpha)).`

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