The gusset plate is subjected to the forces of three members. Determine the tension force in member C and its angle θ for equilibrium. The forces are concurrent at point O. Take F=8 kN.

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We need to find the angle that force T makes with the positive x-axis first. We call this the angle beta, β. This is depicted in the free-body diagram.
Free-body diagram:
Solving for the values of angles α and β.
Knowing that the sum of angles α and β is 90°, we can solve for the β.
Equations of Equilibrium:
Summation of forces in the x-direction:
Summation of forces in the y-direction:
Now, we have two equations with two unknowns. We shall solve the unknowns by solving these equations simultaneously. We can use our calculator, or we can solve this manually using the method of substitution.
Using equation (1), solve for T in terms of F.
Now, substitute this equation (3) to equation (2) to solve for F:
Substitute the value of F to equation (3) to solve for T:
Therefore, and .
Free-body diagram of the roller:
Equations of Equilibrium:
Note that if we take the sum of forces in the x-direction, there are two unknown forces involve, but if we take the sum of forces in the y-direction, there is only one unknown force involve.
Summation of forces in the y-direction:
Summation of forces in the x-direction:
Therefore, the normal reactions NB and NC on the bearing at its contact points B and C for equilibrium are 163.1759 N and 104.8874 N, respectively.
Free-body Diagram:
Equilibrium Equation:
Summation of forces in the x-direction:
Summation of forces in the y-direction:
We now have two equations. Divide Eq (1) by (2)
We know that :
Substituting this result to equation (1), we have
Free-body diagram:
Equations of Equilibrium:
The summation of forces in the x-direction:
The summation of forces in the y-direction:
We came up with 2 equations with unknowns and . To solve the equations simultaneously, we can use the method of substitution.
Using equation 1, solve for in terms of .
Now, substitute this equation (3) to equation (2).
Substitute the solved value of to equation (3).
Therefore, the answers to the questions are:
Free-body diagram:
Equations of Equilibrium:
Take the sum of horizontal forces considering forces to the right positive, and equate to zero.
Take the sum of vertical forces considering upward forces positive, and equate to zero.
Now, we have two equations with two unknowns and . So, we have a system of two equations. We can solve this using algebra, or we can directly use our calculator with this capability. The answers are
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