Question

the truss section shown here is in equilibrium. it is being acted on by three forces.
vector b represents a force that is known to be correct, and its direction is as shown. the magnitude of vector b is 1414 lbs. vectors a and c are guesses about the other forces acting on the truss.
to ensure equilibrium, force c must be in the x and y directions.
the magnitude of force c must be
to ensure equilibrium, force a must be in the direction.
the magnitude of force a must be
member a (horizontal) is in
member c (diagonal) is in

Explanation:

Step1: Analyze the equilibrium condition in the y - direction

For equilibrium in the y - direction, \(\sum F_y = 0\). Let the magnitude of force \(C\) be \(C\). The y - component of force \(C\) is \(C\sin45^{\circ}\), and force \(B = 1414\) lbs (acting in the positive y - direction). Since \(\sum F_y = C\sin45^{\circ}-B = 0\), then \(C=\frac{B}{\sin45^{\circ}}\). Substituting \(B = 1414\) lbs and \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\approx0.707\), we get \(C=\frac{1414}{0.707}=2000\) lbs. The y - component of force \(C\) acts in the negative y - direction (opposite to force \(B\)) to balance the forces in the y - direction.

Step2: Analyze the equilibrium condition in the x - direction

For equilibrium in the x - direction, \(\sum F_x = 0\). The x - component of force \(C\) is \(C\cos45^{\circ}\), and force \(A\) acts in the negative x - direction. Since \(C = 2000\) lbs and \(C\cos45^{\circ}=2000\times\frac{\sqrt{2}}{2}\approx1414\) lbs, and \(\sum F_x = C\cos45^{\circ}-A = 0\), then \(A = 1414\) lbs. Force \(C\) has a positive x - component (to the right) and a negative y - component (downward). Force \(A\) acts in the negative x - direction.

Step3: Determine the state of members

For member \(A\), since the force \(A\) is pushing on the member (compressive force), it is in compression. For member \(C\), the force \(C\) is pulling on the member (tensile force), so it is in tension.

Answer:

• Force \(C\) must be in the \(+\) \(x\) and \(-\) \(y\) directions.
• The magnitude of force \(C\) must be \(2000\) lbs.
• Force \(A\) must be in the \(-\) \(x\) direction.
• The magnitude of force \(A\) must be \(1414\) lbs.
• Member \(A\) (horizontal) is in compression.
• Member \(C\) (diagonal) is in tension.