Question

If two forces acting at a joint are not along the straight line, then for the equilibrium of the joint

• A. one of the forces must be zero
• B. each force must be zero
• C. forces must be equal and of the same sign
• D. forces must be equal in magnitude but opposite in sign

Hint:

This is a fundamental rule used in truss analysis:
If a joint has only two non-collinear members and no external load acts on it, both members are zero-force members ($F_1 = F_2 = 0$).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the condition of force magnitudes required to maintain static equilibrium at a joint where only two non-collinear (not along a straight line) forces are acting.

Step 2: Key Formula or Approach:
For any joint to be in static equilibrium, the vector sum of all forces acting on it must be zero:
\[ \sum \mathbf{F} = 0 \implies \mathbf{F_1} + \mathbf{F_2} = 0 \]
This requires:
\[ \sum F_x = 0 \quad \text{and} \quad \sum F_y = 0 \]

Step 3: Detailed Explanation:

• Let the joint be the origin of a coordinate system. Let $\mathbf{F_1}$ act along the x-axis, and $\mathbf{F_2}$ act at an angle $\theta$ (where $\theta \neq 0^\circ$ and $\theta \neq 180^\circ$ because they are non-collinear).

• Resolving the forces along the x and y axes:
- Along the y-axis:
\[ \sum F_y = F_2 \sin\theta = 0 \] Since the forces are non-collinear, $\sin\theta \neq 0$. This forces:
\[ F_2 = 0 \]
- Along the x-axis:
\[ \sum F_x = F_1 + F_2 \cos\theta = 0 \] Substitute $F_2 = 0$ into this equation:
\[ F_1 + 0 = 0 \implies F_1 = 0 \]

• Therefore, the only way for a joint with two non-collinear forces to be in static equilibrium is if both forces are exactly zero.

Step 4: Final Answer:
For equilibrium, each force must be zero.