Statics of particles: For a joint acted upon by only two forces and in equilibrium, what must be the relationship between those two forces?

Introduction / Context:
The two-force member and two-force particle conditions are foundational in statics. When only two forces act on a joint or a rigid body and it is in equilibrium, those forces have a strict geometric and magnitude relationship.

Given Data / Assumptions:

• Only two forces act on the joint or particle.
• Equilibrium requires sum of forces to be zero.
• No moments can be resisted at a point particle; hence forces must be collinear to avoid a net moment.

Concept / Approach:
For equilibrium, the vector sum of the two forces must vanish. Two non-zero vectors can sum to zero only if they are collinear and of equal magnitude but opposite direction. Any non-collinearity would produce a non-zero resultant or a couple, violating equilibrium for a particle or a two-force member.

Step-by-Step Solution:
1) Let the forces be F1 and F2. Equilibrium demands F1 + F2 = 0.2) Therefore, F2 = -F1, which implies equal magnitudes and opposite directions.3) Because they are exact negatives, they must lie along the same line of action (collinear) to avoid any moment effect.

Verification / Alternative check:
Free-body diagrams of a two-force member (e.g., a link) show the forces at its ends along the member axis, equal and opposite, matching this rule.

Why Other Options Are Wrong:
Zero-force options contradict the assumption of two non-zero forces.Same direction cannot satisfy equilibrium unless magnitudes are zero.Perpendicular forces cannot sum to zero without a third balancing force.

Common Pitfalls:
Overlooking the collinearity requirement; confusing conditions for concurrent three-force systems with two-force cases.

Final Answer:
Forces must be equal in magnitude, opposite in direction, and collinear