Questions: Equilibrium of Particles in 2D

Question 1

You draw a free body diagram of a particle and identify three unknown force magnitudes. You write ΣFx = 0 and ΣFy = 0. What can you conclude?

• A. You can solve for all three unknowns using the two equations plus a moment equation
• B. The problem is statically indeterminate — two equilibrium equations cannot solve for three unknowns
• C. One unknown must equal zero, so you effectively have two unknowns
• D. You need to use ΣF = 0 in vector form to get a third equation

Answer: (B)

In 2D statics, a particle has exactly two equilibrium equations: ΣFx = 0 and ΣFy = 0. Two equations can solve for at most two unknowns. Three unknowns make the problem statically indeterminate — additional information (like a compatibility condition or material stiffness) is required. Option A is wrong because moment equations apply to rigid bodies, not particles, which have no moment arm. Option D is wrong because the vector equation ΣF = 0 is equivalent to the two scalar equations — it doesn't provide a third independent equation.

Question 2

Solving ΣFx = 0 gives T₁ = −45 N for a cable tension. What is the correct interpretation?

• A. An error was made — cable tension cannot be negative, so you must flip the direction and re-solve
• B. The cable is in compression with magnitude 45 N
• C. The cable pulls with magnitude 45 N in the direction opposite to what was assumed on the FBD
• D. The particle is not in equilibrium because forces cannot be negative

Answer: (C)

A negative value means the force acts in the direction opposite to what you assumed when drawing the FBD. This is not an error — it is the math correcting your initial assumption. The cable still has magnitude 45 N; you simply drew its arrow pointing the wrong way. You never re-solve after flipping the arrow (that would waste effort and risk new sign errors). Option B is wrong because cables cannot be in compression — they can only pull. If the math gives a negative cable tension, it means the assumed cable was oriented incorrectly, not that the cable is compressed.

Question 3

Tilting your x-y axes to align with a surface does not change the physics of the problem — it only changes the algebra, and both axis orientations give the same final answer.

• T. True
• F. False

Answer: True

Equilibrium is a physical condition (ΣF = 0) that is independent of the coordinate system you choose. The same forces must sum to zero regardless of how you orient your axes. However, axis choice profoundly affects the algebra: aligning one axis along a known direction (like an inclined surface) often eliminates unknown components from one equation, reducing a coupled two-equation system to two sequential single-unknown equations. The physics is unchanged; only the computational path differs.

Question 4

If you solve the equilibrium equations and get a negative value for a cable tension, you should flip the arrow on the FBD and re-solve the equations with the corrected diagram.

• T. True
• F. False

Answer: False

Flipping the arrow and re-solving is unnecessary extra work that introduces opportunities for new sign errors. The correct practice is to assign unknown forces an assumed positive direction at the start, then let the algebra determine the sign. A negative result simply means the force acts opposite to your assumption — interpret it as such. The only reason to redraw the FBD is if you need a clean diagram to present; even then, you just relabel the arrow's direction based on the result, you don't re-solve.

Question 5

Why is drawing the free body diagram correctly the most critical step in a 2D equilibrium problem — more important than the equilibrium equations themselves?

Model answer: The FBD defines the physics of the problem. It determines which forces are present, their directions, and their geometry. If you include forces acting on other bodies, omit a reaction force from a constraint, or misidentify the direction of a force, the equilibrium equations will solve the wrong problem — and they will solve it correctly, giving you a confidently wrong answer. The equations themselves are just arithmetic once the FBD is set. The FBD is where physical understanding translates into a mathematical model.

A common failure mode is drawing the FBD hastily, then spending effort debugging the algebra when the error was in the diagram. Every missed constraint force or included external force corrupts the entire solution. The FBD is the step that requires physical reasoning about what objects exert what forces on the isolated particle — everything after is mechanical.