Statics — Three Coplanar Forces in Equilibrium At a point O, three coplanar forces P, Q, and R act. If the system is in equilibrium, which statement(s) must hold true?
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AThey can be represented (magnitude and direction) by the three sides of a triangle taken in order.
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BTheir vector sum is zero: P + Q + R = 0.
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CThey satisfy Lami’s theorem: P/sin A = Q/sin B = R/sin C.
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DAll of the above
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E—
Answer
Correct Answer: All of the above
Explanation
Introduction / Context: Equilibrium of three concurrent coplanar forces is a foundational topic in engineering mechanics. Several equivalent characterizations exist, including the triangle law of forces and Lami’s theorem. This question asks which statements must hold when three forces are in equilibrium at a point.
Given Data / Assumptions:
- Three forces P, Q, R acting at a single point O in one plane.
- System is in static equilibrium (no resultant force).
Concept / Approach: For three concurrent forces to be in equilibrium, the vector polygon formed head-to-tail must close (triangle law), yielding a zero resultant. Lami’s theorem provides a scalar relation among magnitudes and the angles between forces when exactly three non-collinear forces are in equilibrium.
Step-by-Step Solution:
Triangle law: Draw P, then Q from the head of P, then R from the head of Q. If the tip of R returns to the start, P + Q + R = 0. Vector condition: Equilibrium requires the resultant to vanish, hence P + Q + R = 0. Lami’s theorem: For three forces in equilibrium, P/sin α = Q/sin β = R/sin γ, where α, β, γ are the angles opposite P, Q, R respectively.Verification / Alternative check: Resolving along any two perpendicular directions yields two independent equations. Solving gives a unique relation among magnitudes and angles equivalent to Lami’s theorem, confirming consistency of all three statements.
Why Other Options Are Wrong: Each of (a), (b), and (c) is individually correct; picking only one omits complete characterization. Hence 'All of the above' is the best answer.
Common Pitfalls: Misapplying Lami’s theorem to more than three forces; forgetting that angles in Lami’s theorem are those between the other two forces (opposite to the listed force).
Final Answer: All of the above.