Resultant of Two Equal Forces — Magnitude Formula Two equal forces each of magnitude P act at an angle θ to each other. What is the magnitude of their resultant?
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A2 P sin(θ/2)
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B2 P cos(θ/2)
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C2 P tan(θ/2)
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D2 P cot(θ/2)
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E—
Answer
Correct Answer: 2 P cos(θ/2)
Explanation
Introduction / Context: Vector addition of two forces is a staple of statics and dynamics. For equal magnitudes with angle θ between them, the resultant has a compact expression that is widely used in engineering and physics.
Given Data / Assumptions:
- Force magnitudes: P and P.
- Angle between forces: θ.
Concept / Approach: Use the law of cosines for vectors: R^2 = P^2 + P^2 + 2 P P cos θ. Then take the square root to obtain R. Trigonometric half-angle identities lead to a simple form involving cos(θ/2).
Step-by-Step Solution:
R^2 = 2P^2 (1 + cos θ). Use identity: 1 + cos θ = 2 cos^2(θ/2). Hence R = √(2P^2 * 2 cos^2(θ/2)) = 2P cos(θ/2).Verification / Alternative check: Special cases: θ = 0 ⇒ R = 2P (collinear same direction). θ = 180° ⇒ R = 0 (equal and opposite), consistent with formula since cos 90° = 0.
Why Other Options Are Wrong: Forms with sin, tan, or cot of θ/2 do not satisfy the cosine-law derivation and fail at boundary checks (e.g., θ = 0 or 180°).
Common Pitfalls: Forgetting the half-angle identity or misapplying parallelogram/triangle construction with wrong included angle.
Final Answer: 2 P cos(θ/2).