Flashcards on Forces in Equilibrium and Lami's Theorem
Forces in Equilibrium & Lami's Theorem: A Student Guide
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Statics: Lami's Theorem and Force Equilibrium
10 cards
Card 1
Question: State Lami's theorem for three non-zero forces in equilibrium acting at a point when none of the angles between them is 0° or 180°.
Answer: If three non-zero forces A, B and C are in equilibrium and the angles between them α, β, γ are not 0° or 180°, then ‖A‖ sinα = ‖B‖ sinβ = ‖C‖ sinγ.
Card 2
Question: Write the vector equilibrium condition for three forces A, B and C acting at a point.
Answer: A + B + C = 0.
Card 3
Question: How can you derive scalar equations relating magnitudes and angles from A + B + C = 0 using the dot product?
Answer: Dot both sides with each force to get: (A+B+C)·A=0 → ‖A‖^2 + ‖B‖‖A‖cosγ + ‖C‖‖A‖cosβ = 0, and similarly for dotting with B and C, producing the system
Card 4
Question: Explain part (i) of the exercise: how does Lami's theorem solve the dot-product system when α, β, γ ≠ 0°,180°?
Answer: When none of the angles are 0° or 180°, the system from the dot products is satisfied by the relations ‖A‖ sinα = ‖B‖ sinβ = ‖C‖ sinγ (Lami's theorem)
Card 5
Question: What special cases must be considered for the dot-product equilibrium system if one angle equals 0° or 180°?
Answer: If one angle is 0° or 180°, the assumptions behind Lami's theorem fail; the system must be re-examined because forces are collinear (0°) or oppositely
Card 6
Question: In the example problem, given ‖A‖=15 N, ‖B‖=20 N and γ=90°, what Lami relations apply?
Answer: 15 sinα = 20 sinβ = ‖C‖ sin90°, so 15 sinα = 20 sinβ and 15 sinα = ‖C‖·1.
Card 7
Question: How is the angle relation between α and β obtained when one angle γ = 90° and three angles around a point sum to 360°?
Answer: Since α + β + γ = 360°, with γ=90° we get α + β = 270°, so β = 270° − α.
Card 8
Question: Show how to solve for α given 15 sinα = 20 sin(270° − α).
Answer: Use the identity sin(270° − α) = sin270° cosα − cos270° sinα, substitute and rearrange to get tanα = −15/20, so α = arctan(−15/20). Adjust by 180° to
Card 9
Question: Using the computed α ≈ 143.13°, how is ‖C‖ determined in the example?
Answer: From Lami: ‖C‖ = 15·sin90°/sinα = 15 / sin(143.13°) = 25 N (rounded to two decimal places as given).
Card 10
Question: What verification exercise is suggested after finding ‖C‖ by Lami's theorem in the example?
Answer: Exercise 2 asks to verify by another method that the magnitude of force C found by Lami's theorem (25 N) is correct.