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2-1 (e) The moment theorem (Varignon)

2-1 (e) The moment theorem (Varignon)

Point 5 of 12 — Parallel Coplanar Forces: Resultant and EquilibriumQuestion bank: 0 questions (0 with pictures)

05 2-1 (e) The moment theorem (Varignon)

Theorem

  • The algebraic sum of the moments of a system of parallel coplanar forces about any point in their plane equals the moment of their resultant about the same point.
  • Write ∑MO=MO(R).

Proof for two like forces

  • Draw the common perpendicular through O meeting the lines of action at A, B and C (for R).
  • Moment sum about O: −F1⋅OA−F2⋅OB (with OA=OC−AC and OB=OC+CB in the appropriate arrangement).
  • Using F1⋅AC=F2⋅CB the sum becomes −(F1+F2) OC=−R⋅OC, which is the moment of R.
  • For unlike forces the same method gives (F1−F2) OC=R⋅OC.
  • A system of many forces is handled by combining forces in pairs.

Consequence

  • If the resultant is zero, the sum of moments about any point is zero or constant (the system is either in equilibrium or equivalent to a couple).
  • The theorem is the tool for locating the line of action of a resultant: take moments about a convenient point.
✍️ Worked examples

Example 5:

  • Parallel forces act perpendicular to a line at the points A, C, E, B, which are 0, 4, 9 and 16 cm from A.
  • The force at A is 5 N, at C it is 6 N, at E it is 4 N in the opposite sense to the force at C, and at B it is 2 N in the same sense as the force at C.
  • Find the algebraic sum of the moments about A (take the sense of the force at C as positive).

Solution:

  • The 5 N force at A has zero moment about A.
  • Moments about A: 6×4−4×9+2×16=24−36+32=20 N·cm.

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