Couples and their moments
Answers are for parents and teachers.
1
Two forces of 35 N form a clockwise couple with arm 12 cm. (a) Find its moment (anticlockwise positive). (b) What force gives the same moment with arm 30 cm and the same sense?
(a) M = −35 × 12 = −420 N·cm. (b) F = 420 / 30 = 14 N.
2
F₁ = (a, 3) N at A(−2, 4) and F₂ = (−5, b) N at B(1, −1) form a couple (m). Find a, b, the moment (anticlockwise positive) and the arm to two decimal places.
F₂ = −F₁ gives a = 5, b = −3. The vector from B to A is (−3, 5) and F₁ = (5, 3): M = (−3)(3) − 5 × 5 = −9 − 25 = −34 N·m (clockwise). |F| = √(5² + 3²) = √34, so L = 34 / √34 = √34 ≈ 5.83 m.
3
A plate has a couple of 14 N with arm 0.3 m (anticlockwise) and a couple of 9 N with arm 0.4 m (clockwise). A third couple has forces of G newtons and arm 0.2 m. Find G and its sense for equilibrium (anticlockwise positive).
4.2 − 3.6 = +0.6 N·m, so the third couple must be −0.6 N·m: clockwise, 0.2G = 0.6, G = 3 N.
4
In rectangle ABCD, AB = 20 cm and BC = 12 cm. Forces of 5 N act from A to B and from C to D. Forces of Q newtons act from C to B and from A to D. Find Q for equilibrium (anticlockwise positive).
+5 × 12 − 20Q = 0, so Q = 60 / 20 = 3 N.
5
In two sentences, explain why one single force can never balance a couple.
A couple has resultant zero, so with one more force the total force is that force, not zero. Equilibrium needs total force zero and total moment zero, so only another couple (equal moment, opposite sense) can balance a couple.