Theorem of moments, forces along rectangle sides, and two-point conclusions
Week 3 · Moments: the theorem of moments for forces meeting at a point, forces along the sides of a rectangle or square, and equal or opposite moments about two points · 3 lessons
3
Equal moments about two points means the resultant is parallel to the line joining them; opposite moments means it passes through the midpoint
Applied Mathematics Sec 3 Term 1, Unit 1, Lesson 1, Part 2
GoalUse equal or opposite moments about two points to find the direction of the resultant, and use a zero moment to place its line of action.
New words
parallel to AB · never meets the line AB; its two ends keep the same distance from ABmidpoint of AB · the point halfway between A and B
Explanation
The moment of the resultant R about a point is R times the signed distance from that point to the line of action of R. Take two points A and B. If the moments about A and B are equal and R is not zero, then A and B are at the same signed distance from the line of action, on the same side of it: so the line of action is parallel to AB. If the moments are opposite (equal in size, opposite in sign), A and B are at equal distances on opposite sides of the line, so the line cuts AB at its midpoint. If the moment about a point C is zero, then C lies on the line of action (or R = 0). These three facts hold for forces at one point, and next week we extend them to any coplanar forces. They help to find a direction or a point without ever drawing the resultant. Sign convention: anticlockwise moments are positive and clockwise moments are negative.
Worked examples
Forces F₁ = (1, 3) N and F₂ = (4, −2) N act at P(0, 2). R = (5, 1). About B(−1, 4): P from B is (1, −2), M = 1 × 1 − (−2) × 5 = 11 N·m. About C(4, 5): P from C is (−4, −3), M = (−4) × 1 − (−3) × 5 = 11 N·m. The moments are equal, so the line of action of R is parallel to BC. Check: BC = (5, 1), the same direction as R.
The same forces and R = (5, 1) at P(0, 2). About B(7, 6): P from B is (−7, −4), M = (−7) × 1 − (−4) × 5 = 13 N·m. About C(13, 2): P from C is (−13, 0), M = (−13) × 1 − 0 × 5 = −13 N·m. The moments are opposite, so the line of action of R passes through the midpoint of BC, which is (10, 4). Check: from P the point (10, 4) is 10 across and 2 up, and (10, 2) = 2 × (5, 1): yes, it lies on the line.
A force F = (a, b) N acts at A(2, 3). Its moment about B(0, 0) is +10 N·m and about C(4, 0) it is +2 N·m. Find F. About B: 2b − 3a = 10. About C: A from C is (−2, 3), so −2b − 3a = 2. Adding the two equations: −6a = 12, a = −2. Then 2b = 10 + 3a = 4, b = 2. So F = (−2, 2) N. Check with the theory: the moments are not equal or opposite, so the line is not parallel to BC and does not pass through the midpoint of BC.
Class activity
Draw two points B and C on squared paper and a force arrow from a third point A. Extend the line of the arrow and measure the perpendicular distances from B and from C to it. Move the arrow, keeping its direction parallel to BC: the two distances stay equal. Then place the line through the midpoint of BC and see that the distances are equal but on opposite sides.
Practice
1
The moments of a resultant about A and about B are both +18 N·m. What can you say about its line of action?
The moments are equal, so the line of action is parallel to AB.
2
The moments about A and B are +5 N·m and −5 N·m. What can you say?
The moments are opposite, so the line of action passes through the midpoint of AB.
3
A force (2, k) N has equal moments about O and about B(6, 3). Find k.
Equal moments mean the force is parallel to OB = (6, 3), so (2, k) = (2, 1) and k = 1.
4
F = (3, 1) N acts at A(2, −1). Find its moments about B(0, 0) and C(3, 2) (anticlockwise positive). Is the line of action parallel to BC, or does it pass through the midpoint of BC?
About B: 2 × 1 − (−1) × 3 = 5. About C: A from C is (−1, −3), M = (−1) × 1 − (−3) × 3 = 8. The moments are neither equal nor opposite, so neither statement holds.
5
The sum of the moments of a system of forces about C is 0, and the resultant is not zero. What does this tell you about C?
