Egypt Secondary 3 Applied Mathematics, Week 3: Forces along the sides of a rectangle or a square: showing that the resultant passes through a given point, and finding an unknown force

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

Forces along the sides of a rectangle or a square: showing that the resultant passes through a given point, and finding an unknown force

Applied Mathematics Sec 3 Term 1, Unit 1, Lesson 1, Part 2

GoalTake moments of forces acting along the sides of a rectangle or square to show that the resultant passes through a point, and to find an unknown force.

New words

force along a side · a force whose line of action is a side of the figure, named by its direction: AB means from A to Brectangle ABCD · a four-sided figure with four right angles; AB and DC are opposite sides, and so are BC and AD

Explanation

A force along a side of a rectangle has a very easy moment about any point E: its arm is simply the perpendicular distance from E to that side, so |M| = F × (distance from E to the side). If E lies on the line of the side, the moment is 0. To get the sign without coordinates, follow the arrow of the force with your eyes: if E is on your left, the force turns anticlockwise about E (positive); if E is on your right, it turns clockwise (negative). (A rightward force below E has E on its left, and indeed it turns anticlockwise.) To show that the resultant passes through E, add the moments of all the forces about E: if the total is 0, then by the theorem of moments the moment of the resultant about E is 0, so its line of action passes through E (the resultant is not zero, since we can see the forces do not cancel). To find an unknown force, use the same condition and solve for it. A force along a diagonal needs its arm, or its two components along the sides, each at the same point.

Worked examples

Rectangle ABCD has AB = 12 cm and BC = 9 cm. Forces 8 N, 5 N, 2 N and 1 N act along AB, BC, DC and AD (directions A to B, B to C, D to C and A to D). Show that the resultant passes through E on BC with BE = 3 cm. Moments about E (anticlockwise positive): the 8 N force along AB is 3 cm below E, with E on its left: +8 × 3 = +24. The 5 N force along BC passes through E: 0. The 2 N force along DC is 6 cm above E, with E on its right: −2 × 6 = −12. The 1 N force along AD is 12 cm to the left of E and points up, so E is on its right: −1 × 12 = −12. The total is 24 + 0 − 12 − 12 = 0. The resultant (10, 6) N, of size about 11.66 N, passes through E.

Square ABCD has side 8 cm. Forces 5 N, 4 N and 1 N act along AB, BC and DC (A to B, B to C, D to C), and a force F acts along AC (from A to C). The resultant passes through E on BC with BE = 3 cm. Find F. Moments about E: AB: E is 3 cm above and on the left, +5 × 3 = +15. BC: 0. DC: 5 cm below the line, E on the right: −1 × 5 = −5. The diagonal force F has components F/√2 along both axes and acts at A, which is 8 cm left of E and 3 cm below it: moment = (−8)(F/√2) − (−3)(F/√2) = −5F/√2. Sum: 15 − 5 − 5F/√2 = 0, so F = 10√2 / 5 = 2√2 ≈ 2.83 N. The resultant is (5 + 1 + 2, 4 + 2) = (8, 6), so R = 10 N.

Class activity

Draw a rectangle 12 cm by 9 cm on squared paper, mark E on BC 3 cm from B, and draw the four force arrows of the first example on their sides (1 cm for 1 N would be too long, so use 1 cm for 2 N). For each force measure the distance from E to its side and write the moment with its sign. Check that the total is zero.

Practice

Square ABCD has side 5 cm. A force of 6 N acts along AB (from A to B). Find its moment about D (anticlockwise positive).

In the same square, a 6 N force acts along BA (from B to A). Find its moment about C (anticlockwise positive).

Rectangle ABCD has AB = 8 cm and BC = 5 cm. A force of 4 N acts along BC (from B to C). Find its moment about A (anticlockwise positive).

Rectangle ABCD has AB = 8 cm and BC = 5 cm. Forces 6 N, 5 N and 6 N act along AB, BC and DC (A to B, B to C, D to C). Show that the resultant passes through the midpoint E of BC.

Square ABCD has side 6 cm, with centre O. Forces 2 N, 3 N and 4 N act along AB, BC and CD (A to B, B to C, C to D). Find the sum of their moments about O (anticlockwise positive).