Systems equivalent to a couple
Answers are for parents and teachers.
1
A light rod AB, 30 cm long, has 20 N upward at A, 12 N downward at C (AC = 10 cm) and 8 N downward at B. Show that the system is a couple and find its moment (anticlockwise positive). Check your answer by taking moments about B.
R = 20 − 12 − 8 = 0. About A: −12 × 10 − 8 × 30 = −360 N·cm. About B: 20 × (−30) − 12 × (−20) = −600 + 240 = −360 N·cm. A clockwise couple of 360 N·cm.
2
On a light rod AB, 12 cm long, 14 N acts upward at A, P newtons downward at C (AC = 4 cm) and Q newtons downward at B. The system is a clockwise couple of 120 N·cm. Find P and Q.
P + Q = 14 and 4P + 12Q = 120. With P = 14 − Q: 56 + 8Q = 120, so Q = 8 and P = 6.
3
Triangle ABC has AB = 20 cm, BC = 21 cm, CA = 29 cm (right angle at B), anticlockwise. Forces of 40 N, 42 N and 58 N act along AB, BC, CA in order. Find the moment of the couple.
m = 40 / 20 = 2 N per cm; area = ½ × 20 × 21 = 210 cm². M = 2 × 210 × 2 = +840 N·cm.
4
Equal forces act along the sides of an equilateral triangle of side 4 cm (anticlockwise, in order) and form a couple of 24√3 N·cm. Find each force.
Area = (√3 / 4) × 16 = 4√3 cm². 2 × 4√3 × m = 24√3, so m = 3 N per cm and each force is 3 × 4 = 12 N.
5
A square ABCD of side 5 cm has equal forces along its sides in order, going round clockwise, forming a couple of 100 N·cm. Find each force and the moment with its sign (anticlockwise positive).
Area = 25 cm². 2 × 25 × m = 100, so m = 2 N per cm and each force is 2 × 5 = 10 N. Going round clockwise: M = −100 N·cm.