Theory question 118
Vectors(a) The position vectors of points A, B and C are i + 5j, 3i + 9j, and −i + j respectively. (i) Show that points A, B and C are collinear. (ii) Determine the ratio |AB| : |BC|
(b) A uniform beam XY of mass 10kg and length 24m is hunged horizontally from a cross bar by two vertical inextensible strings, one attached to X and the other at a point M, 4m away from Y, a mass of 50kg is suspended at a point N which is 8m from X. If the system remains in equilibrium, calculate the tensions in the strings [Take g = 10ms−2]
Show the model answer
(a)(i) AB = 2i + 4j, BC = −4i − 8j = −2AB; parallel with common point B, so A, B, C are collinear
(ii) |AB| : |BC| = √20 : √80 = 1 : 2
(b) Weights: beam 100 N at 12 m from X, load 500 N at 8 m from X; M is 20 m from X
Moments about X: 20TM = 100 × 12 + 500 × 8 = 5200, TM = 260 N
Resolving: TX + TM = 600, TX = 340 N
Take moments about one string to remove its tension, then resolve vertically.