## A T-shaped balancing scale

QUESTION 1 A T-shaped balancing scale is set up to pivot about the origin 0 (Figure 1). Two (nonzero) downward forces F1 and. F2 are applied to points A and B respectively (iiFtii = > 0, II ‘2II = F2 > 0, d > 0). The T-shaped part of the scale is of negligible mass. The aim is to understand the importance of the length it 0 for the equilibrium configurations of the scale.
A ■ Fi
if
• F2
Figure 1: The scales when held fixed in place.
(a) Assume the balancing scale is held fixed in place in the position shown in Figure 1 and not allowed to rotate. Find the 2D position vectors OA and 08 and calculate the resultant moment M around the origin 0 caused by Fi and F2. Hint: A cross product requires 3D vectors: add on empty z component where necessary. (4 marks)
— —d OA = — ( ) OB = (d ) h 1 —h . OA x Fi = —d —h x 00 = 00 , 0 —F1 dF1 I 0 0 0 (T ) A X F2 = (LI —h X ( 0 ) = ( 0 , M = OA x Fi + OB x F2 = ( 0 0 —F2 —dF2 d(F1 — F2)
(2 marks for correct OA, 08; 2 marks for correct M )
(b) The same balancing scale is now allowed to rotate freely. It has rotated by an angle 19 when it reaches equilibrium (Figure 2). Write a 2 x 2 rotation matrix R that represents rotation by an angle 0 and use it to determine the new vectors OA and 08 in Figure 2.
(4 marks)

0
Figure 2: The scales when allowed to rotate to equilibrium.