. (a) A number of particles with masses m1, m2, m3,··· ,mn are situated at the points with position vectors r1, r2, r3,··· ,rn relative to an origin O. The center of mass G of the particles is defined to be the point of space with position vector
R= m1r1+m2r2+m3r3+···+mnrn /m1 +m2+m3+···+mn .
Show that if a different origin O were used, this definition would still place G at the same point of space.
(b) An object of mass 40kg is supported in equilibrium by four cables. The forces, in Newtons, exerted by three of the cables, F1, F2 and F3, are given in terms of the unit vectors, i, j and k as F1 = 80i + 20j + 100k, F2 = 60i − 40j + 80k and F3 = −50i − 100j + 80k. The unit vectors i and j are perpendicular and horizontal and the unit vector k is vertically upwards.
i. Find F4, the force exerted by the fourth cable, in terms of i, j and k. Also find its magnitude to the nearest Newton.
ii. Find the angle between F1 and F4.