Conservation of Angular momentum Since the angular momentum is defined with respect to a point in space, we will consider several cases, using a different reference point for each case Conservation of Angular Momentum about a Fixed Point o The angular momentum of a system of particles about a fixed point, O, is the sum of the angular momentum of the individual particle The time variation of Ho can be written as Ho=∑(1xm1)+∑(n1xm1)=0+∑(n1x(F1+∑f)=∑xF) 1,j≠ Here, we have used two facts. First, the cross product of two parallel vectors is zero. Second, ri x fii+ riXi=(ri-ri)x fi=0, and, therefore, the internal forces have no net effect on the total angular momentum change of the particle system. Thus, we have that M (8) here M I(ri x Fi) is the total moment, about O, of the applied external forces. Note that this moment needs to be computed as the sum of the individual moments. In general MofrG x F, unless the line of action of the the external forces resulant, F, passes through G. Conservation of Angular Momentum about G The angular momentum about G is given by HG (r m:0 Taking the time derivative of equation 2, we obtain rG+r=ug+U Inserting this expression into equation 9, we obtain HG=∑(rxm(a+)=∑(rxm;v), (rix mirG)=-iG x 2i mir=0(see equation 3). We note that equations 9 and 11 give us alternative representations for HG. Equation 9 is called the absolute angular momentum(since it involves absolute velocities, vi), whereas equation 11 is called the relative angular momentum (since it involves velocities, v, relative to G). It turns out that when G is chosen to be the origin for the relative velocitiesConservation of Angular Momentum Since the angular momentum is defined with respect to a point in space, we will consider several cases, using a different reference point for each case. Conservation of Angular Momentum about a Fixed Point O The angular momentum of a system of particles about a fixed point, O, is the sum of the angular momentum of the individual particles, HO = Xn i=1 (ri × mivi) . (7) The time variation of HO can be written as, H˙ O = Xn i=1 (r˙i × mivi) +Xn i=1 (ri × miv˙i) = 0 + Xn i=1 (ri × (Fi + X j=1, j6=i fij )) = Xn i=1 (ri × Fi) . Here, we have used two facts. First, the cross product of two parallel vectors is zero. Second, ri × fij + rj × fji = (ri − rj ) × fij = 0, and, therefore, the internal forces have no net effect on the total angular momentum change of the particle system. Thus, we have that H˙ O = MO , (8) where MO = Pn i=1(ri × Fi) is the total moment, about O, of the applied external forces. Note that this moment needs to be computed as the sum of the individual moments. In general MO 6= rG × F, unless the line of action of the the external forces resulant, F, passes through G. Conservation of Angular Momentum about G The angular momentum about G is given by, HG = Xn i=1 (r ′ i × mivi) . (9) Taking the time derivative of equation 2, we obtain vi = r˙ i = r˙ G + r˙ ′ i = vG + v ′ i . (10) Inserting this expression into equation 9, we obtain HG = Xn i=1 (r ′ i × mi(r˙ G + r˙ ′ i )) = Xn i=1 (r ′ i × miv ′ i ) , (11) since Pn i=1(r ′ i × mir˙ G) = −r˙ G × Pn i=1 mir ′ i = 0 (see equation 3). We note that equations 9 and 11 give us alternative representations for HG. Equation 9 is called the absolute angular momentum (since it involves absolute velocities, vi), whereas equation 11 is called the relative angular momentum (since it involves velocities, v ′ i , relative to G). It turns out that when G is chosen to be the origin for the relative velocities, 3