When the body is rotating about a fixed point O, we can write Io= IG +mr2 and T==mu2+=(o-mrG)o-2 sInce UG wr The above expression is also applicable in the more general case when there is no fixed point in the motion provided that O is replaced by the instantaneous center of rotation. Thus, in general We shall see that, when the instantaneous center of rotation is known, the use of the above expression does simplify the algebra considerably. Work Recall that the work done by a force F, over an infinitesimal displacement, dr, is dW=F. dr. If Fto denotes the resultant of all forces acting on particle i, then we can write dw:= ftot du Ir:=mi dt. dr:=miU, dv=d( mv2)=d(T), where we have assumed that the velocity is measured relative to an inertial reference frame, and, hence Ftotal= mia;. The above equation states that the work done on particle i by the resultant force Tota is equal to the change in its kinetic energy The total work done on particle i, when moving from position 1 to position 2, is (Wi)1-2 dwi and, summing over all particles, we obtain the principle of work and energy for syst T1+∑(W)1-2=T2 The force acting on each particle will be the sum of the internal forces caused by the other particles, and the erternal forces. We now consider separately the work done by the internal and external forces Internal forces We shall assume, once again, that the internal forces due to interactions between particles act along the lines joining the particles, thereby satisfying Newton's third law. Thus, if fii denotes the force that particle j exerts on particle i, we have that fi is parallel to ri-Ty, and satisfies fi=-fj i Let us now look at two particles, i and j, undergoing an infinitesimal rigid body motion, and consider the f;·mr+f·drWhen the body is rotating about a fixed point O, we can write IO = IG + mr2 G and T = 1 2 mv2 G + 1 2 (IO − mr2 G)ω 2 = 1 2 IOω 2 , since vG = ωrG. The above expression is also applicable in the more general case when there is no fixed point in the motion, provided that O is replaced by the instantaneous center of rotation. Thus, in general, T = 1 2 IC ω 2 . We shall see that, when the instantaneous center of rotation is known, the use of the above expression does simplify the algebra considerably. Work Recall that the work done by a force, F, over an infinitesimal displacement, dr, is dW = F · dr. If F total i denotes the resultant of all forces acting on particle i, then we can write, dWi = F total i · dri = mi dvi dt · dri = mivi · dvi = d( 1 2 miv 2 i ) = d(Ti) , where we have assumed that the velocity is measured relative to an inertial reference frame, and, hence, F total i = miai . The above equation states that the work done on particle i by the resultant force F total i is equal to the change in its kinetic energy. The total work done on particle i, when moving from position 1 to position 2, is (Wi)1−2 = Z 2 1 dWi , and, summing over all particles, we obtain the principle of work and energy for systems of particles, T1 + Xn i=1 (Wi)1−2 = T2 . (2) The force acting on each particle will be the sum of the internal forces caused by the other particles, and the external forces. We now consider separately the work done by the internal and external forces. Internal Forces We shall assume, once again, that the internal forces due to interactions between particles act along the lines joining the particles, thereby satisfying Newton’s third law. Thus, if fij denotes the force that particle j exerts on particle i, we have that fij is parallel to ri − rj , and satisfies fij = −fji. Let us now look at two particles, i and j, undergoing an infinitesimal rigid body motion, and consider the term, fij · dri + fji · drj . (3) 2