PHY205 Ch15: // Axis Theorem and Torque 1.Recall main points: Expression of the Kinetic energy of a rigid body in terms of Kcm and Icm Parallel Axis Theorem Torque and Cross Product Angular acceleration from the torque Slide # 1

PHY205 Ch15: // Axis Theorem and Torque 1. Main Points Kinetic energy of a rigid body in terms of Kcm and Icm In the 3 rd slide of Ch 13 we showed that the total kinetic energy of a system is equal to the energy of the Center of Mass (CM) plus the energy RELATIVE to the CM: In our case since we study RIGID bodies, any motion relative to the CM MUST be a rotation around an axis through the CM. And therefore, since : This is a general result that should be used when the work-energy theorem, or the conservation of mech. energy, is used for solving problems involving rigid bodies rotating. Note that any motion can be viewed as the CM motion PLUS a rotation AROUND an axis thru the CM v CM  CM Slide # 2

PHY205 Ch15: // Axis Theorem and Torque 1. Main Points Optional: Direct proof of the preceding result: By definition of the center of mass (see CM chapter) the last term is zero. Note also that we replaced v i/cm by r i/cm  since we are dealing with a rigid body. We obtain the fundamental result: Slide # 3

PHY205 Ch15: // Axis Theorem and Torque 1. Main Points Parallel Axis Theorem via K=Kcm+1/2Iw 2 Using the result on the previous slides, we can arrive at the very useful parallel axis theorem: Slide # 4

PHY205 Ch15: // Axis Theorem and Torque 1. Main Points Torque and Cross product: x y F  r m O The torque is defined as: Right Hand Rule: Slide # 5

PHY205 Ch15: // Axis Theorem and Torque 1. Main Points Angular acceleration from the torque, using Newton’s 2 nd law: Slide # 6