Chapter 8 
                continued 
        Rotational Dynamics 
                         8.4 Rotational Work and Energy 
                                                                                                                                                                                                                                                                              φ                                            
                            Work to accelerate a mass rotating it by angle                                                                                                                                                                                                                                                F
                                                                                                                                                                                                                                                                                                                          
                                 W=F(cosθ)x                                                                                                            x = rφ                                                                                                                                              s
                                                                                                                                                                                                                                                      r                                                                                                            
                                                   = Frφ                                                  Fr =τ (torque)                                                                                                                                                                                                        F  to x
                                                                                                                                                                                                                                                                    φ                                                                 θ =0°
                                                   =τφ
                               DEFINITION OF ROTATIONAL WORK 
                               The rotational work done by a constant torque in  
                               turning an object through an angle is  
                                                                                               W =τφ                                                                                             Requirement: The angle must 
                                                                                                            R                                                                                    be expressed in radians.               
                                                                                                                                                                                                 SI Unit of Rotational Work: joule (J) 
       8.4 Rotational Work and Energy 
          Kinetic Energy of a rotating one point mass 
             K=1mv2
                   2     T
                   1     2   2
                = 2 mr ω
            Kinetic Energy of many rotating point masses 
                         1      2   2     1           2     2    1      2
             K=∑(2mrω )=2(∑mr )ω =2Iω
              
            DEFINITION OF ROTATIONAL KINETIC ENERGY 
            The rotational kinetic energy of a rigid rotating object is 
                                                 1      2                     n
                                       KRot = 2 Iω                     I =      (mr2)i
                                                                            ∑
             Requirement: The angular speed must                             i=1
             be expressed in rad/s. 
             SI Unit of Rotational Kinetic Energy: joule (J) 
             8.4 Rotational Work and Energy 
                Moment of Inertia depends on axis of rotation. 
                Two particles each with mass, m, and are fixed at the ends of a  
                thin rigid rod.  The length of the rod is L. Find the moment of  
                inertia when this object rotates relative to an axis that is  
                perpendicular to the rod at (a) one end and (b) the center. 
                                      r =0,  r = L                                                                                            r = L,  r = L
                                        1               2                                                                                        1      2          2        2
                                                                                                           (b) 
                                                                                                                          (b) 
                                                                                  (a) 
                    (a)                                  2                   2                2                   2                    2              2
                                I = ∑(mr ) = mr +m r =m(0) +m(L) =mL
                                                             i           1 1             2 2
                                           i
                    (b)                                  2                   2                2                          2                         2        1         2
                                I = ∑(mr ) = mr +m r = m(L 2) +m(L 2) = 2mL
                                                              i          1 1             2 2
                                            i