Dynamics of Rigid Bodies 
                
               A rigid body is one in which the distances between constituent particles is constant 
               throughout the motion of the body, i.e. it keeps its shape.   
                
               There are two kinds of rigid body motion: 
                
                     1.  Translational 
                
                
                                                         Rectilinear forces acting. 
                                                         Particles move on straight-
                                                         line paths. 
                
                
                
                     2.  Rotational 
                
                
                                                              Rotational forces, i.e. 
                                                              torques, acting. 
                
                                                              Particles move on circular 
                                                              paths 
                
                
                
                
                
                
               Kinematics of Rigid Bodies 
                
                                     dθ          dω   d2θ       Angle θ  is in 
                      s = Rθ     ω= dt       α= dt = dt2           radians 
                Let α be a constant. 
                                                                dθ
                                  d2θ                            dt = αt +C
                     Then,        dt2 = α   Integrating,  
                                                   
                                          dθ
                Let ω = ω at t = 0,  ω=     =αt+ω
                             0            dt       0
                                                              COMPARE THESE 
                Integrating,  θ=½αt2+ω t+c                    WITH THE 
                                                0
                                                              EQUATIONS FOR 
                Let θ = 0 at t = 0,    θ=½αt2 +ω t            LINEAR MOTION 
                                                  0
                                                         Note:  θ is in radians 
                                                   
                                         Angular Momentum 
                                          
                                                         Angular momentum is the rotational equivalent of linear momentum.  It is a 
                                         conserved quantity 
                                                                                                                                                               A rigid body can be thought 
                                                                                                            y                                                  of as a large number of 
                                                                                                                r                                              particles i, at positions A , 
                                                                                                               ω                                                                                                    i
                                                                                                                                                               with masses m, at position 
                                                                                                                                                                                              i
                                                                                                                                                               vectors from an origin on the 
                                                                                                                                                               axis of r, at distances R  from 
                                                                                                                            A                                                    i                                 i
                                                                                                                               i                               the axis, and with velocities 
                                                                                                                                                               v.
                                                                                                                                                                 i
                                          
                                                                                                                                                                       x 
                                          
                                          
                                          
                                          
                                                     z 
                                          
                                          
                                         The particle at the point 
                                                                                              A has linear momentum p = mv and Angular Momentum 
                                                                                                  i                                                              i   i
                                         about the origin 
                                                                                                              L =r ×p
                                                                                                                  i        i         i                            
                                                                                                                                                             ˆ
                                                                                                                    =mr ×v =mrvn
                                                                                                                            i  i         i          i   i  i
                                                         ˆ
                                         where n is unit vector normal to both r and p. 
                                                                                                                               i              i
                                          
                                         Note that L is not generally parallel to the axis of rotation.   
                                          
                                          
                                         Moment of Inertia 
                                          
                                         The component of the angular momentum in the y-direction, i.e. along th axis of 
                                         rotation, is 
                                                                                                                    L =mrv sinθ  
                                                                                                                       iy            i i i               i
                                         However,  r sin θ= R, so  
                                                                   i            i        i
                                                                                                                          L =m Rv  
                                                                                                                             i           i    i   i
                                         The magnitude of ω is the same for all points, so we drop the index i, and v = Rω.  
                                                                                     i                                                                                                                       i          i
                                         Then 
                                                                                                                         L =mR2ω 
                                                                                                                            iy            i    i
                                         Summing or integrating over all points, 
                                                                                          L =L +L +K= L =ω mR2 
                                                                                             y           1y           2y                   ∑ iy                  ∑ i i
                                                                                                                                             i                      i
                                         This sum, I = ∑m R2  is called the Moment of Inertia , and L = Iω  (c.f. p = mv) 
                                                                                    i    i
                                                                           i
               Moment of Inertia of Potato-Shapes 
                
               The Moment of Inertia depends on the axis of rotation.  
                
               The Angular Momentum is generally not parallel to the axis of rotation.  
                
               For a body of general shape (an asteroid, a potato . . .) there are three mutually 
               perpendicular axes for which the angular momentum is parallel to the axis.  These are 
               called the Principal Axes of intertia and the moments of inertia about them are the 
               Principal Moments of Inertia. 
                
               For bodies of higher symmetry than potatoes, the Principal Axes are generally Axes 
               of Symmetry. 
                
                
                
               Angular Momentum Examples: 
                                                        Two masses m are going 
                                                        round the z-axis, at radius R, 
                                                        in the x-y plane, with speed v 
                
                                                        = Rω.    
                                                        L=r×mv for eachmass
                                                        r = R
                                                        v = Rω                
                
                                                        L =2mR2ω for the pair
                
                
                
                     Direction of L is at right angles to r and v, i.e. in same direction as ω, i.e. 
                                           L=2mR2ω=Iω 
                                             and I = 2mR 
                
                                                         Now incline the masses to the 
                                                         axis.  
                             L                           L=r×mv for eachmass
                                                          r = R
                                                          v = Rωsinϕ
                                                          L =2mR2ωsinϕ for the pair
                
                
               Direction of L is at right angles to r and v, i.e. at ϕ to ω, i.e. it precesses about z.  
                             2  2              2  2
                     L = 2mR sin ϕ ω, and I = 2mR sin ϕ about the z-axis. 
                       z
                                    Calculation of Moments of Inertia 
                                                                                                                                                           R is distance to 
                                                                                                            I = ∑m R2                                         i
                                                                                                                            i    i                         axis of rotation 
                                                                                                                     i
                                     
                                                   If an object is considered to consist of elemental particles of mass dm, then  
                                                                                                              dm=ρdV  
                                                   and the sum becomes an integral over the volume.  If the density ρ is constant, 
                                                   it comes out of the integral and 
                                                                                                 I = ∫R2dm =ρ∫R2dV  
                                                                                                        V                      V
                                     
                                                                       Note that  ∫ R2dV  is a purely geometrical factor
                                                                                          V
                                     
                                     
                                                   Example:  Moment of Inertia of a thin rod rotated about one end.  
                                     
                                     Cross-sectional area 
                                                                                           S, length L, density ρ. 
                                     
                                     Element dV is disc of area S, distasnce from axis x, thickness dx.  
                                                                                  (                                                                   2      )
                                                                                   i.e. dV = Sdx, dm = ρSdx, dI = ρSx dx  
                                                   So we have 
                                                                                                       L 2               1          3      1         2
                                                                                        I = ρS∫ x dx =                      ρSL = ML  
                                                                                                       x=0               3                 3
                                     
                                                   Example:  Moment of Inertia of the same thin rod rotated about its centre.  
                                                                                           L/2                            3               3 
                                                                               =ρ                   2       = 1ρ L −−L                               1          2
                                                                            I        S∫          x dx                 S                        =         ML  
                                                                                          x=−L/2                3         8              8         12
                                                                                                        2
                                                                    In general, I = Mk .  The length k is a characteristic 
                                                                    length of the object, called the Radius of Gyration – 
                                                                    compare with Centre of Gravity.