Rotation

Rotation Kinetic equation of rotating rigid body   = 0 + t  = 0t + f 2 = 02 + 2 Moment of inertia I =  Radius of Gyration I = Mk2 That is, k is the distance of the point mass M from the axis of rotation such that this point mass has the same moment of inertia about that axis as the given body. Theorem of parallel Axes I = ICM + Md2 Theorem of perpendicular Axes Iz = Ix + Iy Rotational kinetic energy Consider a rigid body rotating about a fixed axis with angular velocity w. The body can be supposed to be made up of a large number of particles m1, m2 . . . , mn at distances r1, r2, . . . , rn respectively, from the axis of rotation. Each particle has a kinetic energy. The kinetic energy of the body will be sum of the kinetic energies of all the particles. IF 1, 2, . . . , n be the linear speeds of the particles, respectively, then the kinetic energy of the body can be written as Ek =  Torque  = F × d = Fr sin  In vector notation Angular momentum Relation between t & L Work done by a torque dW = d W =  Relation between Torque and Angular acceleration  = I Work-Energy Theorem W =  Relation between Angular momentum and Angular velocity L = I For slipping body ET =  For spinning body ER =  For rolling body EN =  Rolling without slipping En =  Rolling on an inclined plane Velocity at lower point v =  Acceleration in motion a =  Time of descent t =  Motion of connected mass  Downward acceleration of point mass a =  Tension in string T = mg  Velocity of point mass v =  Angular velocity of rigid body  =  Moment of inertia of different bodies S.No. Body Axis Moment of Inertia 1. Uniform thin bar  (i) Through centre of gravity and perpendicular to length (ii) Through one end and perpendicular to length 2. Rectangular lamina  Passing through its C.G. and perpendicular to its plane of length and breadth.  3. Ring or Hoop (i) Passing through its centre and perpendicular to its plane. (ii) About a diameter  MR2  4. Disc (i) Passing through its centre and perpendicular to its plane. (ii) About a diameter 5. Hollow disc of radii R1and R2 Passing through its centre and perpendicular to its plane.  I =  6. Solid cylinder (i) About its own geometric axis. (ii) Passing through C.G and perpendicular to its geometric axis.  7. Hollow cylinder (i) About its own geometrical axis. (ii) Passing through C.G and perpendicular to its geometric axis. MR2  8. Thin spherical shell (i) About a diameter. (ii) About a tangent. 9. Solid sphere (i) About a diameter. (ii) About a tangent. 10. Diatomic molecule  Passing through centre of gravity and perpendicular to bond length