[理学]Physics 7 rotations of rigid bodies.ppt

Example Atwoods Machine with Massive Pulley:A pair of masses are hung over a massive disk-shaped pulley as shown . Find the acceleration of the blocks. Example A disk of mass M and radius R rotates around the z axis with angular velocity ?i. A second identical disk, initially at rest on the same axis, is dropped onto the first disk. Because of surface friction, the two disks eventually attain a common angular velocity ?f. Find ?f We realize that there are no external torques acting on the two-disk system. Angular momentum will be conserved! O For i-th particle Tangential component The torque about the fixed axis 2. Dynamical Equation for Rotational Motion Compared with Newton’s Second Law： Tnet ext the net external torque about Z-axis I the rotational inertia about Z-axis a the angular acceleration about Z-axis The torque causes angular acceleration; just how much depends on the rotational inertia: Example A light cord is coiled around the rim of a flywheel with radius r =20 cm and rotational inertia I=0.5 kg·m2 . Pull the end of the cord with a force F=98 N . There is no friction between the flywheel and the rotational axis. Calculate: (a) The angular acceleration of the flywheel, and (b) if block of weight P =98 N is hung from the end of the cord instead，what is the angular acceleration of the flywheel. Solution T y Nonslip conditions m2 m1 R M m2g a m1g a For the hanging masses m1g - T1 = m1a T2-m2g = m2a For the pulley T1R - T2R=Ia a=Ra Solution T1 T2 Example A slender uniform rod of length l and mass m rotates from rest at the horizontal position in vertical plane about the axis O. Find the angular speed ? when its angular coordinate is ? . O l m x Solution The torque of gravity acts as if it were applied to the center of mass dmg O l m ? x C We also can use the conservation of mechanical energy Example A disk rotates on a table with inertial angular speed ?0. The coefficient of friction between the disk and the table is ?. Find the