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Chapter 9: Rotational Motion Rigid body instead of a particle Rotational motion about a fixed axis Rolling motion (without slipping) Rotational Motion Angular Quantities Kinematical variables to describe the rotational motion: Angular position, velocity and acceleration l ( rad ) R d lim t 0 t dt ( rad/s ) ave d lim t 0 t dt 2 ( rad/s ) ave Rotational Motion “R” from the Axis (O) Solid Disk Solid Cylinder Rotational Motion Linear and Angular Quantities l (1) t (like l v t ) R l ( R )t v R dv d (2) a tan R dt dt a tan R (3) a rad v 2 (R)2 2R R R Rotational Motion atan arad Kinematical Equations Conversion : x , v , a 1 2 (1) 0 0t t 2 (2) 0 t (3) 2 ( - 0 ) 2 2 0 Note : constant Rotational Motion Chapter 10: Rotational Motion (II) Rigid body instead of a particle Rotational motion about a fixed axis Rotational dynamics Rolling motion (without slipping) Rotational Motion Angular Quantities: Vector Kinematical variables to describe the rotational motion: Angular position, velocity and acceleration z Vector natures l ( rad ) R d ˆ ˆ k k ( rad/s ) dt d ˆ ˆ k k ( rad/s 2 ) dt R.-H. Rule y x Rotational Motion Rotational Dynamics: t (a) ax la (b) lb m I a x F cos F cos m a x F ; l F l F l I t (torque) F l (magnitude ) Rotational Motion Note: t = F R sin Lever arm : l or R (Perpendic ular distance ...) t F ( R sin ) F R t (F sin ) R F R Rotational Motion Note: sign of t t 1 F1 ( R1 sin 90 ) (50.0 N)(0.300 m) 15.0 N m t 2 F2 ( R2 sin 60 ) (50.0 N)(0.500 m)(0.866) 21.7 N m t net t 1 (c.c.w ) t 2 (c.w.) t 1 (1) t 2 (1) (15.0 N m) - (21.7 N m) - 6.7 N m 6.7 N m (c.w.) Rotational Motion Rotational Dynamics: I F m a m ( R ) t (torque) F R (m R 2 ) (a) I m R 2 (b) I m i Ri 2 (moment of inertia (moment of inertia for single particle) for a group of particles) m2 m1 Rotational Motion m3 Rotational Dynamics: I I I cm (moment of intertia about the c.m.) d I I cm Md 2 (parallel - axis theorem) Rotational Motion Parallel-axis Theorem d I I cm Md 2 1 2 2 MR 0 MR 0 2 3 2 MR 0 2 Rotational Motion Parallel-axis Theorem 2 1 l 1 l 2 2 Ig If M M l M M l 4 3 2 12 2 Rotational Motion Example 1 Calculate the torque on the 2.00-m long beam due to a 50.0 N force (top) about (a) point C (= c.m.) (b) point P Calculate the torque on the 2.00-m long beam due to a 60.0 N force about (a) point C (= c.m.) (b) point P Calculate the torque on the 2.00-m long beam due to a 50.0 N force (bottom) about (a) point C (= c.m.) (b) point P Rotational Motion Example 1 (cont’d) Calculate the net torque on the 2.00-m long beam about (a) point C (= c.m.) (b) point P Rotational Motion