2. Newton' s second law for rotation ( For system of particles Fig 9-8 P ∑F 2r BEER ∑ 212 0
2. Newton’s second law for rotation Fig 9-8 m2 m1 0 x y F1t F1 F1R → F2 → F2R → F2t (For system of particles) m2 m1 → P → T1r → T2r → T1 → T2 1 r x y 0 2 r
That is∑2=∑:+∑z2 C∑F1)+C∑F2,) (9-7) For each particle f=ma lt Substituting them into Eq 9-7,we obtain ∑τ:=∑F+∑F1 m111+m2a22 2 -mr a +m,r2 a m la 2 m, r +m (9-8)
That is (9-7) For each particle Substituting them into Eq(9-7), we obtain 1 1 2 2 1 2 ( F )r ( F )r t t z z z = + = + F1t = m1 a1t F2t = m2 a2t 2 2 2 2 z 1 1 z 2 2 2 2 1 1 z 2 z 2 2 2 1 1 1 1 t 1 2 2 t 2 z 1 t 1 2 t 2 Iα I m r m r (m r m r )α m r α m r α m a r m a r τ ( F )r ( F )r = = + = + = + = + = + , (9-8)
The a are the same for both particles the total rotational inertia of this two-particle system 1=m11+ (9-9) The obvious extension to a rigid obiect consisting of N particles rotating about the same aX/ss l=mr+mr+ +mIN ∑m,n2 (9-10)
The are the same for both particles ,the total rotational inertia of this two-particle system: (9-9) The obvious extension to a rigid object consisting of N particles rotating about the same axis is (9-10） z 2 2 2 2 1 1 I = m r + m r 2 2 2 2 2 2 1 1 n n N N m r I m r m r m r = = + + +
Fig. 9-8 For system of particles, what kinds of force induce torque? Torque of Internal Forces f1 12 f,十×f f1 Mn=(-乓2)×f1=12×f1=0 Torque of Internal Forces is zero!!! ensionst andt, have also no torque about o
•Torque of Internal Forces: 1 1 2 2 M r f r f In = + 1 2 f f = − 1 2 1 1 2 1 M (r r ) f r f In = − = =0 1 f 2 f Ｏ 1 r 2 r 12 r Torque of Internal Forces is zero!!! For system of particles, what kinds of force induce torque? m2 m1 → P → T1r → T2r → T1 → T2 1 r x y 0 2 r Fig. 9-8 •Tensions and have also no torque about o. → T1 → T2
Thus the torque about o is due only to the external force p Thus we can rewrite the Eq(9-8)as ext.z (9-11) This is the rotational form of newton's Second law Notes: >Tex,: I,a. must be calculated about same axIs For rotations about a single axis i is scalar If many external forces act on the system We add up the torques due to all the external forces a bout that same axis
Thus the torque about o is due only to the external force . Thus we can rewrite the Eq(9-8) as (9-11) This is the rotational form of Newton’s Second law. ext z z , = I → P Notes: , I, must be calculated about same axis. For rotations about a single axis, I is scalar. If many external forces act on the system, we add up the torques due to all the external forces about that same axis. ext,z z