Can we drop the internal forces consider particles 1

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Can we drop the internal forces? Consider particles 1 and 2 (all other pairs are similar): ~ r 1 ~ F 21 + ~ r 2 ~ F 12 = ~ r 1 ~ ( - F 12 ) + ~ r 2 ~ F 12 = ( ~ r 2 - ~ r 1 ) ~ F 12 y z r 1 r 2 x ~ r 2 - ~ r 1 Does not vanish generally! But vanishes if the forces between 1 and 2 are directed along the line between them. Most forces are like this. ~ F 21 ~ F 12
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Phys 2A - Mechanics Summary: for a collection of particles ~ net = d ~ L dt The net torque is from external forces only (except if internal pairwise forces are not along the line between the pair) Implication: Angular momentum is conserved if ~ net = 0 DEMOS!! L = I 1 ! 1 = I 2 ! 2 ~ L = ~ L w + ~ L c = ~ L 0 + 0 = ( - ~ L 0 ) + (2 ~ L 0 )
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Phys 2A - Mechanics Example: Inelastic Collision of rotating disks A horizontal disk rotates at 33 rpm about a perpendicular axis through its center. A second coaxial disk is held a small distance above the first. It is then dropped. As a result of the inelastic collision the two disks now spin together. The moments of inertia about the common axis of rotation of the upper and lower disks are 24 kg.m 2 and 12 kg.m 2 , respectively. What is the final common rate of rotation? ANS: Angular momentum conservation can be used (no external torque). Then Take the z-axis along the axis of rotation, ie, vertical. Recall and we only need z component here (so drop the subscript). ~ L f = ~ L i L z = I ! L i = I 1 ! 1 ,i + I 2 ! 2 ,i L f = I 1 ! 1 ,f + I 2 ! 2 ,f set ! 2 ,i = 0 set ! 2 ,f = ! 1 ,f = ! f ) ( I 1 + I 2 ) ! f = I 1 ! 1 ,i ) ! f = I 1 I 1 + I 2 ! 1 ,i = 24 24 + 12 33 rpm = 22 rpm
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Phys 2A - Mechanics Rigid Body-fixed axis, revisited As we have seen, torque is really a vector For a rigid body rotating about a fixed axis of rotation, take the coordinate system to have the z-axis (including the origin) on the axis of rotation. z = ( ~ r ~ F ) z (You can check that this gives with r the distance from the axis -not from the origin- and F the magnitude of the projection of the force on the xy-plane). rF sin z
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Phys 2A - Mechanics Angular Momentum for Rigid Body with fixed axis We already saw L z = I !
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