Or in vector notation l mr 2 ˆ k v r x y phys 2a

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or, in vector notation, ~ L = mR 2 ! ˆ k ! v r x y
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Phys 2A - Mechanics v r x y ~ L = mR 2 ! ˆ k Note: In this example, we can think of the particle motion as rotating about a fixed axis corresponding to the z-axis of coordinates. Then the moment of inertia is and the angular momentum can be given in terms of this as I = mR 2 ~ L = I ! ˆ k For a rigid body rotating about the z-axis, L z = I ! Generalization: Given without proof. But it is plausible that one obtains this by adding over many particles, as that of the example.
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Phys 2A - Mechanics Example: A 7 kg particle is moving on the xy-plane in a trajectory parallel to the x-axis, at y=3 m, in the positive x-direction, at 2 m/s. What is its angular momentum? ANS: Since at all times and are in the xy-plane, the cross product is along the z-axis, but this time in the negative z-direction (by the right hand rule). The magnitude of the angular momentum vector is ~ r ~ v x y ~ r ~ v L = mrv sin = mr ? v = (7)(3)(2) kg.m.m/s = 42 kg.m 2 /s And we can write ~ L = - (42 kg.m 2 /s) ˆ k Moral: Do not need “circular” motion for non-zero angular momentum. r ?
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Phys 2A - Mechanics Conservation of Angular Momentum Recall: conservation of (linear) momentum follows from ~ F net = d ~ P dt Compute, for a single particle that is ~ net = d ~ L dt d ~ L dt = d dt ( ~ r ~ p ) = d ~ r dt ~ p + ~ r d ~ p dt = ~ v ( m ~ v ) + ~ r ~ F net = 0 + ~ net This gives a conservation law: if the net external torque on a particle is zero, its angular momentum is conserved (constant in time)
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Phys 2A - Mechanics v r x y In our previous examples... ~ L = mR 2 ! ˆ k is constant ~ net = 0 since ~ F net is centripetal, hence parallel to ~ r x y ~ r ~ v ~ L = - mr ? v ˆ k is constant ~ net = 0 since ~ F net = 0 r ?
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Phys 2A - Mechanics For a collection of particles, we can copy from above: d ~ L dt = d dt X i ~ r i ~ p i = X i d ~ r i dt ~ p i + ~ r i d ~ p i dt = X i ~ r i ~ F i, net In the last line we would like to separate external from internal forces, and keep only external forces (for a net external torque).
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