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I am now stuck on the impacts of choice of the coordinate system on the change in angular momentum and I would really appreciate it if someone can give me some help on this.

Consider the following example, there is a stick with mass $m$ and length $l$ which is initially at rest. Suppose a mass travels perpendicular to the stick and hits the stick at one of its ends in a very short time (then it disappears) and then the stick will have both translational and angular momentum.

However, I am very confused that if I choose the fixed point which coincides with the point of hit as the origin of our coordinate system, then the torque will be zero because the lever is $0$, which implies that the angular momentum will be conserved. Then after the collision, there is no external forces, and the angular momentum will be also conserved. In other words, the angular momentum doesn't change because the impact is so fast and I choose the point of impact as origin. I am wondering why this argument is not correct?

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  • $\begingroup$ This may help - Toppling of a cylinder on a block $\endgroup$
    – mmesser314
    Nov 12, 2023 at 0:29
  • $\begingroup$ I think torque may be zero if you choose origin to be at the point of the impact, but the motion will not change - the other equations will simply become more complex. How would you describe the motion of this rod? The only sane way I can think of is to have equation of motion for the center of rod, and then another equation for its rotation. In this system there would be torque. You could however choose to describe motion in terms of position of the point of the rod that is struck, and then the orientation of the rod from that position (like an angle). .... $\endgroup$
    – Cryo
    Nov 12, 2023 at 9:50
  • $\begingroup$ In that system, you would not have torque, probably, but system of equations will be far more complex, and you would still observe the same motion of the rod, now much more obscured by algebra $\endgroup$
    – Cryo
    Nov 12, 2023 at 9:51
  • $\begingroup$ To obtain the stick translation velocity and the angular velocity $~\omega~$, after the collision , you have to choose a coordinate system at the center of mass of the stick. thus the angular momentum of the stick after the collision is $~L=I_{\rm cm}\,\omega~$ which is conserved $\endgroup$
    – Eli
    Nov 12, 2023 at 16:40

1 Answer 1

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When you choose the fixed point which coincides with the point of hit as the origin of our coordinate system, the angular momentum of the rod is $$\vec{L}= M (\vec{R}\times \vec{V}_{cm}) + I_{cm}\vec{\omega}$$ Where $\vec{R}$ is position vector of CM of the rod. These two will have to exactly cancel each other out to ensure $\vec{L}=0$ and angular momentum is conserved.

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