Why can orbital and spin angular momenta be added?

Justifying the addition of these two types of angular momentum in classical mechanics, using the example of a planet orbiting a star.
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About me: I studied Physics at the University of Cambridge, then stayed on to get a PhD in Astronomy. During my PhD, I also spent four years teaching Physics undergraduates at the university. Now, I'm working as a private tutor, teaching Physics & Maths up to A Level standard.
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[@MissPiggyM976]

Very insightful, thanks!

[@DrBenYelverton]

Thank you!

[@feiqu4817]

Actually you posted a good question to help people free from dogmatically thinking

[@r2k314]

Besides the math, to me this makes sense. What's harder to understand is: how the angular velocities of objects revolving about different axes can be added to plug in equations, how angular velocities can be projected on axes they don't spin around, and how the angular momentum vector can point in a direction which is not coincident with an spin axis. But my physical intuition sucks.

[@DrBenYelverton]

All good ideas for future videos! That last point especially can be pretty unintuitive.

[@AdityaKumar-bm9mh]

Thanks sir . Had a question lets say if this planet is forced to spin about an axis that is not its centre of mass . Will this equation work at any instant ? Assuming R vetor as the position vector of centre of rotation wrt to the sun at the centre .and in the final expression v_i .m_i is the linear momentum wrt to centre of mass but r_i is the wrt to the axis of rotation .

[@DrBenYelverton]

The spin angular momentum has to be measured about the centre of mass, otherwise the two sums that we set to zero in the video would no longer be zero and you'd get a more complicated expression.

[@Nxck2440]

Is this also true for atoms? in class we learned about how electron orbitals have some angular momentum (like p or d orbitals have nonzero angular momentum quantum number) and spin angular momentum. If these spins are both nonzero they can add together and increase the magnetic moment of the atom?

[@DrBenYelverton]

Yes, the orbital angular momentum and spin can be added in quantum mechanics too, so this does work for atoms! However, the justification in the video is no longer applicable in that case because the spin of an electron is an intrinsic property rather than a sum over particles as in the video.

[@feiqu4817]

But i will assume that this question is not completed. it is accurally to say “ in what way/ with which prepositions orbital and spin angular momenta be added”

[@kekso2205]

But if web say the planet is moveing and an orbit is elypse than R changes with respect to time (in the case of earth). Or with respect to its position on an orbit. Is this thinking right?

[@DrBenYelverton]

It certainly is! Maybe it was a bit misleading for me to call it a "constant" at 4:10 in the video. What I meant was that it's constant with respect to the sum over particles (i.e. doesn't depend on i), not that it doesn't change over time. The position vectors and velocities all vary with time but our reasoning is still equally valid.

[@feiqu4817]

because the motion can be observed in a summable way