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Neither spin nor orbital angular momentum are conserved on their own, we need both to have a fully conserved quantity. Why and How?

Take the example of the earth. Earth moves in an orbit around the sun which is its orbital angular momentum. Earth also spins on its axis which is spin angular momentum. So in order to conserve momentum the sum of both the momenta must bee equal and offset each other. But neither orbital angular momentum nor spin angular momentum is conserved on its own. Why?

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In short, whenever there is anything that couples to the spin. If you the spin really is isolated, then it may well be conserved independently, but this will generally not be the case. Unfortunately, that's about as much as you can say about the general case.

For the specific case of the spin of the Earth, then indeed it is approximately conserved over short timescales, but over periods of several thousand years it does change.

In this specific case, as with all spin-affecting gravitational interactions, the culprit is tidal forces, and these exert a torque whenever the body deviates from spherical symmetry. The Earth, for example, bulges at the equator, which is not aligned with the orbital plane; moreover, the bulge on the Sun-facing side is more strongly attracted to the Sun than the bulge on the night side, which means that there is a residual torque that tries to align the spin and orbital angular momenta. And, since there is a torque acting on it, the spin is no longer conserved independently.

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So in order to conserve momentum the sum of both the momentum must equal and offset.

No. We define the system as a closed system, i.e. no external torques. Newtonian and Lagrangian mechanics then informs us that the total angular momentum of the system is a conserved quantity (so is total energy). Since there are only two sources of angular momentum, their sum has to equal the total angular momentum and thus be conserved. I am restating what you said but putting it in the right order.

You can draw the parallel to energy, as anna v pointed out. The total energy of the system is conserved and since the only two energies present are kinetic and potential, their sum equals the total energy and is conserved.

tl;dr In a closed system, orbital angular momentum can be exchanged with spin orbital momentum in the same way kinetic and potential energies can be exchanged, and balance out to give a conserved quantity.

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