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I came across a question in which an insect walks towards the center of a rotating disc and its angular velocity increases due to conservation of angular momentum.

How can I solve this without conservation of angular momentum, with only using basic concepts? I tried to solve it in reverse.

I solved for the angular velocity using conservation of angular momentum, followed by change in kinetic energy and differentiated it to get force. I was getting a weird expression for it. It would be a great help to solve this problem.

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    $\begingroup$ is not balance of angular momentum not fundamental enough? It's one of the first equations that can be derived starting from Newton's second law $\endgroup$
    – basics
    Commented Mar 16, 2023 at 10:13
  • $\begingroup$ Yes but still if I try to solve using torque provided by friction or work energy theorem can I solve it $\endgroup$
    – Siddarth Gupta
    Commented Mar 16, 2023 at 10:40
  • $\begingroup$ try to have a look here: physics.stackexchange.com/a/733999/343955. Maybe it is not exactly what you need, but it can give you a hint $\endgroup$
    – basics
    Commented Mar 16, 2023 at 12:34
  • $\begingroup$ Does this answer your question? Coriolis force and conservation of angular momentum $\endgroup$
    – Joe Iddon
    Commented Mar 16, 2023 at 13:27
  • $\begingroup$ Likely you are supposed to ignore the change of angular momentum of the disc. That is, the disc is so massively much larger than the insect that it changes only a negligible amount. This would be a fairly good approximation. Plus, it's easier, and would give you something to compare to if you do the full solution. $\endgroup$
    – Boba Fit
    Commented Mar 16, 2023 at 13:32

2 Answers 2

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Yes, the kinetic energy will change as the insect will have to do some work against the centrifugal force to keep moving towards the centre, and that work will get converted into the kinetic energy. So here you can actually solve this with conservation of energy rather than using conservation of momentum.

Here it is,

$\frac{1}{2}I_1 \omega_1^2+\frac{1}{2}I_2 \omega_1^2+F_cx=\frac{1}{2}I_3 \omega_2^2+\frac{1}{2}I_2 \omega_2^2$

Where here, $I_1$ is the moment of inertia for that insect initially, $I_2$ is the moment of inertia for disc and $I_3$ is the final moment of inertia of that insect. Also, $\omega_1$ and $\omega_2$ are the initial and final angular speeds.

Careful, insect and the rotating disc will have the same angular speed only when it's confirmed that there is no slipping between them.

Now coming towards what is $F_cx$ here, so it is the amount of work done by that insect to reach the desired point towards the centre, it's assumed to be $F_c$ as there is no mentioning here, if that insect has some net acceleration towards the centre or not. So if it is moving with a very small constant speed then it's absolutely fine to write that.

But if you really want to do this in a much quicker way then you will have to solve this by conserving momentum only as it is itself a fundamental method to solve for these types of questions.

Hope this helps.

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  • $\begingroup$ But how will you calculate Fc here as it will be mw^2x but w itself is variable $\endgroup$
    – Siddarth Gupta
    Commented Mar 16, 2023 at 12:02
  • $\begingroup$ @SiddarthGupta Fun fact:- believe me or not, this will get complicated ahead, that's why it's always preferred to solve such problems with the help of conservation of momentum instead, but still of you want to go further with the answer I posted above, then to find relation between $\omega$ and $x$ you will again have to use conservation of angular momentum's general equation and then you will differentiate that to get relation between $d\omega$ and $dx$, which you can now finally use that when solving for the work of that insect. $\endgroup$
    – Tejas Dahake
    Commented Mar 16, 2023 at 12:34
  • $\begingroup$ (Obviously that would be an instaneous work so you will have to take care of variables along with) clear? $\endgroup$
    – Tejas Dahake
    Commented Mar 16, 2023 at 12:35
  • $\begingroup$ But it again comes down to conservation of angular momentum. And also I tried this but the answer will not match $\endgroup$
    – Siddarth Gupta
    Commented Mar 16, 2023 at 13:57
  • $\begingroup$ Yes, because you can't do anything else here, conservation of angular momentum is what the fundamental rule. Because without that how will you relate their angular speeds? And coming to the error you are getting here. That is most probably happening because we've assume no slipping between the insect and the disc. Yes? That means all the L.H.S of energy will be getting converted into the R.H.S andsince you are using calculus here so the chances of getting answer different one might get higher. But again I will warn you that we have considered the scenario in which there is no loss of energy. $\endgroup$
    – Tejas Dahake
    Commented Mar 16, 2023 at 14:09
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How can I solve this without conservation of angular momentum with only using basic concepts? - With considerable difficulty.

If angular momentum of the ant & disc system about the centre of the disc is to be conserved then there can be no external torques applied to the system although external radial forces can be applied.

Applying conservation of angular momentum one finds that if the ant moves towards the centre of the disc the angular speed of the system increases because the momentum of inertia of the system has decreased, and if the ant moves away from the centre of the disc the angular speed of the system decreases because the momentum of inertia of the system has decreased.
Then comparing the kinetic energy of the system before and after the ant has moved one finds that when the ant moves inwards the kinetic energy of the system increases and when the ant moves outwards the kinetic energy of the system decreases.

One then might imagine that the difference was due to the work done by/on a radial external force but that neglects work done by forces internal to the system which in this case might be frictional forces.
As the ant travels inwards or outwards and have the same angular velocity as the disc its tangential velocity must change.
An external radial force cannot do this and it is the internal frictional forces which change the angular velocities of the ant (and of the disc simultaneously).

The problem now is to evaluate the work done by the internal frictional forces.
In moving inwards or outwards the angular speed of the ant (and the disc) cannot change instantaneously as that would imply there was an infinite acceleration produced by an infinite force.
Thus, there must be slippage (relative movement) between the ant and the disc before the ant and disc have the same angular speed.
To evaluate the work done by the frictional forces one must know the value of the frictional forces and their displacement during the period of slippage.
That is the sum which is very difficult to do without any further information about the system and even with some extra assumptions it is much harder to do the sums than if using the conservation of angular momentum.

I am sure that you have met may other systems in which frictional/dissipative internal forces are involved, eg non-elastic collision problems, where use of energy conservation or the work-energy theorem are not used?

Note An added complexity to the system can be added by making the ant flexible and then to have the internal forces within the ant doing work.

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  • $\begingroup$ For your information, I would say that friction force is not doing any work here to make that ant reach to it's desired point. Ant will apply torque with its muscles to make it move forward, so hence friction will only help that ant to make it hold still till it puts its legs forwards. And please clarify what exactly do you mean by external forces here. $\endgroup$
    – Tejas Dahake
    Commented Mar 16, 2023 at 13:34
  • $\begingroup$ OK, I forgot the ant is flexible but the force the ant exerts on the disc which changes the angular speed of the disc and the force the disc exerts on the ant which changes the angular speed of the disc are tangential forces and each of these forces does work. Both these forces are internal forces. The external radial force would be applied to the spindle of the disc so that the centre of the disc does not move whilst the ant is moving across the disc. I also did say that the internal forces might be frictional forces, $\endgroup$
    – Farcher
    Commented Mar 16, 2023 at 13:45
  • $\begingroup$ OK, so you were explaining about how is that ant making the disc (and also how disc is making that ant) change its angular speed. I mean what forces are causing them to behave like that. Yes? $\endgroup$
    – Tejas Dahake
    Commented Mar 16, 2023 at 13:54
  • $\begingroup$ The stretching/contacting of the ant until all the ant is moving at its new angular speed. The whole point about this is that no matter by what mechanism the ant moves from an initial position to a final position the start and end conditions (angular speed of ant and disc) are the same. The only way the angular momentum of the disc can change is for a torque to be applied to it by the ant and the only way the angular momentum of the ant can change is for a torque to be applied to it by the disc. In the ant & disc system those are internal torques. $\endgroup$
    – Farcher
    Commented Mar 16, 2023 at 14:04
  • $\begingroup$ OK, I thought you were saying ant as an external agent, that's why I asked you for some clarification regarding your answer. $\endgroup$
    – Tejas Dahake
    Commented Mar 16, 2023 at 14:13

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