Suppose a horizontal disc fixed in the center with a vertical shaft passing perpendicular to the plane, is rotating at some angular speed and there is an insect sitting initially at center. The insect then starts moving radially outwards. From conservation of angular momentum of insect+disc system, we can state that the angular speed of disc reduces as the moment of inertia of the insect with respect to the axis increases. My question here is that, if we see only the disc without the insect as the system(or as a single rotating object), the angular speed of it changed and hence, some external torque has definitely acted upon the disc. What is the force that produced this torque?

I tried to think about it - If we say that the insect is moving along $x$ axis and the axis of rotation is the $z$ axis, the weight of the insect should not cause any torque because the cross product of distance and insect's weight lies in $y$ axis while we need a torque about z axis. Therefore, there must be a force along the y axis, what can be such a force? Can anyone help me out in analyzing the motion and dynamics out here?

  • $\begingroup$ U cant call a single object as a system $\endgroup$ – Anusha Jan 11 at 9:34
  • $\begingroup$ @Anusha May be terminologically I used incorrect words. But, I meant to say be it an object or a rotating system, why does the angular speed change - who exerted the torque? $\endgroup$ – Sam Jan 11 at 15:08
  • $\begingroup$ if torque would have been exerted then you couldn't have used conservation of angular momentum since u are using it here so net torque is 0 $\endgroup$ – Anusha Jan 11 at 17:08
  • $\begingroup$ In the insect+disc system, there is no external torque. Hence, in disc+insect system angular momentum is conserved. But see with respect to the disc neglecting insect, the moment of inertia of disc remains same but still the angular speed changes. You will see angular momentum of just the disc changes definitely. $\endgroup$ – Sam Jan 11 at 17:28
  • $\begingroup$ If the fly is forced to walk on a groove, then the side load provides the torque, and it should be felt by the pivot also. $\endgroup$ – JAlex Jan 11 at 21:14

Therefore, there must be a force along the y axis, what can be such a force?

Friction is that force. It speeds the bug as it moves to tangentially faster moving parts of the ring. Resulting in an equivalent and opposite force slowing the ring down. In the disc's frame friction opposes the fictitious Coriolis force.

Tangentially faster? Isn't the ring slowing down too? Here is why the angular momentum of the bug always increases even though the ring slows down.

Let the total angular momentum of the system be 'L' which it =$I_{system}\omega_{system}$ so the angular velocity of the bug as a function of distance from the axis(r):$$ \frac{L}{\frac{MR^2}{2}+Mr^2}$$ Assuming ring and bug to have the same mass for ease of calculation.

The angular momentum of the bug will be$$ \frac{L}{\frac{MR^2}{2}+Mr^2}Mr^2$$$$ \frac{Lr^2}{\frac{R^2}{2}+r^2}$$ Let the disc have a radius of 5 and the total angular momentum 10 all in its SI units. Graphing our result:

The blue line represents r=5 which is the radius of the disc and it's where the bug stops, note how the angular momentum constantly increases. Now a Graph with how the angular momentum of the disc varies(green):enter image description here

  • $\begingroup$ Well thanks,but I have a small doubt. As the insect is progressing away from the center, there is definitely a constantly increasing friction acting radially inwards(whose torque is zero); in absence of this force there wouldn't be sufficient force for circular motion and the insect will fly out tangentially. When we normally walk on road, we give friction that acts only along one axis. In a rotational reference frame, would there be friction along two axes or just one? If there is tangential friction, what is its significance - I mean what would the motion be without that tangential friction? $\endgroup$ – Sam Jan 11 at 17:49
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    $\begingroup$ In the absence of frictional force, the insect will stay where it is. It will fly out tangentially if the force is suddenly removed. Two axes if the disc is accelerating (retarding in our case). The last query (bug moving on the disc with only radial friction) seems very interesting to me can't wrap my head around it. Unfortunately, I have a test tomorrow don't have time to give it much thought , might have to pull off an all-nighter. You could ask it as a separate question if you would like. $\endgroup$ – JustJohan Jan 11 at 18:12
  • $\begingroup$ The bug would start to slide (or stumble, perhaps) sideways relative to what it would consider the straight radial path. This is the Coriolis effect. $\endgroup$ – Kristoffer Sjöö Mar 13 at 13:48

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