What's the source of increase in rotational energy of carousel?

[Karagoz]

Homework Statement

What's the source of increase in rotational energy of a carousel when two persons in that carousel move into centrum, assuming the angular momentum is conserved?

Relevant Equations

w: angular speed
I: inertia
Rotational energy: 1/2 * I * w^2

A carousel has the shape of a circular disc with radius 1.80 m and a mass of 300 kg. There are two people with masses of 30 and 45 kg out on the edge while carousel rotates with the angular speed 0.6 rad / s.
The people move towards the center of the carousel

Calculations show that the rotational energy increase after these two people move towards the center of the carousel. It's calculated by this:

The first question is: what will be the angular velocity now. The angular velocity will be 0.9 rad/s.

Since angular momentum is conserved: Inertia_0 * w_0 = Inertia_1 * w_1

Inertia_0 = 729 (when the persons are at the edge)

Inertia_1 = 486 (when the persons are in the centrum)

w_0 = 0.6 rad/s (as mentioned above)

w_1 = Inertia_0*w_0 / Inertia_1 = 0.9 rad/s.

The second question is the change in rotational energy. That's simple:

1/2*Inertia_1*w_1 - 1/2*Inertia_0*w_0 = 65 j.

But the question I can't answer is, what's the source of that energy?

In this link where they show a similar example (the example is a rotating ice dancer pulling her hands inwards and increasing her rotational energy):
https://opentextbc.ca/universityphysicsv1openstax/chapter/11-2-conservation-of-angular-momentum/

They say the following:

The source of this additional rotational kinetic energy is the work required to pull her arms inward. Note that the skater’s arms do not move in a perfect circle—they spiral inward. This work causes an increase in the rotational kinetic energy, while her angular momentum remains constant. Since she is in a frictionless environment, no energy escapes the system. Thus, if she were to extend her arms to their original positions, she would rotate at her original angular velocity and her kinetic energy would return to its original value.

It says the source of the additional rotational kinetic energy is the work required to pull her arms inward.

There's work required to pull her arms inward, but there's also work required to pull her arms outward too. How come when the work of her pulling her arms inwards the rotational kinetic energy increases, but the work of her extending her arms outward decreases the rotational kinetic energy?

I can't understand how it works. How does the rotational kinetic energy increase?

And does total energy increases? Or only rotational kinetic energy?

 

[Orodruin]

The force required to keep the persons from falling off the carousel is the centripetal force. When the persons move inward, they get a velocity component inwards that is parallel to that force and therefore does positive work. When they move outward, the displacement instead has a component in the opposite direction and the work is negative.

Things become a bit simpler in a rotating frame. The work done is because the centrifugal potential is larger near the center so moving inwards is similar to moving up in a gravitational field. The torque providing the additional angular momentum is provided by the Coriolis force on the persons when they move in. When they move out the Coriolis force reverses direction.

Reminds me of one of my favourite physics limericks:

On a merry-go-round in the night
Coriolis was shaken by fright
Despite how he walked
’Twas like he was stalked
By some fiend always pushing him right

 

[Karagoz]

The torque providing the additional angular momentum is provided by the Coriolis force on the persons when they move in. When they move out the Coriolis force reverses direction

Maybe you meant the angular velocity (as angular momentum is conserved).

As I understand, because of Coriolis effect, moving inwards on the carousel makes it look like it's moving into the same direction as the rotation, hence increases the speed and energy of the rotation.
And again because of Coriolis effect, moving outwards on the carousel makes it look like it's moving into the opposite direction as the rotation, hence increases the speed and energy of the rotation,

But isn't Coriolis force a fictive force, (like centrifugal force) and so can't do any work?

 

[Steve4Physics]

There's work required to pull her arms inward, but there's also work required to pull her arms outward too.

If she let her arms dangle (so the muscles were relaxed, not 'working') what would happen to her arms?

 

[Orodruin]

Maybe you meant the angular velocity (as angular momentum is conserved).

Not in the instantaneous corotating frame.

 

[Karagoz]

If she let her arms dangle (so the muscles were relaxed, not 'working') what would happen to her arms?

Her arms would move outwards. It requires force to move things inwards in a rotating object. Slipping things move them outwards. So isn't that some sort of potential energy when you keep things at the centrum in a rotating object?

 

[jbriggs444]

Maybe you meant the angular velocity (as angular momentum is conserved).

As I understand, because of Coriolis effect, moving inwards on the carousel makes it look like it's moving into the same direction as the rotation, hence increases the speed and energy of the rotation.
And again because of Coriolis effect, moving outwards on the carousel makes it look like it's moving into the opposite direction as the rotation, hence increases the speed and energy of the rotation,

But isn't Coriolis force a fictive force, (like centrifugal force) and so can't do any work?

It is not that fictitious forces cannot do work. It is that the Coriolis force acts at right angles to the motion that produces it and, so, does no work relative to the rotating frame that is being considered.

However, if one is constraining an object to follow a radial path and if that object is subject to a Coriolis force then that object must also be subject to some real force -- the force that constrains it to the path. That real force exists regardless of the frame of reference that one adopts. That real force can do work relative to a frame of reference where the object is moving tangentially. So an object drawn inward on a radial track on a rotating carousel can do work on the carousel as it is drawn inward.

On the other hand if we include the object as part of the carousel+object system then that force is an internal contact force between two components that have no relative motion in the direction of the force. So that tangential force can do no net work on the system as a whole.

Now, as to fictitious forces doing work, suppose that we have a free object (no external real forces) that is motionless somewhere in a rotating frame. Suppose further that it is not sitting at dead center. That object is subject to a fictitious centrifugal force. It will begin moving radially (as measured from the rotating frame). It will gain kinetic energy (as measured from the rotating frame). The fictitious centrifugal force will have done work (as assessed in the rotating frame).

Yes, as judged from an inertial frame, this force-free object will still have the same velocity it started with and the same kinetic energy it started with. No work will have been done. But that's fine. Both kinetic energy and work can vary depending on the frame of reference you adopt.

There is one useful thing that is invariant: The sum of the work done by the two sides of a third-law real force pair. Like the force pair for a person pulling a ball toward a hub on a string and the ball pulling back. Positive for reeling in. Negative for reeling out.

 

[Karagoz]

On the other hand if we include the object as part of the carousel+object system then that force is an internal contact force between two components that have no relative motion in the direction of the force. So that tangential force can do no net work on the system as a whole.

But in this case, the persons and the carousel is perceived as one object. And in the calculations the object's rotational kinetic energy increases.

The rotational inertia when the persons stand at the edge is different (is higher) from the rotational inertia when the persons stand at the centrum. This change in rotational inertia causes change in angular speed (since the angular momentum is conserved). Angular speed increases.

Since rotational kinetic energy is measured by: 1/2*(rotational inertia)*(angular speed)^2
Rotational inertia is reduced, but angular speed increased. This change in total causes the rotational kinetic energy of the object to increase.

 

[Steve4Physics]

Her arms would move outwards. It requires force to move things inwards in a rotating object. Slipping things move them outwards. So isn't that some sort of potential energy when you keep things at the centrum in a rotating object?

Yes her arms would move outwards. If her arms did not move outwards, this would be the result of an inwards force (a centripetal force) on her arms; the force would be exerted by her muscles/tissue.

Thinking in terms of ‘potential energy’ isn’t helpful here.

It is important to understand why thing tend to ‘move outwards’.

Suppose you are standing in a bus with a slippery floor while the bus goes round a bend. To an observer in the bus, you are ‘thrown outwards’. Apparently there is a force making you accelerate towards the side of the bus.

But an external observer on the ground will see what is really happening. You are moving in a straight line (because you slide in a straight line – Newton’s 1sdt law) while the bus moves in a curve.

 

[haruspex]

But in this case, the persons and the carousel is perceived as one object. And in the calculations the object's rotational kinetic energy increases.

Yes, but as has been pointed out, that work comes from chemical energy in the muscles. True, some of that is mediated by the tangential force to pass to the carousel its share of the work done, but the equal and opposite force acts on the person. So in the carousel+person system there is no net force arising from this pair, and no net work done by them.