How Does Moving Towards the Center Affect a Merry-Go-Round's Kinetic Energy?

[carsof]

Homework Statement

A child is initially sitting near the outer rim of a revolving merry-go-round. Suddenly, the child moves towards the center of the merry-go-round (while it is still revolving). For the merry-go-round+child system, let the symbols L and K refer to the magnitude of the angular momentum (about the center of the merry-go-round) and rotational kinetic energy, respectively.

Consider the following statements:

Ia. L is conserved Ib. L increases Ic. L decreases

IIa. K is conserved IIb. K increases IIc. K decreases

Which of these statements are true? (The explanation is for the choice of ’II’)

Homework Equations

So, I know L is conserved/constant when dl/dt=0. And I know dl/dt=0 when net torque =0. But, how can I tell from reading this problem that the net torque is zero?
when I draw a diagram of the merry go round and the child on it and make my axis the center of the merry go round, I get Net torque = -mgR (where m is mass of child and R is radius of merry go round). I just don't get how to tell that the net torque in certain problems =0 (and when it doesn't).

Also, I thought for this problem that initially, the merry go round is rotating (has Krot), and the child is not moving w respect to merry go round (K=0). Then, the child is moving (has K trans). So, I don't see how K rotational of the system increases ?
I'm just so confused :(

The Attempt at a Solution

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[kuruman]

Net torque = -mgR

The torque on the child is not mgR. It is generated by friction not by gravity. Regardless of that, your system is child + merry go round. There is no net torque acting on that system because it is isolated (assuming no friction, air resistance, etc.)

From L = Iω and K_rot = (1/2)Iω², you can easily show that K_rot = L²/(2I). What happens to I when the child moves towards the center while L stays constant?