Find the force as a function of radius to pull a ribbon

[Kanda ryu]

Homework Statement

A disc is free to rotate about an axis passing through its center and perpendicular to its plane. The moment of inertia of the disc about its center is I. A light ribbon is tightly wrapped over it in multiple layers. The end of the ribbon is pulled out at a constant velocity 'u'. Let the radius of the ribboned disc be 'R' at any time and thickness of ribbon be 'd' (d<<R).Find the force (F) required to pull the ribbon as a function of radius R.

Homework Equations

The Attempt at a Solution

What confuses me is whether friction is present or not. The question doesn't clearly specify about friction. Assuming friction exists, the question gets difficult to solve. As it is being pulled with uniform velocity, net torque about the center of disc should be zero, so I tried equating net torque at R and and a small change in radius R-dx but it doesn't seem to help .I need a little hint on understanding the question and how to start off with it.

 

[BvU]

As it is being pulled with uniform velocity, net torque about the center of disc should be zero

The radius of disc+layers of ribbon does decrease, albeit slowly. So if the ribbon moves with constant u, the disc still needs some angular aceleration.

 

[Kanda ryu]

So equating torque is completely out of use as angular acceleration would be different as R decreases right? But isn't the ribbon pulled in a way that the ribbon has a constant linear velocity so the ribbon in contact with disc will also have constant velocity. Assuming no slipping between ribbon and disc, disc should also have constant tangential velocity, hence constant angular velocity. If not then how do I get a relation between force and change in radius?
Here is the diagram.

 

[haruspex]

constant tangential velocity, hence constant angular velocity.

With a changing radius??

 

[Kanda ryu]

Alright I get it, there is angular acceleration. Could you give me a hint on starting the question?

 

[BvU]

Assume some $\omega$ and $R$ at $t=0$ (\ for example $\omega_0$ and $R_0$\ ) and start applying your relevant equations ( oops... )

 

[haruspex]

Alright I get it, there is angular acceleration. Could you give me a hint on starting the question?

The usual approach: define some variables that will be functions of time and write some equations relating them to each other and to the given constants.